Source: Chapters 9 and 10
Tags: electric current, resistance, resistivity, Ohm's law, drift velocity, power, EMF, Kirchhoff's rules, series resistors, parallel resistors, RC circuit, time constant, superconductor
Difficulty: Intermediate
Prerequisites: Chapters 5-8 notes (electric field, potential, potential difference, capacitance).
This is where electrostatics becomes electrodynamics: charges start moving. Current is the flow of charge, resistance is what opposes it, and Ohm's law ties them together. Chapter 10 then introduces full circuits with batteries, series and parallel resistors, Kirchhoff's rules for more complex networks, and RC circuits where a capacitor charges or discharges through a resistor. These chapters are the practical heart of PHYS 212 and connect directly to every lab you will do involving circuits.
Current is the rate of charge flow, driven by a potential difference. Resistance opposes flow and depends on the material and geometry of the conductor. Ohm's law (V = IR) is the central relationship. In circuits, resistors combine in series and parallel, Kirchhoff's rules handle multi-loop problems, and RC circuits describe how capacitors charge and discharge exponentially over time.
Electric current (I)
The rate at which charge flows past a point: I = ΔQ/Δt. Measured in amperes (A). Conventional current flows from high to low potential (the direction positive charges would move). Think of it as: the traffic rate of charges through a wire.
Drift velocity (v_d)
The average velocity of free charge carriers in a conductor under an applied field. It is very slow (typically fractions of a mm/s), even though the field's effect propagates at nearly the speed of light.
Current density (J)
Current per unit cross-sectional area: J = I/A = nqv_d, where n is the number density of free charges.
Resistivity (ρ)
A material property measuring how strongly a material opposes current. Units: Ω·m. It depends on temperature: ρ = ρ₀[1 + α(T - T₀)], where α is the temperature coefficient. In simple terms, resistivity is the material's intrinsic resistance to current flow.
Resistance (R)
The opposition to current in a specific conductor: R = ρL/A, where L is length and A is cross-sectional area. Measured in ohms (Ω).
Ohm's law
V = IR. The potential difference across a resistor equals the current through it multiplied by its resistance. Think of it as: more voltage pushes more current through a given resistance.
Electromotive force (EMF, ε)
The energy per unit charge supplied by a source (battery, generator). It is the potential difference across the source's terminals when no current flows. In simple terms, EMF is the battery's "rated voltage" before internal resistance takes its toll.
Terminal voltage
The actual voltage across a battery's terminals when current is flowing: V_terminal = ε - Ir, where r is the internal resistance.
Kirchhoff's current law (junction rule)
At any junction in a circuit, the total current entering equals the total current leaving. This is conservation of charge.
Kirchhoff's voltage law (loop rule)
Around any closed loop in a circuit, the sum of all potential differences equals zero. This is conservation of energy.
RC circuit
A circuit containing a resistor and a capacitor. The time constant τ = RC governs how quickly the capacitor charges or discharges.
Superconductor
A material whose resistance drops to exactly zero below a critical temperature. All magnetic fields are expelled (Meissner effect).
For current to flow, there must be a complete path from the positive terminal to the negative terminal of a source.
Positive charges move in the direction of the electric field. Negative charges (electrons) move opposite to the field, but conventional current is defined in the direction positive charges would move.
Current density: J = nqv_d = σE, where σ is the electrical conductivity (σ = 1/ρ).
Resistivity is a material property. Resistance depends on the object's shape and material: R = ρL/A.
Both resistivity and resistance increase with temperature in most metals: R = R₀[1 + α(T - T₀)].
A diode is a non-ohmic device: it allows current in only one direction. In forward bias, current flows. In reverse bias, it does not (until breakdown voltage).
V = IR. Applies to any ohmic (linear) device.
Can be rearranged: I = V/R or R = V/I.
Power dissipated by a resistor: P = IV = I²R = V²/R.
Work done on a charge: W = qV.
Energy consumed: E = ∫ P dt. For constant power: E = Pt.
The cost of electricity is based on energy consumed (typically in kilowatt-hours).
Type I superconductors have limited applications because the critical magnetic field needed to destroy superconductivity is low.
Type II superconductors are high-temperature superconductors with much higher critical fields and current densities.
The Meissner effect: magnetic fields are expelled from the interior of a superconductor.
BCS theory: at low temperatures, electrons pair up into Cooper pairs that move without resistance because they avoid collisions with impurities.
A real battery has internal resistance r. Terminal voltage: V = ε - Ir.
When no current flows, V = ε (open-circuit voltage).
Series: R_eq = R₁ + R₂ + R₃ + ... The same current flows through each. Voltage drops add up.
Parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ... The same voltage is across each. Currents add up.
Junction rule: ΣI_in = ΣI_out at any node.
Loop rule: ΣΔV = 0 around any closed loop.
