Hold a magnet still beside a coil, then move it. A detector can distinguish the two situations even if the magnet is strong in both. Induction asks what is changing through the circuit, not simply whether a magnetic field is present.
The key is to specify the loop, its orientation and what changes. We then connect the induced emf to current, force and stored energy using the actual circuit.
What you will learn to do
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
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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: Induction, moving conductors and magnetic energy Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Why can magnetic flux through a loop change even when the magnetic field magnitude stays constant?
Compare your reasoning
The loop’s area or orientation can change. Flux depends on the field’s normal component integrated over the chosen surface, so a field magnitude alone does not specify it.
Revisit the explanation →01 / Induction, moving conductors and magnetic energy
Flux measures the field passing through an oriented surface
Imagine placing a small flat frame in a uniform magnetic field. The amount of field passing through it depends on its area and on how it is tilted. Magnetic flux through that surface is B·A = BA cos θ, where the area vector is normal to the surface and θ is measured from that normal.
A field lying in the plane of the frame has zero flux. A field perpendicular to the plane has maximum flux magnitude. This is why using the angle with the plane directly in a cosine is a common error. First draw the normal, then label the angle.
For a nonuniform field or curved surface, divide the surface into small patches and integrate B·dA. A flux value belongs to a chosen surface and orientation. Reversing the normal reverses its sign without changing the physical field.
For a coil, the flux linkage is the sum of fluxes through its turns. It equals NΦ only when the turns link the same flux. A changing linkage can come from changing field strength, changing area, changing orientation or a combination of them.
Magnetic flux is measured in webers: 1 Wb = 1 T·m². Flux linkage counts the contributions of all turns.
Explain this equation
Read it in words
Magnetic flux adds the part of the magnetic field that crosses a chosen surface. Flux linkage adds that flux over all the turns of a coil.
Symbols and units
- Signed magnetic flux through the chosen surface, or through turn k.Webers (Wb) = tesla square metres.
- Magnetic field, including its local direction.Teslas (T).
- Tiny area vector perpendicular to the surface, with a consistently chosen normal.Square metres (m²).
- The surface bounded by the loop, over which flux is calculated.Surface label; not a separate multiplier.
- Total flux linkage of the coil, summing its turn fluxes.Wb; often written Wb-turn to emphasize the sum.
- Number of turns carrying equal flux in the simplified expression.Dimensionless count.
Signs and operations
The dot selects the field normal to the surface. The integral sums across one surface; Σ sums across turns. Flipping the chosen normal flips the signed flux. A field parallel to a flat surface gives zero flux through it.
When it applies
Use NΦ only when the linked flux is equal for every turn. A single field strength times area also requires a uniform field and a known angle; a strongly nonuniform field must be integrated.
Make it concrete
A uniform 0.20 T field perpendicular to a 0.010 m² loop gives 0.0020 Wb per turn. For 50 equal turns, linkage is 0.10 Wb-turn. Rotating the normal to 90° from the field makes the flux 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.
Equation (LaTeX): \Phi_B=\int_S\mathbf B\cdot d\mathbf A,\qquad\lambda=\sum_k\Phi_k=N\Phi\ \text{for equal turn fluxes}
Read it as: Magnetic flux adds the part of the magnetic field that crosses a chosen surface. Flux linkage adds that flux over all the turns of a coil.
Symbols:
\Phi_B,\ \Phi_k: Signed magnetic flux through the chosen surface, or through turn k.; Webers (Wb) = tesla square metres.
\mathbf B: Magnetic field, including its local direction.; Teslas (T).
d\mathbf A: Tiny area vector perpendicular to the surface, with a consistently chosen normal.; Square metres (m²).
S: The surface bounded by the loop, over which flux is calculated.; Surface label; not a separate multiplier.
\lambda: Total flux linkage of the coil, summing its turn fluxes.; Wb; often written Wb-turn to emphasize the sum.
N: Number of turns carrying equal flux in the simplified expression.; Dimensionless count.
Signs and operations: The dot selects the field normal to the surface. The integral sums across one surface; Σ sums across turns. Flipping the chosen normal flips the signed flux. A field parallel to a flat surface gives zero flux through it.
Conditions: Use NΦ only when the linked flux is equal for every turn. A single field strength times area also requires a uniform field and a known angle; a strongly nonuniform field must be integrated.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
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.WORKED EXAMPLE
Find the angle that belongs in the flux expression
A flat 0.030 m² loop is in a uniform 0.40 T field. The field makes 30° with the plane of the loop and has a positive component along the chosen normal. Find its flux.
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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: Induction, moving conductors and magnetic energy
Focus: Flux measures the field passing through an oriented surface
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
\Phi_B=\int_S\mathbf B\cdot d\mathbf A,\qquad\lambda=\sum_k\Phi_k=N\Phi\ \text{for equal turn fluxes}
Meaning: Magnetic flux is measured in webers: 1 Wb = 1 T·m². Flux linkage counts the contributions of all turns.
