Unit 4 Magnetic Fields: Flux, Induction, and Inductors (AP Physics C: E&M)

Magnetic flux ($\Phi_B$)

A scalar measure of how strongly a magnetic field threads a chosen surface (depends on surface area and orientation); not the magnetic field itself or a force.

Weber (Wb)

Unit of magnetic flux; 1 Wb = 1 T·m².

Flux through a flat surface in a uniform field

$\Phi_B = BA \, \cos \theta$, where $\theta$ is the angle between the magnetic field direction and the surface’s area (normal) vector.

Area vector (A⃗)

A vector perpendicular to a surface with magnitude equal to the surface area; its chosen direction sets the sign convention for flux.

Angle $\theta$ in $\Phi_B = BA \, \cos \theta$

The angle between $\text{B}$ and the area vector (surface normal), not the angle between $\text{B}$ and the plane of the loop.

Flux as a dot product

$\Phi_B = \text{B} \bullet \text{A}$, which automatically accounts for orientation and sign.

Positive vs. negative magnetic flux

Flux is positive if $\text{B}$ points generally along $\text{A}$, and negative if $\text{B}$ points opposite $\text{A}$.

Magnetic flux integral form

$\Phi_B = \int \text{B} \bullet d\text{A}$, used for non-uniform fields and/or curved surfaces by summing tiny area contributions.

Uniform-field reduction of the flux integral

When $\text{B}$ is uniform and the surface is flat, $\Phi_B = \int \text{B} \bullet d\text{A}$ reduces to $\Phi_B = BA \, \cos \theta$.

Flux linkage

For a coil with $N$ identical turns experiencing the same flux per turn, total linked flux is $N\Phi_B$.

Faraday’s law of induction (single loop)

Induced emf is $\epsilon = -\frac{d\Phi_B}{dt}$; a changing magnetic flux produces an emf around the loop.

Faraday’s law for $N$ turns

For a coil, $\epsilon = -N \frac{d\Phi_B}{dt}$; induced emf scales with the number of turns.

Lenz’s law

The induced current produces a magnetic field that opposes the change in magnetic flux (the meaning of the minus sign in Faraday’s law).

Common Lenz’s law misconception

The induced field does not always oppose the external field; it opposes the change in flux (e.g., if external flux decreases, induced field may point the same way to maintain it).

Ways to change magnetic flux

Change $B$, change the loop area $A$ (including moving into/out of a field region), or change $\theta$ by rotating the loop.

Rate of change of flux $\frac{d\Phi_B}{dt}$

The quantity that determines induced emf magnitude; a large flux alone is not enough—flux must be changing.

Inductor

A circuit element (often a coil) designed so current produces linked magnetic flux; it resists changes in current because changing current changes flux and induces an opposing emf.

Self-inductance

When a changing current in a circuit induces an emf in the same circuit due to its own changing magnetic flux.

Inductance (L) definition via flux linkage

Defined by $N\Phi_B = LI$ (for linear/typical AP situations): linked flux is proportional to the current producing it.

Henry (H)

Unit of inductance; 1 H = 1 V·s/A.

Weber (Wb)

Unit of magnetic flux; $1 \text{ Wb} = 1 \text{ T} \bullet \text{m}^2$.

Inductance of an ideal long solenoid

$L = \frac{\mu_0 N^2 A}{\text{ℓ}}$, where $N$ is turns, $A$ is cross-sectional area, and $\text{ℓ}$ is solenoid length.

Energy stored in an inductor

Unit of inductance; $1 H = 1 \frac{V \bullet s}{A}$.

Mutual inductance ($M$)

Coupling between two coils where changing current in one induces emf in the other; defined by $N_2\Phi_{B2} = M I_1$ and gives $\epsilon_2 = -M \frac{dI_1}{dt}$.

RL circuit time constant ($\tau$)

For a series RL circuit, $\tau = \frac{L}{R}$; sets the exponential timescale for current growth/decay.