Magnetic force on a charge

01 / Magnetic forces, selected velocities and current-produced fields

A magnetic field acts on the component of motion across it

A point charge q moving with velocity v in a magnetic field B experiences magnetic force q(v × B). The magnitude is |q|vB sin θ, where θ is the angle between velocity and field. Motion parallel or antiparallel to B produces no magnetic force.

Determine the cross-product direction using a right-handed coordinate system, then reverse it if the charge is negative. The charge sign changes the force direction, not the nonnegative magnitude. A dot symbol commonly represents a vector out of the page, while a cross represents one into the page.

The tesla is the SI field unit: 1 T = 1 N/(A·m) = 1 N·s/(C·m). Substituting electric-field units N/C for B would make the force dimensions incorrect. Electric and magnetic fields are related parts of electromagnetism, but their force laws differ.

With both fields present, the total electromagnetic force is q(E + v × B). An electric force can act on a stationary charge; a magnetic force on that point charge requires motion across the field. Other forces such as gravity or collisions must be included if they are not negligible.

\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B),\qquad |\mathbf F_B|=|q|vB\sin\theta

The examples use the nonrelativistic point-particle model and prescribed fields. Radiation losses and collisions are neglected unless stated.

WORKED EXAMPLE

Apply the cross product before the charge sign

A particle moves along positive x in a uniform magnetic field along positive z. Determine the magnetic-force direction for a positive charge and for a negative charge.

Choose an approach

Use x̂ × ẑ = −ŷ, then multiply by the sign of q.

The velocity–field cross product points along negative y.
$\mathbf v\times\mathbf B=-vB\hat{\mathbf y}$
Positive and negative charge signs give opposite responses.
$\mathbf F_{q>0}\parallel-\hat{\mathbf y},\qquad\mathbf F_{q<0}\parallel+\hat{\mathbf y}$

What this gives usThe positive charge deflects toward negative y and the negative charge toward positive y.

Check the result another way

The force is perpendicular to both x and z, and reversing either velocity or field would reverse the cross product.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Apply the cross product before the charge sign
Problem: A particle moves along positive x in a uniform magnetic field along positive z. Determine the magnetic-force direction for a positive charge and for a negative charge.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: A magnetic field acts on the component of motion across it
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B),\qquad |\mathbf F_B|=|q|vB\sin\theta
Meaning: The examples use the nonrelativistic point-particle model and prescribed fields. Radiation losses and collisions are neglected unless stated.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Magnetic forces, selected velocities and current-produced fields

Magnetic force changes direction without directly changing particle speed

The instantaneous power delivered by a force is F·v. For magnetic force on a point charge, q(v × B)·v = 0 because a cross product is perpendicular to v. Therefore magnetic force alone does no work on that particle.

In a static prescribed magnetic field with no other forces and negligible radiation losses, the particle’s kinetic energy and speed remain constant. Its momentum vector can still change because its direction changes. No work is not the same as no acceleration.

This statement does not mean every circuit or moving wire involving magnetic fields has zero mechanical power. A complete apparatus can exchange energy through electric fields, sources and external forces. The point-particle magnetic power identity is precise and should not be stretched into an incomplete device energy account.

If an electric field also acts, it can change kinetic energy through qE·v. A velocity selector has zero net force only for the selected velocity and direction; particles that depart from that condition can exchange energy with the electric field.

P_B=\mathbf F_B\cdot\mathbf v=q(\mathbf v\times\mathbf B)\cdot\mathbf v=0

This is the magnetic contribution to point-particle power. Other forces and radiative effects are separate.

STOP & THINK

A charge accelerates in a magnetic field while magnetic force is its only significant force. What can change?

Its speed must increaseIts velocity direction can change while speed stays fixedIts kinetic energy must become zero

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields / Magnetic force changes direction without directly changing particle speed
Problem: A charge accelerates in a magnetic field while magnetic force is its only significant force. What can change?
Choices:
1. Its speed must increase
2. Its velocity direction can change while speed stays fixed
3. Its kinetic energy must become zero

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Magnetic force changes direction without directly changing particle speed
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
P_B=\mathbf F_B\cdot\mathbf v=q(\mathbf v\times\mathbf B)\cdot\mathbf v=0
Meaning: This is the magnetic contribution to point-particle power. Other forces and radiative effects are separate.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Magnetic forces, selected velocities and current-produced fields

Perpendicular entry produces uniform circular motion

When v is perpendicular to a uniform B, the magnetic force remains perpendicular to velocity with constant magnitude |q|vB. It supplies the centripetal force mv²/r, giving r = mv/(|q|B).

The angular frequency is v/r = |q|B/m and the period is 2πm/(|q|B). In the nonrelativistic model, the period is independent of speed even though radius grows with speed. This follows because a faster particle travels a proportionally larger circumference.

The orbit centre lies in the initial force direction, one radius from the entry point. Locating it geometrically is often the easiest way to solve entry and exit questions for a bounded magnetic region.