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.02 / Induction, moving conductors and magnetic energy
A changing linkage produces an emf
Faraday’s law relates an induced emf to the rate of change of magnetic flux linkage. An emf is energy supplied per unit charge around an oriented path; its unit is the volt. It is not a force measured in newtons, despite the historical name electromotive force.
Choose the positive loop direction and surface normal together using the right-hand rule. Fingers following the positive circulation have a thumb along the positive normal. With that convention, induced emf is −dλ/dt. An average change gives the average emf; an instantaneous emf needs the derivative.
A nonzero emf does not by itself guarantee a current. A closed conducting circuit, its resistance and its inductance determine the current. An open coil can have an induced terminal voltage without a sustained conduction current around the interrupted path.
For a stationary, purely resistive loop in the quasistatic approximation, current is emf divided by resistance. If self-inductance matters, its contribution must be included in the circuit equation rather than using I = emf/R at every instant.
The minus sign uses a consistent right-hand relation between loop circulation and surface normal.
Explain this equation
Read it in words
The induced emf is the negative rate of change of flux linkage. Over a finite interval, the average emf is the negative linkage change divided by the elapsed time.
Symbols and units
- Instantaneous induced emf, or its average over the stated interval. Emf is energy supplied per unit charge, despite the historical name “force”.Volts (V).
- Signed magnetic flux linkage, including all coil turns.Wb, or Wb-turn.
- Time.Seconds (s).
- Instantaneous rate: the slope of the linkage–time graph at a point.Wb/s = V.
- Change in linkage divided by a positive elapsed time; an interval average.Wb/s = V.
Signs and operations
The minus sign is Lenz’s law with the positive loop direction tied to the surface normal by the right-hand rule. An increasing positive flux gives a negative emf around that loop. It opposes the change in flux, not necessarily the magnetic field itself.
When it applies
For a fixed coil, changing magnetic flux induces a circulating electric field. For a moving material loop, the flux rule describes total emf including the magnetic force on moving charges. A current additionally requires a conducting circuit and its electrical response.
Make it concrete
If linkage rises uniformly from 0.020 to 0.050 Wb-turn in 0.10 s, average emf is −0.30 V. The same average can come from a curved graph with a changing instantaneous emf.
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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.
Equation (LaTeX): \mathcal E=-\frac{d\lambda}{dt},\qquad\overline{\mathcal E}=-\frac{\Delta\lambda}{\Delta t}
Read it as: The induced emf is the negative rate of change of flux linkage. Over a finite interval, the average emf is the negative linkage change divided by the elapsed time.
Symbols:
\mathcal E,\ \overline{\mathcal E}: Instantaneous induced emf, or its average over the stated interval. Emf is energy supplied per unit charge, despite the historical name “force”.; Volts (V).
\lambda: Signed magnetic flux linkage, including all coil turns.; Wb, or Wb-turn.
t: Time.; Seconds (s).
\frac{d\lambda}{dt}: Instantaneous rate: the slope of the linkage–time graph at a point.; Wb/s = V.
\frac{\Delta\lambda}{\Delta t}: Change in linkage divided by a positive elapsed time; an interval average.; Wb/s = V.
Signs and operations: The minus sign is Lenz’s law with the positive loop direction tied to the surface normal by the right-hand rule. An increasing positive flux gives a negative emf around that loop. It opposes the change in flux, not necessarily the magnetic field itself.
Conditions: For a fixed coil, changing magnetic flux induces a circulating electric field. For a moving material loop, the flux rule describes total emf including the magnetic force on moving charges. A current additionally requires a conducting circuit and its electrical response.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
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.WORKED EXAMPLE
Calculate an average emf with an explicit orientation
A 100-turn coil has area 0.020 m² per turn. A uniform field normal to the coil increases from 0.10 T to 0.40 T in 0.050 s. Choose the outward field direction as positive normal and counterclockwise as positive circulation. Find the average induced emf.
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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: Induction, moving conductors and magnetic energy
Focus: A changing linkage produces an emf
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
\mathcal E=-\frac{d\lambda}{dt},\qquad\overline{\mathcal E}=-\frac{\Delta\lambda}{\Delta t}
Meaning: The minus sign uses a consistent right-hand relation between loop circulation and surface normal.
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.03 / Induction, moving conductors and magnetic energy
The induced response opposes the change in flux
Lenz’s law is a physical way to interpret Faraday’s minus sign: the induced current’s magnetic effect opposes the change in flux that drives it. It does not always oppose the original magnetic field.
If an outward field is increasing, a passive closed loop responds with an inward induced field, which means clockwise current as viewed from the outward-normal side. If that outward field is decreasing, the response is outward and the current is counterclockwise. The original field direction is the same in both situations; its change is different.
