Difficulty: Intermediate | Prerequisites: Electrostatics and capacitor basics (Part 1 notes), right-hand rule
This block of material bridges electrostatics and electrodynamics. RC circuits introduce time-dependent behaviour, where current and voltage change exponentially. Magnetic force tells you what happens when charges move through magnetic fields, and electromagnetic induction (Faraday's law) is the principle behind generators, transformers, and most of modern electrical engineering. You need to be comfortable with the right-hand rule and exponential decay before tackling AC circuits.
RC circuits charge and discharge exponentially with time constant tau = RC. Moving charges in a magnetic field feel a force perpendicular to both their velocity and the field (use the right-hand rule or cross product). A conductor moving through a magnetic field develops a motional EMF. Inductors resist changes in current; at steady state in a DC circuit, they act as ideal wires.
RC circuit
A circuit containing a resistor and capacitor in series with a voltage source. The capacitor charges or discharges exponentially. In simple terms, it is the simplest circuit that has a built-in time delay.
Time constant (tau = RC)
The time it takes for the voltage or current in an RC circuit to reach about 63% of its final value (charging) or drop to about 37% of its initial value (discharging). After five time constants, the process is essentially complete.
Magnetic force (Lorentz force)
The force on a charge moving through a magnetic field: F = qv x B. The force is always perpendicular to both v and B, so it changes the direction of motion but never the speed.
Cross product (v x B)
A vector operation that gives a result perpendicular to both input vectors. The right-hand rule determines the direction: point fingers along v, curl toward B, and your thumb points in the direction of F for a positive charge.
Motional EMF
The voltage induced across a conductor moving through a magnetic field: EMF = BLv, where L is the length of the conductor and v is its speed perpendicular to B. Think of it as the magnetic field doing work on the charges inside the moving bar.
Faraday's law of induction
The induced EMF in a loop equals the negative rate of change of magnetic flux through the loop. In simple terms, a changing magnetic environment creates a voltage.
RL circuit
A circuit with a resistor and inductor in series. The inductor opposes sudden changes in current, so current rises or falls exponentially with time constant tau = L/R.
Self-inductance (L)
A measure of how much a coil resists changes in current through it, measured in henrys (H). A larger inductance means the coil fights harder against current changes.
Ampere's law
The line integral of B around a closed loop equals mu_0 times the enclosed current. Like Gauss's Law for electric fields, it is most useful when symmetry allows you to pull B out of the integral.
At t = 0 (switch just closed), the capacitor has no charge, so it acts like a short circuit. All the battery voltage drops across the resistor: I(0) = V/R.
As the capacitor charges, the current decreases exponentially: I(t) = (V/R) e^(-t/RC).
The voltage across the capacitor rises: V_C(t) = V(1 - e^(-t/RC)).
After a long time (t >> RC), the capacitor is fully charged, current is zero, and V_C = V.
A fully charged capacitor (initial voltage V_0) is disconnected from the battery and discharged through R.
Voltage decays: V(t) = V_0 e^(-t/RC).
The voltage drops to V_0/e (about 37% of the initial value) after one time constant tau = RC.
This decay is the basis for measuring unknown resistance: if you know C, V_0, and the time to reach a certain voltage, you can solve for R.
F = qv x B. The force is perpendicular to both v and B.
For a positive charge moving in the +x direction through a field in the +y direction: F = q(v x-hat x B y-hat) = qvB(x-hat x y-hat) = qvB z-hat. The force is in the +z direction.
For a negative charge, reverse the direction.
The magnetic force does no work (it is always perpendicular to displacement), so it changes direction but not speed.
A straight conductor of length L moving at speed v perpendicular to a uniform field B develops EMF = BLv across its ends.
This follows from the magnetic force on charges inside the conductor: the force separates positive and negative charges to opposite ends of the bar.
Example: L = 0.5 m, v = 10 m/s, B = 2 T gives EMF = (2)(0.5)(10) = 10 V.
When the switch is first closed, the inductor opposes the change in current, so initially I = 0 and the full battery voltage appears across L.
Current rises exponentially: I(t) = (V/R)(1 - e^(-Rt/L)).
At steady state (t >> L/R), dI/dt = 0, so the inductor voltage V_L = L(dI/dt) = 0. The inductor behaves as an ideal wire with zero resistance, and I = V/R.
For an infinitely long cylindrical shell carrying current I uniformly distributed between inner radius a and outer radius b:
In the region a < r < b, only the fraction of current enclosed within radius r contributes.
The enclosed current is I_enc = I x (r^2 - a^2)/(b^2 - a^2). This comes from the ratio of the cross-sectional area within r to the total cross-sectional area of the shell.
Applying Ampere's law with a circular loop of radius r: B(2 pi r) = mu_0 I_enc.
Result: B = (mu_0 I / 2 pi r) x (r^2 - a^2)/(b^2 - a^2).
Charge a capacitor to a known voltage V_0, then discharge through the unknown resistor.
Measure the time t for the voltage to drop to some known fraction of V_0.
Since V(t) = V_0 e^(-t/RC), taking the natural log gives t = RC ln(V_0/V(t)).
