A narrow section of a water pipe may suggest how a thin wire behaves, but the analogy has limits. Charges do not get used up as they travel through a resistor. We will track current, voltage and energy separately so that the circuit has one consistent account.
This chapter connects carrier drift, material resistivity, wire shape and circuit networks. It then follows how temperature changes resistance, and why the power response differs for a voltage source and a current source.
What you will learn to do
- Calculate transferred charge from a current history.
- Distinguish electron drift from conventional current and signal propagation.
- Relate current density and field to resistivity.
- Derive geometry and fixed-volume stretching effects on resistance.
- Reduce and reconstruct series–parallel resistor circuits.
- Include source internal resistance and check energy flow.
- Use a temperature coefficient with its proper reference.
- Choose fixed-current or fixed-voltage power reasoning from the actual circuit.
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: Current, resistance and circuit constraints Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. 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.
If charge is not accumulating at a junction, how must incoming and outgoing currents compare?
Compare your reasoning
Their signed sum is zero: total current entering equals total current leaving. This follows from charge conservation, not from an assumption that all branches have the same resistance.
Revisit the explanation →01 / Current, resistance and circuit constraints
Current is a rate of charge transfer through a chosen surface
Choose a cross-section of a conductor and count the signed charge passing through it. Electric current is I = dQ/dt. One ampere means one coulomb per second. For a changing current, transferred charge is the area under the current–time graph, Q = ∫I dt.
Conventional current points in the direction positive charge would move. In a metal the mobile electrons drift in the opposite direction. Both descriptions give the same signed charge transport when carrier charge is included consistently.
A steady current does not consume charge as it passes through a resistor. Charge enters and leaves at the same rate if no net charge accumulates there. Energy is transferred to the material, but that does not mean electrons disappear along the wire.
Current density J describes local current per area. For a uniform current perpendicular to a cross-section, I = JA. More generally I is the surface integral of J·dA, so a nonuniform or angled flow requires the appropriate projection and distribution.
Current is a signed rate through a specified oriented surface. Current density has units A/m².
WORKED EXAMPLE
Use the current graph to count charge
A current rises linearly from zero to 3.0 A over 4.0 s, then remains at 3.0 A for another 2.0 s. Find the total charge transferred.
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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: Current, resistance and circuit constraints
Focus: Current is a rate of charge transfer through a chosen surface
Learning goals:
- Calculate transferred charge from a current history.
- Distinguish electron drift from conventional current and signal propagation.
- Relate current density and field to resistivity.
- Derive geometry and fixed-volume stretching effects on resistance.
- Reduce and reconstruct series–parallel resistor circuits.
- Include source internal resistance and check energy flow.
- Use a temperature coefficient with its proper reference.
- Choose fixed-current or fixed-voltage power reasoning from the actual circuit.
Equations in this section (LaTeX):
I=\frac{dQ}{dt},\qquad Q=\int I\,dt,\qquad I=JA\ \text{for uniform normal flow}
Meaning: Current is a signed rate through a specified oriented surface. Current density has units A/m².
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 / Current, resistance and circuit constraints
A small average drift can carry a substantial current
In a conductor with carrier number density n, each carrier charge q and mean drift velocity v_d, the current density is J = nqv_d. For a single carrier type and uniform flow, the current magnitude is n|q|A|v_d|.
To see the counting, during a short interval dt the carriers in a cylinder of length |v_d|dt and area A cross the chosen section on average. That cylinder contains nA|v_d|dt carriers, so its transported charge divided by dt gives the current.
The drift is a small directed average superposed on microscopic random motion. Random velocities largely cancel in the current average, while the small directional bias remains. Do not substitute an rms thermal or electronic speed for drift speed in I = nqAv_d.
A circuit response does not require one particular electron to travel all the way from the source to a device before that device responds. Establishing electromagnetic fields through the circuit redistributes the motion of charges already present. Drift speed and signal propagation describe different processes.
For electron carriers q = −e, conventional J is opposite to electron drift velocity.
