Physics C: E&M · Unit 4: Magnetic Fields and Electromagnetic Induction · 16 min read · Updated 2026-05-11
Magnetic Fields and Induction — AP Physics C: Electricity and Magnetism
AP Physics C: Electricity and Magnetism · Unit 4: Magnetic Fields and Electromagnetic Induction · 16 min read
1. Biot-Savart Law★★☆☆☆⏱ 3 min
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.
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 $d\vec{B}$.
2. Ampère's Law★★☆☆☆⏱ 3 min
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 complex Biot-Savart integration when symmetry exists, and is one of Maxwell’s four core electromagnetism equations.
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}
3. Lorentz Force on Charges and Wires★★☆☆☆⏱ 3 min
Magnetic fields exert forces on moving charged particles, and by extension on current-carrying wires (collections of moving charges). The magnetic Lorentz force is always perpendicular to both the velocity of the charge and the magnetic field, so magnetic force does no work and can only change direction of motion, not speed.
\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}
4. Faraday's Law of Induced EMF★★★☆☆⏱ 3 min
Faraday’s law of induction describes how a time-varying magnetic flux through a closed loop induces an electromotive force (EMF) that drives current, the foundational principle behind generators and transformers.
\Phi_B = \int \vec{B} \cdot d\vec{A}
Faraday's law states that induced EMF equals the negative rate of change of magnetic flux. The negative sign encodes Lenz's law: the induced current produces a magnetic field opposing the change in flux that caused the EMF. For a coil with $N$ turns, total flux linkage is $N \Phi_B$.
\varepsilon = - \frac{d\Phi_B}{dt}
5. Self-Inductance and LC Circuits★★★☆☆⏱ 4 min
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 (H, $1 H = 1 V \cdot s / A$).
\varepsilon_L = - L \frac{dI}{dt}
The self-inductance of a solenoid is given by $L = \mu_0 n^2 A l$, where $n$ is turns per unit length, $A$ is cross-sectional area, and $l$ is solenoid length. An ideal LC circuit consists of an 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.
\omega = \frac{1}{\sqrt{LC}}
Where $\omega$ is angular frequency in rad/s, and linear frequency $f = \omega/(2\pi)$ in Hz.
Common Pitfalls
Why: Confusion with left-hand rules from other curricula, or mixing up conventional current and electron flow directions.
Why: Overgeneralizing the utility of Ampère's law, assuming it works for all current distributions like Gauss's law.
Why: Only calculating flux for a single loop, ignoring total flux linkage for the entire coil.
Why: Confusing magnetic force with electric force, which can change particle speed.
Why: The standard formula gives $\omega$ in rad/s, but exam questions often ask for frequency in Hz.