Use a three-step direction argument. State the external flux change, state the field direction that would oppose that change, then use the right-hand rule to determine the current direction. This keeps the reasoning tied to the geometry.
The opposition is consistent with energy conservation. Pulling a conductor through a magnetic arrangement can require mechanical work that becomes electrical energy and heat. A sign choice that makes a passive circuit continuously accelerate the mover while also heating the resistor should trigger an energy check.
WORKED EXAMPLE
A decreasing field reverses the induced response
A resistive loop lies in the page. A magnetic field pointing out of the page is getting weaker. Determine the induced current direction, neglecting self-inductance.
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STOP & THINK
A stationary loop sits in a large but time-independent magnetic field. Its area and orientation are fixed. What emf does this flux mechanism produce?
- A large emf because B is large
- Zero because the flux linkage is constant
- An emf proportional to the loop resistance
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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: Induction, moving conductors and magnetic energy Focus: The induced response opposes the change in flux Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
04 / Induction, moving conductors and magnetic energy
A changing magnetic field creates a circulating electric field
Induction in a stationary loop cannot be explained by charges moving with the wire through a magnetic field, because the wire is at rest. The changing magnetic field produces a circulating electric field. Its closed-path integral equals −dΦB/dt.
Unlike an electrostatic field, this induced electric field generally has a nonzero integral around a closed loop. A single-valued electrostatic potential alone cannot describe the entire field. A charge travelling around the loop can gain energy from the changing electromagnetic environment.
For a long cylindrical region with spatially uniform axial field changing at rate dB/dt, cylindrical symmetry makes the induced electric field tangential. For a circle of radius r inside the field region, 2πr Eφ = −πr² dB/dt. Outside the field radius a, the enclosed changing flux uses πa² instead.
These are idealized symmetry results. They describe electric field even if no conducting loop is present. Placing a conductor there changes the physical response and can require including the conductor’s own fields.
The normal field is along +z; positive tangential circulation is counterclockwise viewed from +z. The external ideal field is confined to radius a.
WORKED EXAMPLE
Find an induced electric field without assuming a wire
Within a cylindrical field region of radius 0.20 m, a uniform +z magnetic field increases at 2.0 T/s. Find the induced electric-field magnitude and circulation at radius 0.10 m.
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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: Induction, moving conductors and magnetic energy
Focus: A changing magnetic field creates a circulating electric field
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
E_\phi=-\frac r2\frac{dB}{dt}\ (r<a),\qquad E_\phi=-\frac{a^2}{2r}\frac{dB}{dt}\ (r>a)
Meaning: The normal field is along +z; positive tangential circulation is counterclockwise viewed from +z. The external ideal field is confined to radius a.
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.05 / Induction, moving conductors and magnetic energy
A moving conductor separates charges through the magnetic force
A straight conducting rod moving across a magnetic field contains mobile charges that share the rod’s bulk velocity. The magnetic term q(u × B) pushes positive and negative charge in opposite transverse directions. Charge separation can establish an electric field inside an open rod.
For a rod of length ℓ, moving with speed v perpendicular to both its length and a uniform B, the magnitude of the motional contribution is Bℓv. More generally, use the path integral of (u × B)·dℓ, including the actual local velocity of each piece.
To identify the positive end, determine the force on a positive charge. For a rod along y moving along +x in B along +z, u × B points along −y. The lower end gains positive charge, and in open-circuit equilibrium the internal electrostatic field points upward to balance the magnetic force.
A rotating rod has different speeds along its length. Integrating Bωr dr from an axis at one end gives a magnitude Bωℓ²/2 for perpendicular uniform B. Substituting the tip speed into Bℓv for the entire rod would double the correct result.
A motional emf is a path contribution. The full circuit can also contain electric-field and other source contributions.
WORKED EXAMPLE
Integrate because the speed varies along the rod
A conducting rod of length 0.40 m rotates about one end at angular speed 10 rad/s in a uniform 0.50 T field perpendicular to the rotation plane. Find the magnitude of its pivot-to-tip motional emf.
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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: Induction, moving conductors and magnetic energy
Focus: A moving conductor separates charges through the magnetic force
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
\mathcal E_{\mathrm{motion}}=\int(\mathbf u\times\mathbf B)\cdot d\boldsymbol\ell,\qquad |\mathcal E|=B\ell v\ \text{for perpendicular uniform translation}
Meaning: A motional emf is a path contribution. The full circuit can also contain electric-field and other source contributions.
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.06 / Induction, moving conductors and magnetic energy
A sliding rod connects flux, force and energy
Place a conducting rod across two parallel conducting rails joined by a resistor. When the rod moves, the loop area changes. Faraday’s flux method and the magnetic-force method give the same emf magnitude Bℓv for a uniform perpendicular field.