These expressions use nonrelativistic momentum mv. At sufficiently high speed, the more general radius relation uses momentum p: r = p_perp/(|q|B), and the simple speed-independent period no longer applies. Keep the stated approximation in view.

r=\frac{mv_\perp}{|q|B},\qquad\omega_c=\frac{|q|B}{m},\qquad T=\frac{2\pi m}{|q|B}

Uniform field, nonrelativistic particle, no other significant forces. The radius depends on the perpendicular velocity component.

WORKED EXAMPLE

Calculate radius and period independently

A positive ion has mass 3.2 × 10⁻²⁶ kg and charge 1.6 × 10⁻¹⁹ C. It enters a uniform 0.20 T field perpendicularly at 2.0 × 10⁵ m/s. Find radius and period.

Choose an approach

Balance magnetic and centripetal forces, then use either the angular frequency or circumference.

The curvature radius follows from momentum divided by qB.
$r=\frac{(3.2\times10^{-26})(2.0\times10^5)}{(1.6\times10^{-19})(0.20)}=0.20\ \mathrm m$
The period follows from the mass-to-charge ratio.
$T=\frac{2\pi(3.2\times10^{-26})}{(1.6\times10^{-19})(0.20)}=2\pi\times10^{-6}\ \mathrm s$

What this gives usThe radius is 0.20 m and period approximately 6.28 μs.

Check the result another way

The independent circumference calculation 2πr/v = 2π(0.20)/(2.0 × 10⁵) gives the same period.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Calculate radius and period independently
Problem: A positive ion has mass 3.2 × 10⁻²⁶ kg and charge 1.6 × 10⁻¹⁹ C. It enters a uniform 0.20 T field perpendicularly at 2.0 × 10⁵ m/s. Find radius and period.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Perpendicular entry produces uniform circular motion
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
r=\frac{mv_\perp}{|q|B},\qquad\omega_c=\frac{|q|B}{m},\qquad T=\frac{2\pi m}{|q|B}
Meaning: Uniform field, nonrelativistic particle, no other significant forces. The radius depends on the perpendicular velocity component.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Magnetic forces, selected velocities and current-produced fields

Parallel motion survives while perpendicular motion curves

Resolve velocity into components parallel and perpendicular to a uniform magnetic field. The parallel component experiences no magnetic force and remains constant in the ideal model. The perpendicular component produces circular motion around the field direction.

The resulting path is a helix. Its radius uses v_perp, while its pitch—the displacement along the field in one full turn—is v_parallel T. Using total speed in the radius formula would overestimate the curvature radius when there is a parallel component.

If v_perp = 0, the path is a straight line along the field. If v_parallel = 0, it is a circle. These limiting cases are useful checks on a helical-motion calculation.

A spatially varying magnetic field can change the shape of the path and require additional approximations. The constant-radius, constant-pitch helix is specifically a uniform-field result.

WORKED EXAMPLE

Use different velocity components for radius and pitch

The ion from the previous example has v_perp = 2.0 × 10⁵ m/s and v_parallel = 1.0 × 10⁵ m/s in the same uniform field. Find its helix radius and pitch.

Choose an approach

The unchanged perpendicular component preserves the radius and period; use the parallel component only for axial travel.

The transverse circular motion is the same as before.
$r=0.20\ \mathrm m,\qquad T=2\pi\times10^{-6}\ \mathrm s$
Axial displacement in one period is v_parallel times T.
$h=(1.0\times10^5)(2\pi\times10^{-6})=0.20\pi\ \mathrm m$

What this gives usThe helix radius is 0.20 m and the pitch is about 0.628 m.

Check the result another way

The total speed is √5 × 10⁵ m/s, but substituting it for v_perp would mix axial translation into the circular force balance.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Use different velocity components for radius and pitch
Problem: The ion from the previous example has v_perp = 2.0 × 10⁵ m/s and v_parallel = 1.0 × 10⁵ m/s in the same uniform field. Find its helix radius and pitch.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Parallel motion survives while perpendicular motion curves
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Magnetic forces, selected velocities and current-produced fields

Use the orbit geometry to find where a particle exits

A bounded field region does not automatically contain a complete orbit. Draw the circle determined by the entry point, velocity and force direction, then intersect it with the region boundary. The arc angle travelled determines the residence time.

For a particle entering normally into the half-plane x > 0 through x = 0, with v initially along +x and B along +z, a positive charge curves toward −y. Its circle centre is at (0, −r). It returns to the boundary after a semicircle, displaced by 2r along negative y.

The time for that half-turn is πm/(|q|B), independent of entry speed in the nonrelativistic model. The exit displacement depends on speed through r. A speed change can therefore alter the exit location without altering the half-turn time.

A finite strip, aperture or another force can change which boundary is met first. Check the actual region rather than assuming the same half-circle in every magnetic-deflection question.