When traversing a loop: voltage rises across a battery (from - to +), drops across a resistor in the direction of current.
For N batteries in series: V_terminal = (ε₁ + ε₂ + ... + ε_N) - I(r₁ + r₂ + ... + r_N).
Charging a capacitor through a resistor from a battery (EMF = ε):
Charge: q(t) = Cε(1 - e^(-t/τ)) = Q_max(1 - e^(-t/τ))
Current: I(t) = (ε/R) e^(-t/τ) = I₀ e^(-t/τ)
Time constant: τ = RC
Discharging a capacitor:
Charge: q(t) = Q₀ e^(-t/τ)
Current: I(t) = -I₀ e^(-t/τ)
After one time constant (t = τ), the charge has reached about 63% of its final value (charging) or dropped to about 37% of its initial value (discharging).
Quantity | Formula |
|---|---|
Current | I = ΔQ/Δt |
Drift velocity relation | I = nqv_dA |
Resistance | R = ρL/A |
Ohm's law | V = IR |
Power | P = IV = I²R = V²/R |
Terminal voltage | V = ε - Ir |
Resistors in series | R_eq = R₁ + R₂ + ... |
Resistors in parallel | 1/R_eq = 1/R₁ + 1/R₂ + ... |
RC charging | q(t) = Cε(1 - e^(-t/RC)) |
RC discharging | q(t) = Q₀ e^(-t/RC) |
Time constant | τ = RC |
Every electronic device you use relies on Ohm's law and resistor networks. The RC time constant governs how quickly a camera flash recharges, how touchscreen signals are filtered, and how debouncing works in keyboard circuits. Superconductors are used in MRI machines (the powerful magnets are superconducting coils) and in particle accelerators.
Students often confuse EMF with terminal voltage. EMF is the battery's ideal output. Terminal voltage is always less when current flows, because of internal resistance.
Resistor series/parallel rules are the opposite of capacitor rules. In series, resistances add directly. In parallel, reciprocals add. Do not swap them.
Current is not "used up" as it passes through a resistor. The same current enters and leaves. What changes is the potential (voltage drops across the resistor).
In RC circuits, students sometimes assume the capacitor charges linearly. It does not: the charging rate slows exponentially as the capacitor approaches full charge.
⚠️ Be fluent with Kirchhoff's rules. Multi-loop circuit problems are standard exam fare. Practise setting up the equations systematically.
⚠️ Know both forms of the RC equations (charging and discharging) and what happens at t = 0, t = τ, and t → ∞.
⚠️ Power problems: be ready to use all three forms of P (IV, I²R, V²/R) depending on what information is given.
⚠️ Watch the sign conventions when traversing loops. A battery traversed from - to + is a voltage rise (+ε); a resistor traversed in the direction of current is a voltage drop (-IR).
Fill in the blank: The time constant of an RC circuit is τ = ________.
RC.
True or false: Doubling the length of a wire doubles its resistance.
True (R = ρL/A).
True or false: In a parallel combination of resistors, the equivalent resistance is always greater than the largest individual resistance.
False. It is always less than the smallest individual resistance.
Fill in the blank: After one time constant, a charging capacitor has reached approximately ________% of its final charge.
63%.
Q: A 6 V battery with 0.5 Ω internal resistance drives a current of 2 A through an external circuit. What is the terminal voltage?
A: V = ε - Ir = 6 - (2)(0.5) = 5 V.
Q: Three resistors (2 Ω, 4 Ω, 4 Ω) are connected in parallel. What is the equivalent resistance?
A: 1/R_eq = 1/2 + 1/4 + 1/4 = 1. R_eq = 1 Ω.
Q: An RC circuit has R = 28 kΩ and C = 22 mF. How long does it take for the capacitor to reach 50% of its maximum charge?
A: τ = RC = (28 x 10³)(22 x 10⁻³) = 616 s. For 50%: 0.5 = 1 - e^(-t/τ), so e^(-t/τ) = 0.5, giving t = τ ln 2 ≈ 0.693 × 616 ≈ 427 s.
Q: A 100 W light bulb operates at 120 V. What is its resistance?
A: P = V²/R, so R = V²/P = (120)²/100 = 144 Ω.
Ohm's law and Kirchhoff's rules return in AC circuits (Ch. 15) with the added complexity of reactance and impedance. RC circuits are a special case of the more general RLC circuits (Ch. 14-15). The concept of power dissipation is essential for understanding energy transfer in electromagnetic waves (Ch. 16).
electric current, ampere, drift velocity, current density, Ohm's law, resistance, resistivity, resistor, series circuit, parallel circuit, Kirchhoff's laws, junction rule, loop rule, EMF, electromotive force, terminal voltage, internal resistance, RC circuit, time constant, exponential decay, charging, discharging, power, watt, kilowatt-hour, superconductor, Cooper pair, Meissner effect, BCS theory, diode, PHYS 212