Assume the rail and rod resistance are negligible compared with R and self-inductance can be neglected. Then I = Bℓv/R. The magnetic force on the rod has magnitude BIℓ and opposes the motion for this passive circuit.
An external agent maintaining constant speed supplies power Fv. Substituting the force and current gives Fv = I(Bℓv) = I²R. The external work becomes resistive heat in this ideal steady-motion model.
An open circuit can still develop a terminal voltage but does not carry this sustained loop current. The corresponding resistive magnetic drag is absent in the ideal open-circuit model. Closing the circuit changes the mechanical load.
WORKED EXAMPLE
Check the same apparatus mechanically and electrically
A rod of length 0.50 m slides at 3.0 m/s through a uniform 0.40 T perpendicular field. The closed rail circuit has resistance 2.0 Ω; neglect inductance and other losses. Find emf magnitude, current, drag force and required mechanical power.
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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: Induction, moving conductors and magnetic energy Focus: A sliding rod connects flux, force and energy Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
07 / Induction, moving conductors and magnetic energy
Rotation converts a changing angle into a sinusoidal emf
A coil rotating uniformly in a uniform magnetic field changes the angle between its area normal and B. If θ = ωt and every turn has area A, its flux linkage is NBA cos ωt. Differentiating gives emf NBAω sin ωt in the matched loop convention.
Flux is largest in magnitude when the normal aligns with the field, but its rate of change is zero there. Emf magnitude is largest when the flux passes through zero. Confusing a quantity with its rate of change produces the wrong phase relation.
Increasing turn count, field strength, area or angular speed increases the peak emf in this ideal model. The connected load determines the current. Once current flows, the generator experiences a resisting electromagnetic torque and needs mechanical input.
This model introduces an alternating source. Average signed emf over a full period is zero, but that does not imply zero heating in a resistor because heating depends on current squared.
The time origin is chosen with the coil normal aligned with the field. Changing that origin changes the phase, not the peak amplitude.
WORKED EXAMPLE
Find a generator’s peak emf
A 50-turn coil of area 0.010 m² rotates at 20 rad/s in a uniform 0.30 T field. Find its peak emf.
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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: Induction, moving conductors and magnetic energy Focus: Rotation converts a changing angle into a sinusoidal emf Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. Equations in this section (LaTeX): \lambda=NBA\cos\omega t,\qquad\mathcal E=NBA\omega\sin\omega t Meaning: The time origin is chosen with the coil normal aligned with the field. Changing that origin changes the phase, not the peak amplitude. 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.
08 / Induction, moving conductors and magnetic energy
An inductor opposes changes of its own current
A circuit current creates magnetic flux. In a fixed linear geometry, the linkage due to that current is proportional to it: λself = LI. The proportionality L is the self-inductance, measured in henries, where 1 H = 1 V·s/A.
The induced emf due to changing self-current is −L dI/dt. This opposes the change of current, not necessarily the current direction. When a positive current is decreasing, the induced emf acts to sustain that positive current.
In circuit analysis using the passive sign convention, the terminal voltage drop across an ideal inductor is vL = +L dI/dt: current enters the terminal labelled positive. This is consistent with the induced-emf expression because voltage drop and internal driving emf use different reference conventions.
For finite L and finite voltage, current cannot jump discontinuously: ΔI = (1/L)∫vL dt. A real switched inductor therefore needs a physical current path, often including a diode, capacitance, an arc or other nonideal element. Simply declaring the current instantly zero omits the mechanism and its voltage.
These simple expressions assume constant linear inductance. Magnetic saturation or changing geometry requires a fuller linkage relation.
WORKED EXAMPLE
Distinguish an induced emf from a terminal voltage drop
Current entering the positive-labelled terminal of a 0.20 H ideal inductor rises from 1.0 A to 4.0 A uniformly in 0.10 s. Find the passive terminal voltage and the self-induced emf in the current direction.
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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: Induction, moving conductors and magnetic energy
Focus: An inductor opposes changes of its own current
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
\mathcal E_{\mathrm{self}}=-L\frac{dI}{dt},\qquad v_L=L\frac{dI}{dt}\quad\text{(passive voltage drop)}
Meaning: These simple expressions assume constant linear inductance. Magnetic saturation or changing geometry requires a fuller linkage relation.
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.09 / Induction, moving conductors and magnetic energy
The magnetic energy follows by integrating the input power
For an ideal linear inductor, the terminal power absorbed is vLI = LI dI/dt. Integrating from zero current to I gives stored magnetic energy LI²/2. The energy is nonnegative and does not depend on the sign of current.
The factor one half appears because the current changes during the buildup. Multiplying the final linkage LI by the final current without integrating would count twice the stored energy.
For a long vacuum or air-core solenoid with negligible fringe field, combining L = μ₀N²A/ℓ with B = μ₀NI/ℓ gives energy density B²/(2μ₀). Materials with nonlinear magnetic response require additional care; the vacuum expression is not a universal material law.