If V drops to V_0/e (about 37%), then t = RC, so R = t/C.
Example: C = 10 microfarads, V_0 = 100 V, voltage drops to 37 V in 2.0 s. Since 37 V is approximately 100/e, the time constant tau = RC = 2.0 s. R = 2.0 / (10 x 10^-6) = 200,000 ohms = 200 kilohms.
I(t) = \frac{V}{R} e^{-t/RC} \quad \text{(RC charging current)}V_C(t) = V\left(1 - e^{-t/RC}\right) \quad \text{(RC charging voltage)}V(t) = V_0 \, e^{-t/RC} \quad \text{(RC discharge)}\vec{F} = q\vec{v} \times \vec{B} \quad \text{(magnetic force)}\mathcal{E} = BLv \quad \text{(motional EMF)}I(t) = \frac{V}{R}\left(1 - e^{-Rt/L}\right) \quad \text{(RL charging)}\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \quad \text{(Ampere's law)}B = \frac{\mu_0 I}{2\pi r} \cdot \frac{r^2 - a^2}{b^2 - a^2} \quad \text{(cylindrical shell, } a < r < b\text{)}Constants: mu_0 = 4 pi x 10^-7 T m/A.
RC circuits are the basis of camera flash timing, debounce circuits in keyboards, and the smoothing filters in power supplies. Motional EMF is the operating principle of electromagnetic braking in trains and roller coasters. Ampere's law for cylindrical geometries is essential for designing coaxial cables and MRI solenoids.
Students often think the current at t = 0 in an RC charging circuit is zero. It is at its maximum, V/R, because the uncharged capacitor initially behaves like a wire.
Students confuse the direction of the magnetic force with the direction of the magnetic field. The force is perpendicular to both v and B; it is not along B.
Students forget that magnetic force does no work. It changes direction, not speed. If a problem says "the particle speeds up," look for an electric field, not a magnetic one.
Students mix up RL and RC time constants. For RC: tau = RC. For RL: tau = L/R. The inductor and capacitor swap positions in the formula.
⚠️ RC circuit problems at t = 0 test whether you know the capacitor acts like a short circuit (I = V/R) or the inductor acts like an open circuit (I = 0).
⚠️ Magnetic force direction problems require the right-hand rule or explicit cross-product calculation. A wrong sign on the charge flips the answer.
⚠️ Motional EMF is a favourite numerical question: EMF = BLv, plug in the numbers.
⚠️ Ampere's law for partial current enclosure (cylindrical shells) tests whether you can correctly compute the enclosed current fraction.
⚠️ RL steady-state behaviour: at long times, the inductor is a wire, not an open circuit.
True or false: at t = 0 in an RC charging circuit, the current is zero.
False. The current starts at V/R and decays from there.
Fill in the blank: the magnetic force on a moving charge is always perpendicular to ___ and ___.
The velocity v and the magnetic field B.
True or false: at steady state, an inductor in a DC RL circuit acts as an open circuit.
False. It acts as an ideal wire (zero resistance).
Fill in the blank: motional EMF for a bar of length L moving at speed v in field B is ___.
BLv.
Q: An RC circuit has V = 10 V, R = 20 kilohms, C = 4 microfarads. What is the current immediately after the switch is closed?
A: I = V/R = 10 / 20,000 = 0.5 mA.
Q: A positive charge +q moves in the +x direction through a magnetic field in the +y direction. What is the direction of the magnetic force?
A: F = q(v x B) = q(x-hat x y-hat) = q z-hat. The force is in the positive z direction.
Q: A conducting bar of length 0.5 m moves at 10 m/s perpendicular to a 2 T magnetic field. What is the induced EMF?
A: EMF = BLv = (2)(0.5)(10) = 10 V.
Q: In a DC RL circuit that has been closed for a very long time, what does the inductor do?
A: It acts as an ideal wire with zero resistance. The steady-state current is V/R.
Q: A cylindrical shell carries current I between radii a and b. Derive B at radius r (a < r < b).
A: Enclosed current = I(r^2 - a^2)/(b^2 - a^2). Ampere's law gives B = mu_0 I(r^2 - a^2) / [2 pi r (b^2 - a^2)].
Q: A 10 microfarad capacitor is charged to 100 V and discharged through an unknown R. The voltage drops to 37 V in 2.0 s. What is R?
A: 37 V is approximately 100/e, so t = tau = RC. R = t/C = 2.0 / (10 x 10^-6) = 200 kilohms.
RC circuits feed directly into RLC oscillations and AC circuit analysis (Part 3 notes). The magnetic force is the foundation for the Hall effect and for understanding how current-carrying wires interact. Faraday's law and motional EMF lead to transformers and the full set of Maxwell's equations, which unify electricity and magnetism.
RC circuit, time constant, tau, exponential decay, capacitor charging, capacitor discharging, magnetic force, Lorentz force, cross product, right-hand rule, motional EMF, Faraday's law, electromagnetic induction, RL circuit, inductor, self-inductance, Ampere's law, cylindrical shell, enclosed current, resistance measurement, PHYS 212, electricity and magnetism, E&M final review