WORKED EXAMPLE
Count carriers without confusing drift and random speed
A wire carries 0.40 A uniformly through area 0.50 mm². Its mobile-electron density is 5.0 × 10²⁸ m⁻³. Use e = 1.6 × 10⁻¹⁹ C. Find drift-speed magnitude.
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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: Current, resistance and circuit constraints Focus: A small average drift can carry a substantial current Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. Equations in this section (LaTeX): \mathbf J=nq\mathbf v_d,\qquad |I|=n|q|A|\mathbf v_d| Meaning: For electron carriers q = −e, conventional J is opposite to electron drift velocity. 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 / Current, resistance and circuit constraints
A material relation connects field to current density
For an isotropic ohmic material at fixed conditions, J = σE, where σ is electrical conductivity. Equivalently E = ρJ, where ρ = 1/σ is electrical resistivity. Here ρ denotes resistivity, not mass density.
A simple relaxation-time picture gives electron drift velocity approximately −eτE/m, with τ an effective momentum-relaxation time. Substituting into J = −nev_d gives σ = ne²τ/m. This model connects a larger directed response to larger carrier density or longer relaxation time.
The relaxation-time expression is a simplified microscopic model. Real materials can have temperature-dependent carrier densities, scattering mechanisms and band effects. The macroscopic ohmic relation is valid only over a range where current density is approximately proportional to field at fixed physical conditions.
A diode, a strongly heated filament or a material at very high field may not have a constant ratio V/I over the operating range. Defining a resistance at a particular operating point does not prove that the device obeys a linear Ohm’s law everywhere.
The last expression is the elementary electron relaxation-time model. Resistivity units are Ω·m, distinct from resistance units Ω.
WORKED EXAMPLE
Compare drift in two sections of the same wire path
A steady current passes through two sections made of the same material at the same temperature. The second cross-sectional area is twice the first, with the same carrier density. Compare drift speeds and current densities.
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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: Current, resistance and circuit constraints
Focus: A material relation connects field to current density
Learning goals:
- Calculate transferred charge from a current history.
- Distinguish electron drift from conventional current and signal propagation.
- Relate current density and field to resistivity.
- Derive geometry and fixed-volume stretching effects on resistance.
- Reduce and reconstruct series–parallel resistor circuits.
- Include source internal resistance and check energy flow.
- Use a temperature coefficient with its proper reference.
- Choose fixed-current or fixed-voltage power reasoning from the actual circuit.
Equations in this section (LaTeX):
\mathbf J=\sigma\mathbf E,\qquad\rho=\frac1\sigma,\qquad\sigma\approx\frac{ne^2\tau}{m}
Meaning: The last expression is the elementary electron relaxation-time model. Resistivity units are Ω·m, distinct from resistance units Ω.
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 / Current, resistance and circuit constraints
Resistance combines material response with geometry
For a uniform wire of length L and area A with approximately uniform longitudinal field and current density, V = EL and I = JA. Substituting E = ρJ gives R = V/I = ρL/A.
Resistance is a property of the specific object and current path. Resistivity describes the material under specified conditions. A longer wire has more resistance; a larger cross-section offers more parallel conducting paths and lowers resistance.
Area changes must be handled geometrically. For a circular wire A = πr², so doubling radius reduces resistance by four at fixed length and resistivity. Doubling diameter has the same area factor, but substituting diameter as radius introduces a separate factor-of-four error.
If a wire is stretched at approximately constant material volume and resistivity, its area decreases as its length increases. Then R scales as L², not just L. This assumption neglects temperature changes, damage and any change in intrinsic resistivity due to strain.
Uniform cross-section and material, fixed physical conditions and an approximately uniform longitudinal current distribution.
WORKED EXAMPLE
Calculate a wire resistance with a small cross-section
A uniform wire has resistivity 2.0 × 10⁻⁸ Ω·m, length 2.0 m and area 0.50 mm². Find its resistance.
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WORKED EXAMPLE
Track both dimensions when a wire is stretched
A wire is stretched uniformly to three times its original length, conserving material volume and resistivity. What is its final-to-initial resistance ratio?