An inductor can return stored energy when current decreases. Whether that energy reaches a load, heats a resistor or is lost in a switching event depends on the complete circuit. Inductors store energy; their winding resistance dissipates energy.
The first formula assumes constant linear L. The second is magnetic-field energy per unit volume in vacuum.
WORKED EXAMPLE
Track the energy released by a falling current
A 0.40 H ideal inductor’s current decreases from 3.0 A to 1.0 A. How much magnetic energy is released?
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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: Induction, moving conductors and magnetic energy
Focus: The magnetic energy follows by integrating the input power
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
U_L=\frac12LI^2,\qquad u_B=\frac{B^2}{2\mu_0}\ \text{in vacuum}
Meaning: The first formula assumes constant linear L. The second is magnetic-field energy per unit volume in vacuum.
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.10 / Induction, moving conductors and magnetic energy
A resistor and inductor set the timescale for current change
Connect a series resistor R and ideal inductor L to a constant source Vs, starting with zero current. Kirchhoff’s voltage equation is Vs = RI + L dI/dt. At the first instant the current is zero, so the full source voltage appears across the inductor and the initial slope is Vs/L.
The current approaches Vs/R exponentially with time constant τ = L/R. After one time constant it reaches 1 − e⁻¹, about 63.2%, of its final value. The resistor voltage rises with the current while the inductor voltage falls.
If the source is removed and the inductor is left in a closed resistive discharge loop, current decays as I0e^(−t/τ). Magnetic energy decays as e^(−2t/τ), so its decay time is half the current time constant. Integrating I²R over the discharge gives the initial magnetic energy.
The resistance in these expressions is the total resistance of the actual current path, including winding resistance where relevant. Opening the path altogether is a different switching problem from closing a resistor across the inductor.
Rise starts from zero current with a constant source; decay uses a closed resistive loop and no driving source.
Compare current buildup and magnetic-energy buildup
- Current
- 0
- Magnetic energy
- 0
WORKED EXAMPLE
Find a timescale, an intermediate current and final energy
A 10 V ideal source is switched onto a series 5.0 Ω resistor and 0.20 H inductor with zero initial current. Find the time constant, current after one time constant and final stored energy.
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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: Induction, moving conductors and magnetic energy
Focus: A resistor and inductor set the timescale for current change
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
I_{\mathrm{rise}}=\frac{V_s}{R}(1-e^{-t/\tau}),\qquad I_{\mathrm{decay}}=I_0e^{-t/\tau},\qquad\tau=\frac LR
Meaning: Rise starts from zero current with a constant source; decay uses a closed resistive loop and no driving source.
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.11 / Induction, moving conductors and magnetic energy
A solid conductor can contain closed current paths
A metal sheet is not one wire loop, but it contains many possible closed conducting paths. When the flux through these paths changes, a circulating electric field drives currents within the bulk. These are eddy currents. No external lead needs to complete the circuit because the paths close inside the material.
A fixed permanent magnet moving relative to a passive conductor can lose mechanical energy to these currents. Their field reacts on the motion, and their resistance converts electrical energy into heat. The induced response opposes the flux change; in this passive braking arrangement that produces a resisting mechanical force.
Lenz’s law does not say that every induced force always opposes every motion in every driven system. An externally powered changing field can transfer energy to a moving conductor. State where the input energy comes from before using a braking picture.
Slots and laminations obstruct large circulating conduction paths. Thin insulated laminations in a suitable orientation can reduce eddy-current heating while still providing a magnetic path. Hysteresis loss comes from material magnetization history and is not eliminated merely by cutting the conductor.
WORKED EXAMPLE
Connect a braking measurement to heating
At low speed, a passive magnetic brake has approximately Fdrag = −cv with c = 0.80 kg/s. A carriage is pulled steadily at 0.50 m/s. Neglect other losses. Find required pull and heating power.
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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: Induction, moving conductors and magnetic energy Focus: A solid conductor can contain closed current paths Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
12 / Induction, moving conductors and magnetic energy
A measured drag law gives a timescale and an energy ledger
The linear eddy-brake model can arise over a limited low-speed range where the induced current is approximately proportional to velocity and the force is proportional to that current. Its coefficient must be measured or derived from a specified geometry and conductivity; it is not fixed by Faraday’s law alone.
For a freely coasting mass with m dv/dt = −cv, velocity decays exponentially with timescale m/c. Integrating that velocity gives a finite limiting travel distance in the ideal model. The speed approaches zero asymptotically; there is no exact finite stop time unless another effect, such as dry friction, is added.
Integrating cv² over all time gives precisely the initial kinetic energy. This is a useful independent check of both the motion law and the claimed destination of the lost mechanical energy. The model also predicts that heating decays twice as fast in the exponent as speed.