WORKED EXAMPLE

Separate travel time from exit separation

The ion with r = 0.20 m and period 6.28 μs enters the ideal half-plane field normally as described. Find the distance between entry and exit points and the time spent in the field.

Choose an approach

The boundary intersections are opposite ends of a diameter.

The displacement magnitude between boundary intersections is twice the radius.
$d=2r=0.40\ \mathrm m$
The particle traverses one half of the full orbit.
$t=T/2\approx3.14\ \mathrm{\mu s}$

What this gives usThe entry and exit are 0.40 m apart, and the particle spends about 3.14 μs in the field.

Check the result another way

The arc length is πr, so πr/v gives the same residence time.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Separate travel time from exit separation
Problem: The ion with r = 0.20 m and period 6.28 μs enters the ideal half-plane field normally as described. Find the distance between entry and exit points and the time spent in the field.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Use the orbit geometry to find where a particle exits
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Magnetic forces, selected velocities and current-produced fields

Crossed fields select a velocity by cancelling two forces

Let a beam travel along positive x, with a uniform electric field along positive y and magnetic field along positive z. Then v × B points along negative y. Electric and magnetic forces cancel when E = vB, giving the selected speed v = E/B.

Both terms carry the same charge q, so the zero-force velocity is independent of mass and charge magnitude, and the same directed velocity cancels the forces for either charge sign. The deflection direction of an unselected particle does depend on the sign.

For a positive charge faster than E/B in this arrangement, the initial magnetic contribution is larger and the net force is toward negative y. For a slower positive charge, it is toward positive y. Reversing charge reverses those deflections.

The selector tests a velocity direction as well as a speed. A particle travelling in the opposite direction has a reversed v × B, so the two forces can add rather than cancel. Collimation and entry direction are part of the model.

\mathbf E+\mathbf v\times\mathbf B=0,\qquad v=\frac EB\ \text{for the stated perpendicular geometry}

The scalar ratio applies only after the vector directions establish opposing electric and magnetic forces.

WORKED EXAMPLE

Find a selected speed and predict an off-speed deflection

A selector has E = 1200 V/m along +y and B = 0.020 T along +z. A beam enters along +x. Find the selected speed and the initial deflection direction of a positive particle entering faster than that speed.

Choose an approach

Use the magnitude balance only after checking the cross-product direction.

The magnetic force is along −y for positive charge moving along +x.
$v_{\mathrm{selected}}=E/B=1200/0.020=6.0\times10^4\ \mathrm{m/s}$
At larger speed, the magnetic contribution exceeds the electric one.
$F_y=q(E-vB)<0\quad\text{for}\quad q>0,\ v>E/B$

What this gives usThe selected speed is 6.0 × 10⁴ m/s. A faster positive particle initially deflects toward negative y.

Check the result another way

N/C divided by N·s/(C·m) gives m/s, confirming the speed units.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Find a selected speed and predict an off-speed deflection
Problem: A selector has E = 1200 V/m along +y and B = 0.020 T along +z. A beam enters along +x. Find the selected speed and the initial deflection direction of a positive particle entering faster than that speed.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Crossed fields select a velocity by cancelling two forces
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
\mathbf E+\mathbf v\times\mathbf B=0,\qquad v=\frac EB\ \text{for the stated perpendicular geometry}
Meaning: The scalar ratio applies only after the vector directions establish opposing electric and magnetic forces.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Magnetic forces, selected velocities and current-produced fields

A second magnetic region can measure mass-to-charge ratio

After a selector fixes speed, a separate uniform analysing magnetic field can bend particles into circles. Measuring radius then gives m/|q| = B_a r/v in the nonrelativistic model.

The selector field and analysing field need not be equal. Label them B_s and B_a. Combining v = E/B_s gives m/|q| = B_aB_s r/E. Reusing one unlabeled B for both regions can hide a mistaken assumption.

The curvature direction identifies the charge sign if the velocity and field directions are known, while radius determines the magnitude ratio m/|q|. Determining mass itself requires knowing the charge state.

Equal kinetic energy is a different preparation from equal selected speed. At equal kinetic energy, momentum scales as √m and the radius dependence changes. Read how the beam is prepared before comparing radii.

WORKED EXAMPLE

Keep the two magnetic fields separate

Particles selected at 6.0 × 10⁴ m/s enter a separate 0.10 T analysing field perpendicularly and follow radius 0.030 m. Find m/|q|.

Choose an approach

Use the analysing field in the circular-motion relation.

Rearrange r = mv/(|q|B_a).
$\frac{m}{|q|}=\frac{B_ar}{v}=\frac{0.10(0.030)}{6.0\times10^4}=5.0\times10^{-8}\ \mathrm{kg/C}$

What this gives usThe measured mass-to-charge magnitude ratio is 5.0 × 10⁻⁸ kg/C.