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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: Current, resistance and circuit constraints Focus: Resistance combines material response with geometry Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. Equations in this section (LaTeX): R=\rho\frac LA Meaning: Uniform cross-section and material, fixed physical conditions and an approximately uniform longitudinal current distribution. 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 / Current, resistance and circuit constraints
Circuit connections determine which current or voltage is shared
Resistors are in series when the same current must pass through them without branching between them. Their voltage drops add, so R_series = ΣR_i. A series equivalent exceeds any positive individual resistance.
Resistors are in parallel when each connects to the same two nodes and therefore has the same voltage. Currents add, giving 1/R_parallel = Σ1/R_i. A parallel equivalent is smaller than the smallest positive branch resistance.
These rules are opposite in form to the familiar capacitor rules because the constitutive relations differ: a resistor has V = IR, while a capacitor relates charge and voltage. Derive the appropriate shared-quantity relation rather than swapping formulas from memory.
Ideal wires connect equipotential nodes in the lumped steady-circuit approximation. Redrawing a wire without adding a component does not create a new node. A branch at an intermediate connection can break a proposed series pairing even if two resistors are drawn one after another.
The components are ideal ohmic resistors under fixed conditions, and the identified topology must actually share the claimed current or voltage.
WORKED EXAMPLE
Find branch currents before replacing them by a total
Resistors of 6 Ω and 3 Ω are connected in parallel across an ideal 12 V source. Find each branch current, the total current and the equivalent 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: Current, resistance and circuit constraints
Focus: Circuit connections determine which current or voltage is shared
Learning goals:
- Calculate transferred charge from a current history.
- Distinguish electron drift from conventional current and signal propagation.
- Relate current density and field to resistivity.
- Derive geometry and fixed-volume stretching effects on resistance.
- Reduce and reconstruct series–parallel resistor circuits.
- Include source internal resistance and check energy flow.
- Use a temperature coefficient with its proper reference.
- Choose fixed-current or fixed-voltage power reasoning from the actual circuit.
Equations in this section (LaTeX):
R_s=\sum_iR_i,\qquad\frac1{R_p}=\sum_i\frac1{R_i}
Meaning: The components are ideal ohmic resistors under fixed conditions, and the identified topology must actually share the claimed current or voltage.
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 / Current, resistance and circuit constraints
Reduce the network and then restore its internal voltages
For a mixed network, reduce a clear parallel or series group first, keeping track of the nodes at its terminals. Find the total current from the source, then reverse the reduction to recover branch voltages and currents.
A series resistor does not usually share the full source voltage with a parallel group beside it. It takes a voltage drop determined by the common total current. The remaining voltage appears across each branch of the parallel group.
Check both charge and energy accounting after reconstructing the circuit. Currents entering a junction should equal currents leaving it in steady state. Voltage drops around a complete loop should balance the source rises under the lumped circuit model.
WORKED EXAMPLE
Recover the voltage across a parallel pair behind a series resistor
A 4 Ω resistor is in series with a parallel combination of 6 Ω and 3 Ω. The full network is connected to an ideal 12 V source. Find source current and both branch currents.
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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: Current, resistance and circuit constraints Focus: Reduce the network and then restore its internal voltages Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. 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 / Current, resistance and circuit constraints
An emf supplies energy per charge; terminal voltage can differ
An emf source supplies energy per unit charge through a non-electrostatic mechanism such as chemical action. Despite its name, electromotive force is measured in volts, not newtons. An ideal source maintains its stated voltage independent of current in the model.
A simple real-source model adds an internal series resistance r. When it delivers current I through an external resistance R, the loop balance is ℰ − Ir − IR = 0, giving I = ℰ/(R + r). Its terminal voltage is ℰ − Ir, below its emf during discharge.
Kirchhoff’s junction rule is steady charge conservation. The loop rule balances potential rises and drops in a lumped circuit with any induced emf accounted for explicitly. Choose current directions and loop traversal directions, then keep resistor and source signs consistent. A negative solved current means the actual direction opposes the chosen arrow.
