Electric Circuits — AP Physics C: E&M Phys C E&M Study Guide

For: AP Physics C: E&M candidates sitting AP Physics C: Electricity & Magnetism.

Covers: EMF, current, resistance and Ohm's Law, Kirchhoff's circuit rules, exponential charging/discharging of RC circuits, RL circuit transient behavior, and power/energy transfer in electric circuits.

You should already know: Calculus (especially integration), Phys C Mechanics or equivalent.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Physics C: E&M style for educational use. They are not reproductions of past College Board papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Is Electric Circuits?

Electric circuits are closed, conductive paths that facilitate the controlled flow of charge to perform work, transfer energy, or process electrical signals. They form the foundational unit of all modern electronics, power systems, and experimental E&M apparatus, and make up ~20% of the AP Physics C: E&M exam weight per the 2024 College Board CED. Standard schematic symbols are used to represent components like sources, resistors, capacitors, and inductors; complex multi-loop configurations are often referred to as electrical networks.

2. EMF, current, resistance — Ohm's law

All circuits rely on three core quantities, linked by the empirical Ohm's Law for ohmic materials:

1. Electromotive Force (EMF, $\epsilon$): The energy per unit charge supplied by a power source (e.g., battery, generator) to move charge around the circuit, measured in Volts (V). Unlike ideal sources, real sources have internal resistance $r$, so the usable terminal voltage is $V_{terminal}=\epsilon -Ir$, where $I$ is the current drawn from the source.
2. Current ($I$): The rate of net charge flow past a point in the circuit, measured in Amperes (A): $I=\frac{dQ}{dt}$. By convention, current is described as the flow of positive charge from high to low potential, opposite the actual direction of electron flow.
3. Resistance ($R$): The opposition to charge flow in a material, measured in Ohms (Ω). For a uniform conductor, $R=\rho \frac{L}{A}$, where $\rho$ is the material's resistivity, $L$ is the conductor length, and $A$ is its cross-sectional area.

Ohm's Law states that for ohmic materials (materials with constant resistance independent of applied voltage), the potential difference across a resistor is proportional to the current flowing through it: $$V=IR$$

Worked Example

A 12V car battery has an internal resistance of 0.05Ω, connected to a 2Ω headlight. Calculate the current through the headlight and the terminal voltage of the battery.

1. Total circuit resistance: $R_{total}=2\Omega +0.05\Omega =2.05\Omega$
2. Current: $I=\frac{\epsilon }{R_{total}}=\frac{12}{2.05}\approx 5.85A$
3. Terminal voltage: $V_{terminal}=12-(5.85\ast 0.05)\approx 11.71V$

Examiner note: Never assume a voltage source is ideal unless explicitly stated; internal resistance is a common hidden test point for this topic.

3. Kirchhoff's rules

Kirchhoff's two rules, derived from fundamental conservation laws, allow you to solve for unknown currents and voltages in any multi-loop circuit:

1. Junction Rule (KCL, Conservation of Charge): The sum of currents entering a circuit junction equals the sum of currents leaving the junction: $$\sum I_{in}=\sum I_{out}$$
2. Loop Rule (KVL, Conservation of Energy): The sum of potential differences across all components around any closed circuit loop equals zero: $$\sum \Delta V=0$$

Application Tips

• Assign arbitrary current directions for each branch; if your final current value is negative, it flows opposite your assigned direction.
• For KVL sign conventions: When moving across a resistor in the direction of current, $\Delta V=-IR$; moving opposite current, $\Delta V=+IR$. When moving from the negative to positive terminal of a battery, $\Delta V=+\epsilon$; moving from positive to negative, $\Delta V=-\epsilon$.

Worked Example

A two-loop circuit has a 10V battery in the left loop, a 5V battery opposing the 10V source in the right loop, a 2Ω resistor in the left loop, 3Ω resistor in the right loop, and 1Ω resistor in the shared middle branch. Find the current through the 1Ω resistor.

1. Assign $I_1$ (clockwise, left loop), $I_2$ (clockwise, right loop), $I_3$ (downwards, shared branch). KCL: $I_1=I_2+I_3$, so $I_3=I_1-I_2$.
2. KVL left loop: $10-2I_1-1I_3=0\to 10-3I_1+I_2=0$
3. KVL right loop: $-5+1I_3-3I_2=0\to -5+I_1-4I_2=0$
4. Solve simultaneous equations: $I_1\approx 3.18A$, $I_2\approx -0.45A$, $I_3\approx 3.63A$. The negative $I_2$ means the right loop current flows counterclockwise.