Check the result another way

If two selected particles have the same charge magnitude, their radii are directly proportional to their masses. That inference would differ for unequal charge states.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Keep the two magnetic fields separate
Problem: Particles selected at 6.0 × 10⁴ m/s enter a separate 0.10 T analysing field perpendicularly and follow radius 0.030 m. Find m/|q|.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: A second magnetic region can measure mass-to-charge ratio
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Magnetic forces, selected velocities and current-produced fields

A current element produces a field perpendicular to its geometry

Steady currents produce magnetic fields. In vacuum magnetostatics, the Biot–Savart contribution of a directed current element I dl is dB = (μ₀/4π)I(dl × r̂)/r², where r̂ points from the source element toward the observation point.

The direction of dl is conventional current direction. Determine each cross-product direction before integrating. As with electric fields, magnetic fields superpose as vectors; adding magnitudes is valid only when the contributions align.

A current element is part of a complete steady-current path. When calculating the contribution of a straight section or arc, state whether the rest of the circuit contributes negligibly or is handled separately. A finite isolated wire segment cannot sustain a steady current without a return path.

In numerical examples below, use μ₀/(4π) ≈ 1.0 × 10⁻⁷ T·m/A. The source geometry, distance and angular projection decide how that coefficient enters the integral.

d\mathbf B=\frac{\mu_0I}{4\pi}\frac{d\boldsymbol\ell\times\hat{\mathbf r}}{r^2}

Vacuum magnetostatic current model. The observation vector points from the source element to the field point.

STOP & THINK

At a chosen observation point, a straight current element lies along the line pointing from that element to the point. What is its Biot–Savart contribution there?

Zero because the cross product is zeroMaximum because the distance is along the wireAlways μ₀I/(2πr)

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Topic: Magnetic forces, selected velocities and current-produced fields / A current element produces a field perpendicular to its geometry
Problem: At a chosen observation point, a straight current element lies along the line pointing from that element to the point. What is its Biot–Savart contribution there?
Choices:
1. Zero because the cross product is zero
2. Maximum because the distance is along the wire
3. Always μ₀I/(2πr)

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: A current element produces a field perpendicular to its geometry
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
d\mathbf B=\frac{\mu_0I}{4\pi}\frac{d\boldsymbol\ell\times\hat{\mathbf r}}{r^2}
Meaning: Vacuum magnetostatic current model. The observation vector points from the source element to the field point.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

09 / Magnetic forces, selected velocities and current-produced fields

Integrating a finite segment reveals the long-wire approximation

For a straight current segment from x = −L to x = +L, observe a point a perpendicular distance a above its midpoint. The magnitude of each aligned field contribution is (μ₀I/4π)a dx/(a² + x²)^(3/2).

Integrating gives B = (μ₀I/(2πa))L/√(L² + a²). The bracketed length factor approaches one when L ≫ a, producing the familiar long-straight-wire result μ₀I/(2πa).

The direction circles the wire according to the right-hand rule: thumb along conventional current, curled fingers along B. On opposite sides of the wire, the local field directions are opposite. A field-magnitude comparison alone cannot determine where two wires cancel.

The approximation requires the observation point to be far from the ends relative to its perpendicular distance. If nearby return wires carry the same circuit current, their fields may be comparable and must be included.

B_{\mathrm{segment}}=\frac{\mu_0I}{2\pi a}\frac{L}{\sqrt{L^2+a^2}},\qquad B_{\mathrm{long}}=\frac{\mu_0I}{2\pi a}

Symmetric finite segment of total length 2L; a is perpendicular distance from its midpoint.

WORKED EXAMPLE

Calculate a field with a direction and a distance check

A long straight wire carries 5.0 A upward on the page. Find the field magnitude 0.020 m to its right and state its direction. Use μ₀/(4π) = 1.0 × 10⁻⁷ T·m/A.

Choose an approach

Use perpendicular distance and the curled right-hand rule.

The long-wire coefficient is twice μ₀/(4π).
$B=\frac{2(10^{-7})(5.0)}{0.020}=5.0\times10^{-5}\ \mathrm T$
Upward current and a point to the right give a field into the page.
$B=50\ \mathrm{\mu T}\quad\text{into the page}$

What this gives usThe magnetic field is 50 μT into the page.

Check the result another way

Doubling the perpendicular distance halves the long-wire field. It follows 1/a, not the point-charge inverse-square law.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Calculate a field with a direction and a distance check
Problem: A long straight wire carries 5.0 A upward on the page. Find the field magnitude 0.020 m to its right and state its direction. Use μ₀/(4π) = 1.0 × 10⁻⁷ T·m/A.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Integrating a finite segment reveals the long-wire approximation
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
B_{\mathrm{segment}}=\frac{\mu_0I}{2\pi a}\frac{L}{\sqrt{L^2+a^2}},\qquad B_{\mathrm{long}}=\frac{\mu_0I}{2\pi a}
Meaning: Symmetric finite segment of total length 2L; a is perpendicular distance from its midpoint.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

10 / Magnetic forces, selected velocities and current-produced fields

At a circular arc’s centre, the element contributions share a direction

At the centre of a circular current arc, every element is perpendicular to its radius vector and the distance is the same R. The contributions all point normal to the arc’s plane, with sign set by current circulation. The integral therefore reduces to the arc length Rφ.

