Magnetic Fields and Induction — 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: Biot-Savart law, Ampère's law, Lorentz force on charges and current-carrying wires, Faraday's law of induced EMF, self-inductance, and LC circuit analysis as required by the College Board AP Physics C E&M course and exam description.

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 Magnetic Fields and Induction?

Magnetic fields and electromagnetic induction describe the interactions between moving charges, and the generation of electric fields from time-varying magnetic fields, forming two of the four core pillars of classical electromagnetism defined by Maxwell’s equations. Magnetic field $\vec{B}$ is measured in tesla (T, equivalent to $N/(A\cdot m)$), with related quantities including magnetic flux (in webers, Wb) and inductance (in henries, H). This topic makes up ~40% of the AP Physics C E&M exam, with regular testing in both multiple-choice and free-response sections, including experimental design questions.

2. Biot-Savart Law

The Biot-Savart law is the foundational equation for calculating the magnetic field produced by any steady current-carrying wire, analogous to Coulomb’s law for electric fields but applicable to moving charge distributions. It was derived experimentally in the 19th century by Jean-Baptiste Biot and Félix Savart. The differential form of the law is: $$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$$ Where ${\mu }_0=4\pi \times 10^{-7}T\cdot m/A$ is the permeability of free space, $I$ is the current in the wire, $d\vec{l}$ is an infinitesimal segment of the wire pointing in the direction of current flow, $r$ is the distance from $d\vec{l}$ to the point where you are calculating $\vec{B}$, and $\hat{r}$ is the unit vector pointing from $d\vec{l}$ to that point. The cross product means the direction of $d\vec{B}$ is given by the right-hand rule: point your right index finger along $d\vec{l}$, curl toward $\hat{r}$, and your thumb points in the direction of $d\vec{B}$.

Worked Example

Calculate the magnetic field at the center of a circular loop of wire with radius 0.1 m carrying a 2 A counterclockwise current. For every segment $d\vec{l}$ on the loop, the cross product $d\vec{l}\times \hat{r}$ has magnitude $dl\cdot 1\cdot \sin (90^{\circ })=dl$, and direction out of the page. Integrate around the full loop: $$B=\int dB=\frac{{\mu }_{0}I}{4\pi r^2}{\int }_0^{2\pi r}dl=\frac{{\mu }_{0}I}{4\pi r^2}\cdot 2\pi r=\frac{{\mu }_{0}I}{2r}$$ Substitute values: $$B=\frac{4\pi \times 10^{-7}\cdot 2}{2\cdot 0.1}=4\pi \times 10^{-6}\approx 1.26\times 10^{-5}T$$ The field points out of the page, consistent with the right-hand rule for counterclockwise current. Exam takers are expected to memorize this standard result for circular loops, as well as the result for infinite straight wires ($B={\mu }_{0}I/(2\pi r)$) derived from the Biot-Savart law.

3. Ampère's Law

Ampère’s law is a symmetry-based shortcut for calculating magnetic fields in magnetostatic systems with high geometric symmetry, analogous to Gauss’s law for electric fields. It avoids the complex integration required for the Biot-Savart law when symmetry exists, and is one of Maxwell’s four core equations for electromagnetism. The integral form of Ampère’s law is: $$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{enclosed}$$ The left-hand side is a closed line integral of $\vec{B}$ around an Amperian loop of your choice, and $I_{enclosed}$ is the total net current passing through the area bounded by the loop. Use the right-hand rule to assign sign to enclosed current: curl the fingers of your right hand along the direction of the Amperian loop, and current pointing in the direction of your thumb counts as positive, opposite counts as negative.

Worked Example

Find the magnetic field inside an infinite solenoid with 1000 turns per meter carrying a 3 A current. Choose a rectangular Amperian loop: one side of length $L$ runs parallel to the axis of the solenoid inside the coil, a second side runs parallel to the axis outside the solenoid where $B=0$, and the remaining two sides run perpendicular to the solenoid axis so the dot product $\vec{B}\cdot d\vec{l}=0$ along those segments. The left-hand integral simplifies to $B\cdot L$. The enclosed current is $I_{enclosed}=nLI$, where $n$ is turns per unit length. $$BL={\mu }_{0}nLI⟹B={\mu }_{0}nI$$ Substitute values: $B=4\pi \times 10^{-7}\cdot 1000\cdot 3\approx 3.77\times 10^{-3}T$, uniform across the inside of the solenoid. A common exam trap is attempting to use Ampère’s law for asymmetric geometries like finite-length wires: only use it for infinite wires, solenoids, toroids, and infinite current sheets.