4. RC circuits — exponential charging

RC circuits combine resistors and capacitors, with time-dependent charge and current behavior when connected to or disconnected from a voltage source. For a charging series RC circuit (uncharged capacitor at $t=0$ connected to EMF $\epsilon$), KVL gives the differential equation: $$\epsilon -IR-\frac{Q}{C}=0$$ Substitute $I=\frac{dQ}{dt}$ and integrate from $Q=0$ at $t=0$ to get the exponential charge relation: $$Q(t)=Q_{max}\left(1-e^{-t/\tau }\right)$$ where $Q_{max}=C\epsilon$ is the maximum charge on the capacitor, and $\tau =RC$ is the time constant (time to reach ~63% of maximum charge, units of seconds). The current in the circuit decays exponentially: $$I(t)=I_0{e}^{-t/\tau },I_0=\frac{\epsilon }{R}$$ For a discharging RC circuit (source removed, capacitor connected directly across resistor), the charge relation becomes $Q(t)=Q_0{e}^{-t/\tau }$.

Worked Example

A 10μF capacitor, 100kΩ resistor, and 12V battery are connected in series at $t=0$. Find the charge on the capacitor after 2 seconds, and the time to reach 90% of maximum charge.

1. Time constant: $\tau =10\ast 10^{-6}F\ast 100\ast 10^3\Omega =1s$
2. Maximum charge: $Q_{max}=10\ast 10^{-6}F\ast 12V=1.2\ast 10^{-4}C$
3. Charge at t=2s: $Q(2)=1.2\ast 10^{-4}\ast (1-e^{-2})\approx 1.04\ast 10^{-4}C$
4. Time to 90% charge: $0.9Q_{max}=Q_{max}(1-e^{-t/1})\to t=\ln (10)\approx 2.30s$

Exam tip: You will often be asked to derive these exponential relations from first principles using calculus, so practice the integration step regularly.

5. RL circuits

RL circuits combine resistors and inductors, where inductors oppose changes in current via a back EMF ${\epsilon }_L=-L\frac{dI}{dt}$, with $L$ the inductance measured in Henries (H). For a series RL circuit connected to EMF $\epsilon$ at $t=0$, KVL gives: $$\epsilon -IR-L\frac{dI}{dt}=0$$ Integrate from $I=0$ at $t=0$ to get the current relation: $$I(t)=I_{max}\left(1-e^{-t/\tau }\right)$$ where $I_{max}=\frac{\epsilon }{R}$ is the maximum steady-state current, and the time constant is $\tau =\frac{L}{R}$. When the source is removed and the inductor is connected across a resistor, current decays as $I(t)=I_0{e}^{-t/\tau }$.

Key transient behavior rules for DC circuits:

• At $t=0$: Uncharged capacitors act as short circuits (zero resistance), inductors act as open circuits (infinite resistance, zero current)
• At $t\to \infty$ (steady state): Capacitors act as open circuits (zero current), inductors act as short circuits (zero voltage drop)

Worked Example

A 5H inductor, 10Ω resistor, and 20V battery are connected in series at $t=0$. Find the current after 1 second.

1. Time constant: $\tau =\frac{5H}{10\Omega }=0.5s$
2. Maximum current: $I_{max}=\frac{20V}{10\Omega }=2A$
3. Current at t=1s: $I(1)=2\ast (1-e^{-1/0.5})\approx 1.73A$

6. Power and energy in circuits

Power is the rate of energy transfer in a circuit, measured in Watts (W). For any circuit component with potential difference $V$ across it and current $I$ through it, the power transferred is: $$P=IV$$ For ohmic resistors, substitute Ohm's Law to get alternative forms for power dissipated as heat (Joule heating): $$P=I^{2}R=\frac{V^2}{R}$$ Total energy dissipated by a resistor over time $t$ is $E=Pt=I^{2}Rt$.

Energy is also stored in the electric field of capacitors and magnetic field of inductors:

• Capacitor stored energy: $U_C=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}$
• Inductor stored energy: $U_L=\frac{1}{2}L{I}^2$

Worked Example

A 100Ω resistor carries 0.5A of current for 10 minutes. Calculate the total energy dissipated as heat and the average power.