An emf rise is positive when traversing an ideal source from its negative to positive terminal. Across a resistor in the direction of the assigned current, the potential change is −IR. More complicated networks may require simultaneous junction and loop equations rather than series–parallel reduction.
WORKED EXAMPLE
Distinguish emf, terminal voltage and internal heating
A source of emf 12 V and internal resistance 1 Ω powers an external 5 Ω resistor. Find current, terminal voltage and power dissipated internally.
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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: Current, resistance and circuit constraints Focus: An emf supplies energy per charge; terminal voltage can differ Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. 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.
08 / Current, resistance and circuit constraints
Temperature changes the material resistance and may change the current
Over a limited temperature range, a material’s resistivity can be approximated by ρ(T) = ρ₀[1 + α₀(T − T₀)]. The reference resistivity and coefficient belong to a specified reference temperature T₀. The coefficient has units K⁻¹.
For a wire whose dimensions change negligibly, resistance follows the same linear form R(T) ≈ R₀[1 + α₀(T − T₀)]. If thermal expansion matters, geometry also changes and must be included separately. The simple resistance formula usually neglects that correction.
Many metals have a positive temperature coefficient over ordinary ranges, so resistance rises on heating. Some semiconductors and other materials have different behaviour because carrier density and scattering change differently. Do not treat a positive metallic coefficient as universal.
The approximation is local in temperature. It should not be extrapolated until it predicts negative resistance or used across phase transitions or strong changes in material response. A quoted α should be used with its stated reference and range.
This resistance form assumes the linear resistivity approximation and negligible dimensional changes.
WORKED EXAMPLE
Use the correct reference and interval
A wire has resistance 10 Ω at 20°C and coefficient α₂₀ = 0.0040 K⁻¹ over the stated range. Neglect expansion. Find its resistance at 70°C.
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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: Current, resistance and circuit constraints Focus: Temperature changes the material resistance and may change the current Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. Equations in this section (LaTeX): R(T)\approx R_0[1+\alpha_0(T-T_0)] Meaning: This resistance form assumes the linear resistivity approximation and negligible dimensional changes. 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 / Current, resistance and circuit constraints
A linear law does not give the same fractional coefficient at every reference
If a linear resistance law is represented exactly over a small range, its absolute slope dR/dT is constant. The fractional coefficient at a reference temperature is that slope divided by the resistance at the reference. Therefore changing the reference changes the coefficient.
Starting with R(T) = R₀[1 + α₀(T − T₀)], the coefficient referred to T₁ is α₁ = α₀/[1 + α₀(T₁ − T₀)]. This is a re-expression of the same line, not an additional physical change of material.
For an experimental R–T graph, use the slope and an actual reference resistance to infer α. A graph intercept at 0°C is not automatically the same as a resistance measured at 20°C. Reading a fractional coefficient without naming its reference loses essential information.
The coefficient can also be inferred from two resistance measurements as α₀ = (R₁/R₀ − 1)/(T₁ − T₀), when R₀ is measured at T₀ and the linear approximation is justified. Repeated measurements and a fitted line can reveal departures from that approximation.
WORKED EXAMPLE
Re-express the same line at a new reference temperature
For the preceding wire, R₂₀ = 10 Ω and R₇₀ = 12 Ω. Find the coefficient referred to 70°C while keeping the same linear R–T relation.
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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: Current, resistance and circuit constraints Focus: A linear law does not give the same fractional coefficient at every reference Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. 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.
10 / Current, resistance and circuit constraints
Power formulas must use the constraint that the circuit actually imposes
A potential drop V transfers energy V per unit charge, while current transfers charge at rate I. The electrical power delivered to a passive element is VI with a consistent sign convention. For an ohmic resistor, substituting V = IR gives P = I²R = V²/R.
These expressions are equivalent for the same operating state, but they suggest different trends under different constraints. At fixed current, increasing R increases dissipated power. At fixed voltage, increasing R reduces current and decreases power. State which source condition is maintained before making a comparison.