These formulas assume a constant linear drag coefficient, no driving force and no other resistance. Stronger fields, higher speeds or induced-field screening can invalidate the constant-c approximation.
WORKED EXAMPLE
Check the complete braking event in two ways
A 0.40 kg cart enters an ideal linear brake at 2.0 m/s with c = 0.20 kg/s. Find the time to halve its speed, limiting travel and total heat generated.
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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: Induction, moving conductors and magnetic energy
Focus: A measured drag law gives a timescale and an energy ledger
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
v(t)=v_0e^{-ct/m},\quad x(t)-x_0=\frac{mv_0}{c}(1-e^{-ct/m}),\quad\int_0^\infty cv^2dt=\frac12mv_0^2
Meaning: These formulas assume a constant linear drag coefficient, no driving force and no other resistance. Stronger fields, higher speeds or induced-field screening can invalidate the constant-c approximation.
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.13 / Induction, moving conductors and magnetic energy
Gravity can supply a continuing heat flow
If the same linear brake acts on a descending mass, choose downward as positive and include gravity: m dv/dt = mg − cv. Starting from rest, the speed approaches mg/c rather than zero. At that terminal state kinetic energy is constant, but gravitational potential energy continues decreasing.
Terminal motion is therefore not an absence of energy conversion. Gravity supplies power mgv, the brake transfers cv² into heat, and force balance makes these equal. During the approach, the difference between them increases kinetic energy.
A slotted conductor generally weakens the relevant eddy loops and changes the effective c, but the amount depends on actual geometry. Do not infer a numerical percentage from the presence of slots alone. Compare measured curves or solve the field/current distribution when a quantitative design is required.
WORKED EXAMPLE
Find the continuing power after acceleration has ceased
A 0.10 kg object descends through a region described by constant c = 0.50 kg/s. Take g = 10 m/s² and neglect other forces. Find terminal speed and heating power there.
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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: Induction, moving conductors and magnetic energy Focus: Gravity can supply a continuing heat flow Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
14 / Induction, moving conductors and magnetic energy
Read the slope and keep the sign through an entire flux history
A flux–time graph contains two different kinds of information. Its height tells you the flux at that instant; its slope tells you the induced emf. A line can cross zero flux while keeping a nonzero slope, or sit at a large constant flux with zero emf. For N equal turns, first determine whether the graph labels flux per turn Φ or total linkage NΦ. Multiplying linkage by N again double-counts the turns.
Choose a surface normal and the corresponding positive loop direction by the right-hand rule. Then ε=−N dΦ/dt and, for a constant-resistance closed circuit with negligible inductance, I=ε/R. A decreasing positive flux and a flux becoming more negative can produce the same current direction: both have negative slope. At an ideal sharp corner the one-sided slopes describe the adjacent intervals; a physical change is smoothed over finite time.
Integrating the current gives signed transported charge Q_net=−N(Φ_f−Φ_i)/R. It depends only on the endpoint fluxes under this resistive model. The total amount of charge passing a section in either direction is ∫|I|dt, which adds rather than cancels reversed passages. Dissipated heat ∫I²R dt also adds and depends on how quickly the history is traversed.
Q_net is algebraic transport in the chosen positive direction. Q_through counts both directions positively. These formulas do not imply that net charge accumulates at a point in a closed wire.
Explain this equation
Read it in words
Net signed charge transport equals minus turns divided by resistance times the signed endpoint change in flux per turn.
Symbols and units
- Algebraic charge transported through a cross-sectionC
- Total charge passage counting both directions positivelyC
- Number of identical coil turnsdimensionless
- Total constant circuit resistanceΩ
- Initial and final flux per turn for the chosen normalWb
- Signed conventional currentA
- Times
Signs and operations
Positive loop current and positive area normal obey the right-hand rule. Final minus initial sets ΔΦ; the leading minus sign is Faraday’s sign. Absolute-value bars count either current direction positively.
When it applies
Closed coil, constant resistance, negligible self-inductance and no additional emf sources; identical flux per turn.
Make it concrete
Ten turns with R=2 Ω and ΔΦ=−0.002 Wb give +0.010 C net transport.
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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.
Equation (LaTeX): Q_{\mathrm{net}}=-\frac{N}{R}(\Phi_f-\Phi_i),\qquad Q_{\mathrm{through}}=\int |I|\,dt
Read it as: Net signed charge transport equals minus turns divided by resistance times the signed endpoint change in flux per turn.
Symbols:
Q_{\mathrm{net}}: Algebraic charge transported through a cross-section; C
Q_{\mathrm{through}}: Total charge passage counting both directions positively; C
N: Number of identical coil turns; dimensionless
R: Total constant circuit resistance; Ω
\Phi_i,\Phi_f: Initial and final flux per turn for the chosen normal; Wb
I: Signed conventional current; A
t: Time; s
Signs and operations: Positive loop current and positive area normal obey the right-hand rule. Final minus initial sets ΔΦ; the leading minus sign is Faraday’s sign. Absolute-value bars count either current direction positively.