The centre field is B = μ₀Iφ/(4πR), where φ is in radians. A full loop gives μ₀I/(2R); a semicircle gives μ₀I/(4R). Using degrees directly for φ introduces a unit error.

Straight radial lead segments contribute zero field at the arc centre because dl and r̂ are parallel there. Other lead shapes generally contribute and must be included. Do not erase an arbitrary wire merely because it is called a lead.

For N closely spaced turns with the same radius and current, the centre contributions add to N times the one-loop field when the turn dimensions are adequately approximated as equal.

WORKED EXAMPLE

Include only the segments that contribute at the chosen point

A current of 2.0 A follows a semicircular arc of radius 0.10 m. Its adjoining straight segments run radially relative to the arc centre; the remote return path is assumed to contribute negligibly there. Find the field magnitude at the centre.

Choose an approach

The radial pieces have zero cross product; the semicircle contributes angle π.

Insert the half-circle angle in radians.
$B=\frac{\mu_0I\pi}{4\pi R}=\frac{\mu_0I}{4R}$
Use μ₀ = 4π × 10⁻⁷ T·m/A for this calculation.
$B=\frac{(4\pi\times10^{-7})(2.0)}{4(0.10)}=2\pi\times10^{-6}\ \mathrm T$

What this gives usThe magnitude is approximately 6.28 μT, normal to the arc plane with direction fixed by the current’s circulation.

Check the result another way

A full circular loop of the same radius and current would give twice the centre field.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Include only the segments that contribute at the chosen point
Problem: A current of 2.0 A follows a semicircular arc of radius 0.10 m. Its adjoining straight segments run radially relative to the arc centre; the remote return path is assumed to contribute negligibly there. Find the field magnitude at the centre.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: At a circular arc’s centre, the element contributions share a direction
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

11 / Magnetic forces, selected velocities and current-produced fields

The axial field of a loop checks both a near and far magnetic model

For a circular loop of radius R, symmetry cancels transverse components at an axial observation point z. Integrating the surviving component gives B_z = μ₀IR²/[2(R² + z²)^(3/2)], with sign determined by loop current orientation.

At z = 0, this becomes the centre value μ₀I/(2R). Far along the axis, it falls as 1/|z|³. This differs from the axial field of a charged ring with nonzero total charge, which has a far inverse-square electric field.

The loop’s magnetic dipole moment magnitude is IA, where A = πR², with direction given by a normal right-hand rule. The far-axis expression can then be written μ₀m_dipole/(2π|z|³). This is a far-field approximation, not the exact expression close to the loop.

The point’s location on the axis is part of the derivation. An arbitrary off-axis point generally requires a different integral. As with electric charge distributions, a compact formula comes with geometric conditions.

B_z=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}

Uniform circular loop and axial observation point. The sign follows the chosen current-oriented axis.

Compare a loop’s axial field with its centre value

Inspect a plotted point: z/R = -3

Loop axial field: 0.031623

The normalized field is one at the centre and decreases symmetrically on either side. The far field falls with the cube of axial distance. Lines connect the plotted samples.

WORKED EXAMPLE

Check an axial field ratio without inserting constants

At what fraction of the centre field is the axial field at z = √3R for a circular current loop?

Choose an approach

Divide the axial expression by B(0).

The ratio depends only on z/R.
$\frac{B(z)}{B(0)}=\left[1+(z/R)^2\right]^{-3/2}$
Insert the specified axial distance.
$\frac{B(\sqrt3R)}{B(0)}=(1+3)^{-3/2}=\frac18$

What this gives usThe field there is one eighth of the centre field.

Check the result another way

Both current and radius scale cancel from this dimensionless comparison, as expected.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Check an axial field ratio without inserting constants
Problem: At what fraction of the centre field is the axial field at z = √3R for a circular current loop?

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: The axial field of a loop checks both a near and far magnetic model
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
B_z=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}
Meaning: Uniform circular loop and axial observation point. The sign follows the chosen current-oriented axis.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

12 / Magnetic forces, selected velocities and current-produced fields

Separate field direction, charge sign and speed selection

For velocity along +x and magnetic field along +z, v×B points along −y. If the electric field points along +y, the two field contributions can cancel. The y force is q(E−vB), where E and B denote the positive magnitudes in these declared directions. Multiplying by a negative charge reverses the total force, not the selected speed.

For straight entry perpendicular to both fields, v=E/B makes the force zero for either charge sign and any mass. That is a condition on a velocity vector: sending the same speed along −x reverses v×B and destroys this cancellation. A selector filters velocity; by itself it does not determine mass or charge magnitude. A second field-only region can provide a curvature measurement.