4. Lorentz Force on Charges and Wires

Magnetic fields exert forces on moving charged particles, and by extension on current-carrying wires (which are collections of moving charged particles). This force is called the Lorentz magnetic force, and it is always perpendicular to both the velocity of the charge and the direction of the magnetic field, meaning magnetic force does no work on charged particles and cannot change their speed, only their direction of motion. For a point charge $q$ moving with velocity $\vec{v}$: $${\vec{F}}_B=q\vec{v}\times \vec{B}$$ For a straight current-carrying wire of length $L$: $${\vec{F}}_B=I\vec{L}\times \vec{B}$$ Where $\vec{L}$ points in the direction of current flow. For positive charges, the direction of the force is given by the right-hand rule: point your index finger along $\vec{v}$ (or $\vec{L}$), middle finger along $\vec{B}$, and your thumb points in the direction of the force. Reverse the direction for negative charges like electrons.

Worked Example

A proton ($q=1.6\times 10^{-19}C$, $m=1.67\times 10^{-27}kg$) moves at $3\times 10^{5}m/s$ perpendicular to a uniform 0.5 T magnetic field. Calculate the radius of its circular path. The magnetic force provides the centripetal force required for circular motion: $$qvB=\frac{m{v}^2}{r}$$ Rearrange to solve for $r$: $$r=\frac{mv}{qB}=\frac{1.67\times 10^{-27}\cdot 3\times 10^5}{1.6\times 10^{-19}\cdot 0.5}\approx 6.3\times 10^{-3}m=6.3mm$$ Examiners regularly test this circular motion scenario, so remember that the period of circular motion is independent of particle speed for non-relativistic velocities.

5. Faraday's Law and Induced EMF

Faraday’s law of induction describes how a time-varying magnetic flux through a closed loop induces an electromotive force (EMF) that drives current in the loop, the foundational principle behind generators, transformers, and inductors. First, define magnetic flux through a surface: $${\Phi }_B=\int \vec{B}\cdot d\vec{A}$$ Where $d\vec{A}$ is the normal vector to the infinitesimal area segment $dA$, and flux is measured in webers ($1Wb=1T\cdot m^2$). Faraday’s law states: $$\epsilon =-\frac{d{\Phi }_B}{dt}$$ The negative sign encodes Lenz’s law: the induced EMF drives a current that produces its own magnetic field opposing the change in flux that caused the EMF. For a coil with $N$ turns, multiply the flux by $N$ to get total flux linkage.

Worked Example

A 50-turn square coil of side length 0.2 m is placed with its plane perpendicular to a magnetic field that increases at a rate of 0.1 T/s. Calculate the magnitude of the induced EMF. Total flux linkage: ${\Phi }_{total}=NBA=50\cdot B\cdot (0.2{)}^2=2B$. The induced EMF is the magnitude of the derivative: $$\epsilon =\left|\frac{d{\Phi }_{total}}{dt}\right|=2\cdot \frac{dB}{dt}=2\cdot 0.1=0.2V$$ Common question types include moving conducting rods in magnetic fields, rotating generator loops, and changing magnetic fields: all use the same flux derivative framework, so always start by writing an expression for total flux before differentiating.

6. Self-Inductance and LC Circuits

When current changes in a coil, the resulting change in magnetic flux through the coil itself induces an opposing EMF, a property called self-inductance $L$ (measured in henries, $1H=1V\cdot s/A$). The induced EMF across an inductor is: $${\epsilon }_L=-L\frac{dI}{dt}$$ The self-inductance of a solenoid is given by $L={\mu }_0{n}^{2}Al$, where $n$ is turns per unit length, $A$ is cross-sectional area, and $l$ is solenoid length. An LC circuit consists of an ideal inductor and capacitor connected in a loop with no resistance. Energy oscillates between the electric field of the capacitor ($U_C=q^2/(2C)$) and the magnetic field of the inductor ($U_L=\frac{1}{2}L{I}^2$), with total energy conserved. The angular frequency of oscillation is: $$\omega =\frac{1}{\sqrt{LC}}$$ Where linear frequency $f=\omega /(2\pi )$ in Hz.

Worked Example

An LC circuit has $L=2mH$ and $C=5\mu F$. Calculate the oscillation frequency in Hz. First convert to SI units: $L=2\times 10^{-3}H$, $C=5\times 10^{-6}F$. $$\omega =\frac{1}{\sqrt{2\times 10^{-3}\cdot 5\times 10^{-6}}}=\frac{1}{\sqrt{1\times 10^{-8}}}=10^{4}rad/s$$ $$f=\frac{\omega }{2\pi }\approx 1590Hz$$ Free-response questions regularly test energy conservation in LC circuits, so remember that maximum current occurs when the capacitor is fully discharged, and maximum charge occurs when current is zero.