Conditions: Closed coil, constant resistance, negligible self-inductance and no additional emf sources; identical flux per turn.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
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.| t / s | Φ / mWb |
|---|---|
| 0 | 0 |
| 0.10 | 4 |
| 0.30 | 4 |
| 0.50 | −2 |
WORKED EXAMPLE
A flat interval carries flux but induces no resistive current
The table describes a 10-turn coil with R=2 Ω and negligible inductance. Find the current on each open time interval, the net transported charge and the total charge passage.
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Flux height and slope tell different stories
- Flux per turn / mWb
- 0
STOP & THINK
A fixed 20-turn coil of area 0.010 m² has normal +z. B_z=(0.30−0.10t) T. With R=2 Ω and negligible inductance, find the signed current at t=3 s; positive current is counterclockwise viewed from +z.
Answer in A.
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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: Induction, moving conductors and magnetic energy
Focus: Read the slope and keep the sign through an entire flux history
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
Q_{\mathrm{net}}=-\frac{N}{R}(\Phi_f-\Phi_i),\qquad Q_{\mathrm{through}}=\int |I|\,dt
Meaning: Q_net is algebraic transport in the chosen positive direction. Q_through counts both directions positively. These formulas do not imply that net charge accumulates at a point in a closed wire.
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.15 / Induction, moving conductors and magnetic energy
Follow entry, full immersion and exit as three different regimes
Translate a closed rectangular loop through a stationary strip of uniform magnetic field. Let H be its transverse height, w its length along the direction of travel, and v its constant speed. Take the strip wider than w so there is an interval when the entire loop is inside. Ignore fringing, self-inductance and changes in resistance.
During entry, the overlap area grows at Hv. During full immersion, the overlap area is constant even though the loop moves. During exit, the overlap shrinks at Hv. Thus the emf magnitude is BHv at the two boundaries and zero in the middle. The entry and exit currents oppose opposite flux changes and therefore have opposite senses.
The magnetic force during either boundary crossing opposes the imposed motion. With I=BHv/R, its magnitude is BHI. The external pull maintaining the speed supplies power Fv=I²R. Full immersion gives no net emf and no sustained current in this resistive model; adding the two moving-side emfs as positive magnitudes would invent a current. An appreciable inductance would change these instantaneous current predictions.
WORKED EXAMPLE
Account for heat at both boundaries
A loop has H=0.30 m, w=0.20 m and R=3 Ω. It moves at 2 m/s through a field strip of width 0.80 m, with B=0.50 T into the page. Find the total heat from complete entry and exit.
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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: Induction, moving conductors and magnetic energy Focus: Follow entry, full immersion and exit as three different regimes Learning goals: - Calculate signed flux and distinguish flux from flux linkage. - Apply Faraday and Lenz laws with explicit orientation and time dependence. - Explain an induced circulating electric field without assuming a conductor. - Derive motional emf for translating and rotating rods. - Check a sliding-rod calculation using mechanical and electrical power. - Relate rotating-coil flux and generator emf. - Distinguish an inductor’s passive voltage drop from its self-induced emf. - Derive magnetic storage and solve ideal RL rise and decay. - Explain closed eddy-current paths and distinguish their loss from hysteresis. - Use a stated magnetic-drag model to check motion against energy transfer. 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.
16 / Induction, moving conductors and magnetic energy
Outside the changing field, circulation can still be nonzero
Suppose a spatially uniform B_z(t) is confined to a circular region of radius a in the ideal cylindrical-symmetry model. For a concentric circular path inside that region, the enclosed changing flux grows as r². Outside it, the changing flux is limited to the area πa² even though the path circumference keeps growing.
Symmetry makes the induced electric field tangential and equal in magnitude around each path. Applying the signed circulation law gives E_φ=−r(dB_z/dt)/2 inside and E_φ=−a²(dB_z/dt)/(2r) outside, with positive φ counterclockwise viewed from +z. The field is continuous at r=a. A material conducting ring is not needed for this electric field to exist.
The two expressions agree at the edge. A larger outside circle encloses the same changing flux spread over a longer integration path.
WORKED EXAMPLE
Two radii can have the same induced field magnitude
Let a=0.12 m and dB_z/dt=5 T/s. Compare E_φ at r=0.060 m and r=0.24 m.
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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: Induction, moving conductors and magnetic energy
Focus: Outside the changing field, circulation can still be nonzero
Learning goals:
- Calculate signed flux and distinguish flux from flux linkage.
- Apply Faraday and Lenz laws with explicit orientation and time dependence.
- Explain an induced circulating electric field without assuming a conductor.