Away from selection, use q(E+v×B) at the current velocity. The electric field can change kinetic energy once there is velocity along it. The magnetic part remains perpendicular to the instantaneous velocity and does no direct work. It is therefore unsafe to reuse the pure-magnetic circular trajectory throughout a nonselected crossed-field region.

F_y=q(E-vB),\qquad v_{\mathrm{selected}}=\frac{E}{B}

This scalar form depends on the specified +x, +y and +z axes. E/B has units of speed: (N/C)/(N·s/(C·m)) = m/s.

WORKED EXAMPLE

Reverse the charge without changing the selected speed

A particle enters along +x at 3.0×10⁵ m/s. E=4.0×10⁴ V/m is along +y and B=0.20 T is along +z. Find the y force for q=−1.60×10⁻¹⁹ C.

Choose an approach

First combine the electric and magnetic field contributions; multiply by the signed charge last.

The magnetic contribution points along −y and exceeds the electric contribution.
$E-vB=4.0\times10^4-6.0\times10^4=-2.0\times10^4\ \mathrm{N/C}$
The negative charge reverses the direction of that combined field force.
$F_y=(-1.60\times10^{-19})(-2.0\times10^4)=+3.20\times10^{-15}\ \mathrm N$

What this gives usThe initial force is along +y. The selected speed would instead be 2.0×10⁵ m/s.

Check the result another way

Compute separately: the electric force is −6.40×10⁻¹⁵ N and the magnetic force is +9.60×10⁻¹⁵ N; their sum is +3.20×10⁻¹⁵ N.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Reverse the charge without changing the selected speed
Problem: A particle enters along +x at 3.0×10⁵ m/s. E=4.0×10⁴ V/m is along +y and B=0.20 T is along +z. Find the y force for q=−1.60×10⁻¹⁹ C.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

In the stated perpendicular selector orientation, E=6000 V/m and B=0.30 T. What speed is undeflected?

Your answer (m/s)Use a decimal, a fraction such as 1/3, or scientific notation.

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Topic: Magnetic forces, selected velocities and current-produced fields / Separate field direction, charge sign and speed selection
Problem: In the stated perpendicular selector orientation, E=6000 V/m and B=0.30 T. What speed is undeflected?
Requested answer unit: m/s

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Separate field direction, charge sign and speed selection
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
F_y=q(E-vB),\qquad v_{\mathrm{selected}}=\frac{E}{B}
Meaning: This scalar form depends on the specified +x, +y and +z axes. E/B has units of speed: (N/C)/(N·s/(C·m)) = m/s.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

13 / Magnetic forces, selected velocities and current-produced fields

Find the directions before solving a field-cancellation equation

Magnetic fields add as vectors. Equal magnitudes cancel only when they point in opposite directions. For long parallel wires, divide the line of possible observation points into regions and use the right-hand rule in each. Opposite currents give fields in the same direction between the wires, so no positive solution found by equating magnitudes there can represent a zero field.

Outside the pair the fields oppose. The nearer weak wire may balance the farther strong wire. Use actual distances from the observation point to each wire, solve, then substitute into a signed component equation. A coordinate outside the assumed region invalidates that region’s solution.

WORKED EXAMPLE

Locate the zero outside an antiparallel pair

Two long wires cross the xy plane at x=0 and x=0.15 m. They carry +2 A and −8 A along z. Find the zero-field point on the x axis. Neglect other fields.

Choose an approach

Between the wires the fields add. Try x=−s to the left, where the weaker wire is nearer.

On the left, the field magnitudes can oppose with distances s and s+0.15.
$\frac{2}{s}=\frac{8}{s+0.15}$
The positive distance obtained belongs to the assumed left region.
$2s+0.30=8s\quad\Rightarrow\quad s=0.050\ \mathrm m$

What this gives usThe zero is at x=−0.050 m, outside the pair on the weaker-current side.

Check the result another way

The signed y-field coefficient is 2/(−0.05)−8/(−0.20)=−40+40=0. Between the wires both signed terms are positive.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Locate the zero outside an antiparallel pair
Problem: Two long wires cross the xy plane at x=0 and x=0.15 m. They carry +2 A and −8 A along z. Find the zero-field point on the x axis. Neglect other fields.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Cancellation can also depend on distance from a set of coaxial loops. At a loop’s centre B=μ₀I/(2R), whereas far along its axis B≈μ₀IR²/(2z³). A centre cancellation weights current by 1/R; a far-field cancellation weights it by R². Equal centre fields do not imply that the entire magnetic field vanishes. The dipole moment IπR² is a vector whose sign follows the loop’s right-hand normal.

WORKED EXAMPLE

Zero at the centre can coexist with a far field

Concentric circular loops have radii 0.10 m and 0.20 m. The inner loop carries 1 A with moment along +z. Choose the opposing outer current to cancel the field at the common centre, then find the net magnetic moment.