Electrical energy transferred over a time interval is ∫P dt. For constant power it is Pt. Joule heating raises temperature only according to the thermal balance: heat can also leave to the environment, and material properties may change as the resistor warms.
A measurement can perturb the circuit. An ideal ammeter has zero resistance and is inserted in series; an ideal voltmeter has infinite resistance and is connected in parallel. Real meters have finite internal properties, so their loading must be considered when precision matters.
WORKED EXAMPLE
Resolve opposite-looking power predictions
A resistor’s resistance rises from 10 Ω to 12 Ω. Compare dissipated power when it is driven first by an ideal fixed 6 V source and, in a separate case, by an ideal fixed 0.60 A current source.
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STOP & THINK
How is an ideal voltmeter connected to measure the potential difference across a resistor?
- In series, with zero resistance
- In parallel, with infinite resistance
- Directly across an ideal source as a zero-resistance wire
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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: Current, resistance and circuit constraints Focus: Power formulas must use the constraint that the circuit actually imposes Learning goals: - Calculate transferred charge from a current history. - Distinguish electron drift from conventional current and signal propagation. - Relate current density and field to resistivity. - Derive geometry and fixed-volume stretching effects on resistance. - Reduce and reconstruct series–parallel resistor circuits. - Include source internal resistance and check energy flow. - Use a temperature coefficient with its proper reference. - Choose fixed-current or fixed-voltage power reasoning from the actual circuit. 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.
BRING THE IDEAS TOGETHER
What should stay with you
- Current counts signed charge per time; a resistor transfers energy without consuming charge.
- Electron drift opposes conventional current.
- Resistivity is a material quantity; resistance also depends on geometry.
- A stretched wire can change both length and area.
- Series resistors share current; parallel resistors share node voltage.
- A discharging source’s terminal voltage can be below its emf.
- A temperature coefficient is tied to its reference temperature and approximation range.
- Increasing resistance has opposite power effects under fixed current and fixed voltage.
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 wire’s length doubles while its volume and resistivity remain fixed. What is the resistance ratio R₂/R₁?
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SELF CHECK 2
A resistor is 20 Ω at its reference temperature. With α = 0.002 K⁻¹, what is its resistance after a 25 K increase under the linear model?
Answer in Ω.
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SELF CHECK 3
For the same steady current and carrier density, a wire section has three times the area of another. What is its drift-speed ratio relative to the narrower section? Give at least three significant figures, or an exact fraction.
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SELF CHECK 4
Two resistors connect to the same pair of ideal-wire nodes. What must they share?
- Current
- Potential difference
- Power
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SELF CHECK 5
In a linear R–T model, changing the reference temperature leaves which quantity constant?
- The absolute slope dR/dT
- The fractional coefficient at every reference
- The reference resistance
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SELF CHECK 6
At fixed source voltage, a resistor’s resistance increases. What happens to its ideal dissipated power?
- It increases because I²R contains R
- It decreases because P = V²/R
- It is necessarily unchanged
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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.
Stretching changes two dimensionsIntermediate · Free sample · Hints and guided reasoningCount carriers crossing the wireIntermediate · Hints and guided reasoningUse the stated reference resistanceIntermediate · Hints and guided reasoningSimplify the branches before finding the current.Intermediate · Hints and guided reasoningRead the slope, not a single ratioIntermediate · Hints and guided reasoningTest the clamp before using itAdvanced · 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
For the same material and temperature, how does doubling a wire’s length while keeping its area fixed change its resistance?
Compare your reasoning
Resistance doubles in the uniform-wire model R = ρL/A. Resistivity remains a material property under those conditions; it is not the resistance of the whole specimen.
Revisit the explanation →A few days later: change the situation
Two different resistors are connected in parallel to a fixed voltage. Which dissipates more power, and why can a series comparison give the opposite answer?
Compare your reasoning
In parallel the smaller resistance dissipates more because P = V²/R at equal voltage. In series current is shared, so the larger resistance dissipates more because P = I²R. Identify the fixed quantity first.
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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