1. Average power: $P=(0.5A{)}^2\ast 100\Omega =25W$
2. Total time: $t=10\ast 60=600s$
3. Total energy: $E=25W\ast 600s=15,000J=15kJ$

7. Common Pitfalls (and how to avoid them)

• Pitfall 1: Ignoring internal resistance of batteries when calculating terminal voltage. Why you do it: You default to assuming ideal sources unless explicitly told otherwise. Correct move: Always scan the prompt for internal resistance values first, and use $V_{terminal}=\epsilon -Ir$ instead of assuming $V=\epsilon$.
• Pitfall 2: Mixing up KVL sign conventions leading to incorrect simultaneous equations. Why you do it: You skip writing down explicit sign rules before solving multi-loop problems. Correct move: List your sign rules clearly before setting up KVL equations, and cross-check your result by verifying total power supplied equals total power dissipated.
• Pitfall 3: Using linear relations instead of exponential curves for intermediate time points in RC/RL circuits. Why you do it: You confuse steady-state values ($t=0$ or $t\to \infty$) with transient time points. Correct move: Only use steady-state approximations for $t=0$ and $t\to \infty$; use the exponential time relations for all other time values.
• Pitfall 4: Swapping capacitor and inductor transient behavior. Why you do it: You memorize rules without linking them to physical behavior. Correct move: Remember capacitors store charge so resist changes in voltage, inductors store magnetic flux so resist changes in current.
• Pitfall 5: Using $P=V^2/R$ for non-ohmic components. Why you do it: You assume all components follow Ohm's Law. Correct move: Only use $I^{2}R$ or $V^2/R$ for ohmic resistors; use the general $P=IV$ for all other components.

8. Practice Questions (AP Physics C: E&M Style)

Question 1 (10 points)

A 12V ideal battery is connected in series with a 20kΩ resistor and an uncharged 5μF capacitor at $t=0$. (a) Calculate the time constant of the circuit. (2 points) (b) Find the current through the resistor at $t=0$. (2 points) (c) Find the charge on the capacitor at $t=0.5s$. (3 points) (d) How long does it take for the capacitor to reach 99% of its maximum charge? (3 points)

Solution

(a) $\tau =RC=20\ast 10^3\Omega \ast 5\ast 10^{-6}F=0.1s$ (b) At $t=0$, the uncharged capacitor acts as a short circuit: $I_0=\frac{\epsilon }{R}=\frac{12}{20\ast 10^3}=6\ast 10^{-4}A=0.6mA$ (c) $Q_{max}=C\epsilon =5\ast 10^{-6}\ast 12=6\ast 10^{-5}C$. $t=0.5s=5\tau$: $Q(0.5)=6\ast 10^{-5}\ast (1-e^{-5})\approx 5.96\ast 10^{-5}C$ (d) $0.99Q_{max}=Q_{max}(1-e^{-t/0.1})\to t=\frac{\ln (100)}{10}\approx 0.46s$

Question 2 (15 points)

A series RL circuit has $L=2H$, $R=4\Omega$, connected to a 24V ideal DC source at $t=0$. (a) Calculate the time constant of the circuit. (3 points) (b) Find the maximum steady-state current in the circuit. (3 points) (c) Find the back EMF of the inductor at $t=0$. (4 points) (d) Calculate the total energy stored in the inductor at steady state. (5 points)

Solution

(a) $\tau =\frac{L}{R}=\frac{2}{4}=0.5s$ (b) At steady state, the inductor acts as a short circuit: $I_{max}=\frac{24}{4}=6A$ (c) At $t=0$, current is zero so voltage drop across the resistor is zero: all source voltage is across the inductor, so ${\epsilon }_L=24V$ (opposing the source) (d) $U_L=\frac{1}{2}L{I}_{max}^2=0.5\ast 2\ast 6^2=36J$

9. Quick Reference Cheatsheet

10. What's Next

Mastery of electric circuits is a prerequisite for the remaining AP Physics C: E&M topics of magnetism, electromagnetic induction, and AC circuits, where circuit principles are extended to time-varying magnetic fields and alternating current systems. Many high-weighted exam questions combine circuit analysis with Faraday's Law to solve for induced currents in moving conductive loops, so a strong grasp of Kirchhoff's Rules and transient circuit behavior will directly improve your performance on these complex multi-part problems.

If you are stuck on any part of circuit analysis, from solving multi-loop Kirchhoff problems to deriving exponential RC/RL relations from first principles, you can ask Ollie for step-by-step help, custom practice questions, or clarification of any concept at any time on the homepage. Be sure to also practice with official College Board past papers to familiarize yourself with the exact formatting and mark scheme conventions for AP Physics C: E&M circuit questions.