Choose an approach

Use the centre field to set the current, then apply the different area weighting for magnetic moment.

The larger loop needs a larger opposing current to match the centre field.
$\frac{1}{0.10}=\frac{I_2}{0.20}\quad\Rightarrow\quad I_2=2\ \mathrm A$
The signed magnetic moments weight each current by its own loop area.
$\mu_z=\pi[1(0.10)^2-2(0.20)^2]=-0.07\pi\ \mathrm{A\,m^2}$

What this gives usThe centre field is zero, but the net magnetic moment points along −z.

Check the result another way

For |z| much larger than 0.20 m on the common axis, the leading axial field is μ₀μ_z/(2π|z|³), which is nonzero. The exact loop formula approaches that same coefficient.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Zero at the centre can coexist with a far field
Problem: Concentric circular loops have radii 0.10 m and 0.20 m. The inner loop carries 1 A with moment along +z. Choose the opposing outer current to cancel the field at the common centre, then find the net magnetic moment.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: Find the directions before solving a field-cancellation equation
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

14 / Magnetic forces, selected velocities and current-produced fields

A magnetic strip may send the particle back through its entrance

A straight strip boundary is not automatically an exit. For a positive charge entering along +x into B along +z, the circular centre is below the entry point. Write x=r sinφ and y=r(cosφ−1), with φ increasing from zero. The furthest rightward excursion is only r. A far boundary at x=d can be crossed only when d<r. At d=r the trajectory touches it tangentially in the ideal sharp-boundary model.

If d>r, the particle turns before reaching the far boundary and returns to x=0 at φ=π, displaced by −2r in y. The time is half a cyclotron period. This assumes the field strip is wide enough in y, the motion is nonrelativistic, and edge/fringe fields are neglected. Reversing the charge mirrors the vertical curvature but does not alter the radius or the half-period magnitude.

WORKED EXAMPLE

Check which boundary can actually be reached

A particle has radius r=0.080 m and angular frequency |q|B/m=5.0×10⁵ s⁻¹ in a field strip 0<x<0.12 m. It enters from x=0 perpendicular to the edge. Find the first exit time.

Choose an approach

Compare the strip width with the maximum x excursion before using any straight-line travel time.

The orbit does not reach the far boundary because its maximum x is its radius.
$d=0.12\ \mathrm m>r=0.080\ \mathrm m$
It returns through the entrance after half a revolution.
$t=\frac{\pi}{5.0\times10^5}=2\pi\ \mathrm{\mu s}$

What this gives usThe first exit is back through the entrance after 2π μs, not after crossing the full strip.

Check the result another way

At that phase, x=r sinπ=0 and y=r(cosπ−1)=−0.160 m. Before φ=π, x remains positive, so this is the first return.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Check which boundary can actually be reached
Problem: A particle has radius r=0.080 m and angular frequency |q|B/m=5.0×10⁵ s⁻¹ in a field strip 0<x<0.12 m. It enters from x=0 perpendicular to the edge. Find the first exit time.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Focus: A magnetic strip may send the particle back through its entrance
Learning goals:
- Determine magnetic-force direction including the charge sign.
- Explain zero magnetic work despite a changing velocity vector.
- Compute circular radius, period, helix pitch and finite-region exit geometry.
- Use vector force cancellation for a velocity selector.
- Infer mass-to-charge ratio with a separately labelled analysing field.
- Set up Biot–Savart contributions with a complete-current-path interpretation.
- Derive and check straight-wire, arc and loop-axis fields.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Magnetic force is q(v × B), so charge sign and direction both matter.
Magnetic force alone does no point-particle work but can curve its trajectory.
Only perpendicular velocity enters the orbit radius; parallel velocity sets helix pitch.
A selector fixes a directed velocity through E + v × B = 0.
Mass analysis requires the analysing field, which need not equal the selector field.
Biot–Savart contributions are vectors from specified current elements.
Finite-wire and loop formulas require their stated observation geometry.
A current loop’s far axial magnetic field falls as inverse distance cubed.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

For the same nonrelativistic particle and uniform magnetic field, perpendicular speed doubles. What is the orbit-radius ratio?

Your answerUse a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: For the same nonrelativistic particle and uniform magnetic field, perpendicular speed doubles. What is the orbit-radius ratio?
Requested answer unit: 

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 2

A perpendicular crossed-field selector has E = 900 V/m and B = 0.030 T. What selected speed follows?

Your answer (m/s)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: A perpendicular crossed-field selector has E = 900 V/m and B = 0.030 T. What selected speed follows?
Requested answer unit: m/s

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 3

What is B(√3R)/B(0) on the axis of a circular loop?

Your answerUse a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: What is B(√3R)/B(0) on the axis of a circular loop?
Requested answer unit: 

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 4

A magnetic force alone acts on an ideal nonrelativistic point charge. Which quantity stays constant in the stated no-loss model?