- Derive motional emf for translating and rotating rods.
- Check a sliding-rod calculation using mechanical and electrical power.
- Relate rotating-coil flux and generator emf.
- Distinguish an inductor’s passive voltage drop from its self-induced emf.
- Derive magnetic storage and solve ideal RL rise and decay.
- Explain closed eddy-current paths and distinguish their loss from hysteresis.
- Use a stated magnetic-drag model to check motion against energy transfer.
Equations in this section (LaTeX):
E_\phi=-\frac{1}{2}\frac{dB_z}{dt}\begin{cases}r,&r\leq a,\\a^2/r,&r\geq a.\end{cases}
Meaning: The two expressions agree at the edge. A larger outside circle encloses the same changing flux spread over a longer integration path.
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.BRING THE IDEAS TOGETHER
What should stay with you
- Flux uses the angle with the area normal; linkage sums over turns.
- Induction responds to a change of linkage, not merely a large magnetic field.
- Lenz’s response opposes the change, which may mean supporting the original field.
- Motional emf requires the velocity and geometry of each conductor segment.
- A passive sliding-rod circuit converts external mechanical work into heat.
- Self-induction opposes current change; finite voltage preserves current continuity.
- Linear magnetic energy is LI²/2 and decays twice as fast exponentially as current in an RL discharge.
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
A 40-turn coil of area 0.010 m² per turn sees a perpendicular field change by 0.50 T in 0.10 s. What is the average emf magnitude?
Answer in V.
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SELF CHECK 2
A 0.80 m rod moves at 5.0 m/s perpendicular to itself and to a 0.25 T field. What is its motional-emf magnitude?
Answer in V.
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SELF CHECK 3
A linear 0.40 H inductor carries 3.0 A. What magnetic energy is stored?
Answer in J.
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SELF CHECK 4
A 0.30 H inductor discharges through a total resistance of 6.0 Ω. What is the current time constant in milliseconds?
Answer in ms.
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SELF CHECK 5
An outward magnetic flux through a passive loop is decreasing. What field does its induced current tend to produce?
- Inward, because induced fields always oppose existing fields
- Outward, to oppose the decrease
- No field because decreasing flux cannot induce current
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SELF CHECK 6
During ideal RL discharge, when current falls to half its initial value, what fraction of initial magnetic energy remains?
- One half
- One quarter
- Three quarters
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SELF CHECK 7
For a uniformly rotating coil, when flux linkage is at its maximum, the induced emf is:
- At its positive maximum
- Zero at that instant
- Equal numerically to the flux linkage
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SELF CHECK 8
A 0.60 kg cart coasts under F = −0.30v in SI units. What is the time to halve its speed? Give at least three significant figures, or an exact fraction.
Answer in s.
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SELF CHECK 9
Why can insulated laminations reduce eddy-current heating?
- They obstruct large closed conduction paths
- They remove all magnetic flux by definition
- They guarantee zero hysteresis loss
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SELF CHECK 10
A constant-resistance 5-turn coil has R=10 Ω. Flux per turn changes from +0.008 to +0.002 Wb. Neglect inductance. Find net transported charge in the positive loop direction.
Answer in C.
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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.
A loop is tilted in the fieldIntermediate · Free sample · Hints and guided reasoningSigned transport is not total passageAdvanced · Hints and guided reasoningCount both field boundariesAdvanced · Hints and guided reasoningZero flux can still mean currentIntermediate · Hints and guided reasoningThe changing area stops at the field edgeAdvanced · Hints and guided reasoningThe induced field opposes the changeIntermediate · Hints and guided reasoningAn induced electric field exists outside the changing field regionAdvanced · Hints and guided reasoningThe capacitor starts with only part of the energyAdvanced · Hints and guided reasoningHow large can mutual inductance be?Advanced · Hints and guided reasoningMeasure coupling by reversing one coilAdvanced · Hints and guided reasoningHeat released before the speed halvesIntermediate · Hints and guided reasoningA moving rod can have voltage without a circuit.Intermediate · Hints and guided reasoningFlux changes through fifty turnsIntermediate · Hints and guided reasoningEnergy carried by the magnetic fieldIntermediate · Hints and guided reasoningHow fast does current build?Intermediate · Hints and guided reasoningCOME 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
An induced current opposes a change. Why is that more precise than saying it always opposes the original magnetic field?
Compare your reasoning
If the original flux is decreasing, the induced contribution can reinforce its original direction to oppose the decrease. Identify the change first, then choose the response that opposes it.
Revisit the explanation →A few days later: change the situation
A conductor slows under magnetic braking. Where can its kinetic energy go, and why is a constant opposing force not automatically the right model?
Compare your reasoning
It can become heat through induced currents and resistance. The force depends on geometry, materials and motion; a linear −cv model is an explicit approximation, not a universal consequence of Lenz’s law.
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.
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