7. Common Pitfalls (and how to avoid them)

• Wrong move: Using the left-hand rule instead of the right-hand rule for positive charges, mixing up direction of cross products for Biot-Savart, Lorentz force, and Lenz’s law. Why it happens: Confusion with left-hand rules taught in some secondary curricula, or mixing up current and electron flow directions. Correct move: Always use the right-hand rule for positive charges, reverse force direction for negative charges, and verify direction with Lenz’s law for induction problems.
• Wrong move: Applying Ampère’s law to asymmetric geometries like finite-length wires or non-uniform solenoids. Why it happens: Students overgeneralize the utility of the law, assuming it works for all current distributions like Gauss’s law. Correct move: Only use Ampère’s law for infinite wires, infinite solenoids, toroids, and infinite current sheets; use Biot-Savart for all other cases.
• Wrong move: Forgetting to multiply magnetic flux by the number of turns $N$ for coils in Faraday’s law calculations. Why it happens: Students only calculate flux for a single loop, ignoring flux linkage. Correct move: Always write total flux as $N{\Phi }_B$ for multi-turn coils before taking the derivative for EMF.
• Wrong move: Claiming magnetic force does work on charged particles, or using magnetic force to calculate changes in particle speed. Why it happens: Students confuse magnetic force with electric force, which can change particle speed. Correct move: Magnetic force is always perpendicular to velocity, so work done by magnetic force is always zero; only electric fields or induced electric fields from changing magnetic fields do work on charges.
• Wrong move: Mixing up angular frequency $\omega$ and linear frequency $f$ in LC circuit calculations, leading to off-by-$2\pi$ errors. Why it happens: The standard formula gives $\omega$ in rad/s, but questions often ask for frequency in Hz. Correct move: Always convert between $\omega$ and $f$ using $f=\omega /(2\pi )$, and check units in your final answer to catch mistakes.

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

Question 1

A long straight wire carries a 10 A current upward. A second 0.5 m long wire carrying a 2 A upward current is placed 0.1 m to the right of the first wire. (a) Calculate the magnitude and direction of the magnetic field from the first wire at the location of the second wire. (b) Calculate the magnitude and direction of the force on the second wire.

Solution

(a) For an infinite straight wire, $B=\frac{{\mu }_{0}I}{2\pi r}=\frac{4\pi \times 10^{-7}\cdot 10}{2\pi \cdot 0.1}=2\times 10^{-5}T$. Using the right-hand rule, the field points into the page at a point right of the upward-current wire. (b) Force on the second wire: $F=ILB\sin (90^{\circ })=2\cdot 0.5\cdot 2\times 10^{-5}=2\times 10^{-5}N$. Using the right-hand rule for cross product of upward current and into-page B field, the force points left, attractive between parallel currents.

Question 2

A 100-turn circular loop of radius 0.2 m with resistance 10 Ω is placed perpendicular to a magnetic field that varies as $B(t)=0.2t^{2}T$, where $t$ is in seconds. (a) Find the induced EMF at $t=3s$. (b) Find the induced current at $t=3s$.

Solution

(a) Total flux linkage: ${\Phi }_{total}=NBA=100\cdot 0.2t^2\cdot \pi (0.2{)}^2=0.8\pi t^2$. EMF magnitude: $\epsilon =\left|\frac{d\Phi }{dt}\right|=1.6\pi t$. At $t=3$, $\epsilon =4.8\pi \approx 15.1V$. (b) Induced current: $I=\frac{\epsilon }{R}=\frac{15.1}{10}\approx 1.51A$.

Question 3

An ideal LC circuit has $L=10mH$, $C=10\mu F$, and an initial charge of $1\times 10^{-4}C$ on the capacitor. (a) Find the maximum current in the circuit. (b) Find the time taken for the capacitor to fully discharge for the first time.

Solution

(a) Total initial energy: $U=\frac{Q^2}{2C}=\frac{(1\times 10^{-4}{)}^2}{2\cdot 1\times 10^{-5}}=5\times 10^{-4}J$. At maximum current, all energy is stored in the inductor: $U=\frac{1}{2}L{I}_{max}^2⟹I_{max}=\sqrt{\frac{2U}{L}}=\sqrt{\frac{2\cdot 5\times 10^{-4}}{1\times 10^{-2}}}\approx 0.32A$. (b) Period of oscillation: $T=2\pi \sqrt{LC}=2\pi \sqrt{1\times 10^{-2}\cdot 1\times 10^{-5}}\approx 2\times 10^{-3}s$. First full discharge occurs at $T/4$, so $t=5\times 10^{-4}s=0.5ms$.

9. Quick Reference Cheatsheet

10. What's Next

Magnetic fields and induction form the core of classical electromagnetism, and directly connect to remaining AP Physics C E&M syllabus content including Maxwell’s full set of equations, electromagnetic waves, and AC circuit analysis. A strong grasp of induction is also required for the experimental free-response question, which regularly tests Faraday’s law via hands-on measurements of induced EMF, inductance, or LC circuit oscillation periods. You will also encounter these principles in later physics and engineering courses covering electromechanical systems, wireless communications, and power electronics. To solidify your mastery, work through official College Board past papers to familiarize yourself with common question structures and grading rubrics, and practice applying right-hand rules and symmetry arguments as often as possible. If you get stuck on any concept or practice problem, you can ask Ollie for step-by-step explanations at any time by visiting the homepage, where you can also access more study guides, flashcards, and full practice tests for AP Physics C E&M.