Velocity vectorSpeedDirection of momentum

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: A magnetic force alone acts on an ideal nonrelativistic point charge. Which quantity stays constant in the stated no-loss model?
Choices:
1. Velocity vector
2. Speed
3. Direction of momentum

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 5

For a fixed directed velocity satisfying E + v × B = 0, what happens if q changes sign?

The cancellation still holdsThe forces no longer cancelThe selected speed doubles

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: For a fixed directed velocity satisfying E + v × B = 0, what happens if q changes sign?
Choices:
1. The cancellation still holds
2. The forces no longer cancel
3. The selected speed doubles

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 6

What angle unit must be used in B = μ₀Iφ/(4πR) for an arc-centre field?

Degrees without conversionRadiansAny unit gives the same numerical result

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: What angle unit must be used in B = μ₀Iφ/(4πR) for an arc-centre field?
Choices:
1. Degrees without conversion
2. Radians
3. Any unit gives the same numerical result

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 7

Opposite currents +1 A and −3 A run along z through x=0 and x=0.20 m. Find the x coordinate of their zero field on the x axis.

Your answer (m)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: Opposite currents +1 A and −3 A run along z through x=0 and x=0.20 m. Find the x coordinate of their zero field on the x axis.
Requested answer unit: m

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 8

A particle enters a magnetic strip normally, with orbit radius r. The far boundary is at d=1.4r. What happens in the ideal model?

It reaches the far boundary in time d/v.It returns through the entrance after half a revolution.It stops at x=r.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Magnetic forces, selected velocities and current-produced fields
Problem: A particle enters a magnetic strip normally, with orbit radius r. The far boundary is at d=1.4r. What happens in the ideal model?
Choices:
1. It reaches the far boundary in time d/v.
2. It returns through the entrance after half a revolution.
3. It stops at x=r.

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

A particle enters a uniform magnetic field with velocity partly parallel to the field. Why does its path become a helix rather than a circle?

Compare your reasoning

The perpendicular component produces circular motion, while the parallel component remains unchanged under magnetic force alone. Combining them produces a helix.

Revisit the explanation →

A few days later: change the situation

In crossed electric and magnetic fields, a particle travels undeflected at one selected speed. Does that prove both forces are zero?

Compare your reasoning

No. Their nonzero forces can cancel at the selected direction and speed. For perpendicular fields in the standard selector geometry the speed is E/B. Force balance is not the absence of interactions.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑

What you will learn to do

Bring back one idea

A magnetic field acts on the component of motion across it

Apply the cross product before the charge sign

Magnetic force changes direction without directly changing particle speed

A charge accelerates in a magnetic field while magnetic force is its only significant force. What can change?

Perpendicular entry produces uniform circular motion

Calculate radius and period independently

Parallel motion survives while perpendicular motion curves

Use different velocity components for radius and pitch

Use the orbit geometry to find where a particle exits

Separate travel time from exit separation

Crossed fields select a velocity by cancelling two forces

Find a selected speed and predict an off-speed deflection

A second magnetic region can measure mass-to-charge ratio

Keep the two magnetic fields separate

A current element produces a field perpendicular to its geometry

At a chosen observation point, a straight current element lies along the line pointing from that element to the point. What is its Biot–Savart contribution there?

Integrating a finite segment reveals the long-wire approximation

Calculate a field with a direction and a distance check

At a circular arc’s centre, the element contributions share a direction

Include only the segments that contribute at the chosen point

The axial field of a loop checks both a near and far magnetic model

Compare a loop’s axial field with its centre value

Check an axial field ratio without inserting constants

Separate field direction, charge sign and speed selection

Reverse the charge without changing the selected speed

In the stated perpendicular selector orientation, E=6000 V/m and B=0.30 T. What speed is undeflected?

Find the directions before solving a field-cancellation equation

Locate the zero outside an antiparallel pair

Zero at the centre can coexist with a far field

A magnetic strip may send the particle back through its entrance

Check which boundary can actually be reached

What should stay with you

Try it without the examples

For the same nonrelativistic particle and uniform magnetic field, perpendicular speed doubles. What is the orbit-radius ratio?

A perpendicular crossed-field selector has E = 900 V/m and B = 0.030 T. What selected speed follows?

What is B(√3R)/B(0) on the axis of a circular loop?

A magnetic force alone acts on an ideal nonrelativistic point charge. Which quantity stays constant in the stated no-loss model?

For a fixed directed velocity satisfying E + v × B = 0, what happens if q changes sign?

What angle unit must be used in B = μ₀Iφ/(4πR) for an arc-centre field?

Opposite currents +1 A and −3 A run along z through x=0 and x=0.20 m. Find the x coordinate of their zero field on the x axis.

A particle enters a magnetic strip normally, with orbit radius r. The far boundary is at d=1.4r. What happens in the ideal model?

Take the ideas into a problem

What can you explain later?

Next study session

A few days later: change the situation