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

Magnetic Flux

What magnetic flux is (and what it is not)

Magnetic flux is a measure of “how much magnetic field passes through” a surface. More precisely, it measures how strongly a magnetic field threads an area with a particular orientation. You can think of it like counting how many “field lines” pierce a loop, but the real (math) definition is what matters for AP Physics C.

Flux is not the magnetic field itself, and it’s not a force. It’s a scalar quantity associated with a chosen surface. If you change the surface (its size, its tilt, or even which surface you imagine spanning a loop), you can change the flux.

Flux matters because changing magnetic flux is the direct cause of induced emf (electromotive force) in Faraday’s law. This is the central link between magnetism and circuits: changing magnetic environments create electric effects.

Definition of flux through a flat surface in a uniform field

For a flat surface of area $A$ in a uniform magnetic field of magnitude $B$, the magnetic flux is

$\Phi_B = BA\cos\theta$

Here:

• $\Phi_B$ is magnetic flux, measured in webers (Wb), where $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$.
• $B$ is the magnitude of the magnetic field (tesla).
• $A$ is the area of the surface (square meters).
• $\theta$ is the angle between the magnetic field direction and the area vector (a vector perpendicular to the surface).

A common conceptual trap is using the wrong angle. The formula uses the angle between $\vec B$ and the surface’s normal vector, not the angle between $\vec B$ and the plane of the loop.

Area vector and sign conventions

To talk about flux for a loop (especially when using Faraday’s law), you need a consistent sign.

For a loop, you choose a direction for the area vector $\vec A$ (perpendicular to the surface). For a flat loop, you typically pick one of the two perpendicular directions and stick with it.

Then flux can be written as a dot product:

$\Phi_B = \vec B\cdot\vec A$

If $\vec B$ points in the same general direction as $\vec A$, flux is positive. If $\vec B$ points opposite $\vec A$, flux is negative.

This sign is not “extra detail” — it’s essential for getting Lenz’s law direction questions correct. Many mistakes happen because students compute a positive magnitude $BA\cos\theta$ and forget that the flux could be decreasing or increasing depending on sign.

Flux for non-uniform fields or curved surfaces (the integral idea)

If the field is not uniform, or the surface is curved, you break the surface into tiny pieces and add up the flux contributions. The calculus definition is

$\Phi_B = \int \vec B\cdot d\vec A$

Here $d\vec A$ is a tiny area vector perpendicular to the surface at each point.

On the AP exam, you’re often not asked to evaluate complicated surface integrals, but you are expected to understand what the integral means and how it reduces to $BA\cos\theta$ when $\vec B$ is uniform and the surface is flat.

Flux through coils: multiple turns

If you have a coil with $N$ identical turns, all experiencing the same flux per turn, the **flux linkage** is $N\Phi_B$. Faraday’s law uses this linked flux.

A common error is forgetting the factor of $N$. If a coil has 200 turns, the induced emf can be 200 times larger than for a single loop.

Why changing flux matters: Faraday’s law of induction

Faraday’s law tells you that a changing magnetic flux through a circuit induces an emf around the circuit:

$\varepsilon = -\frac{d\Phi_B}{dt}$

For a coil with $N$ turns:

$\varepsilon = -N\frac{d\Phi_B}{dt}$

Where:

• $\varepsilon$ is the induced emf (volts).
• $\Phi_B$ is the magnetic flux through one turn.
• The minus sign encodes Lenz’s law (direction opposes the change).

This is the big idea: you do not need a battery to get an emf. Nature “creates” an emf whenever the magnetic flux through a loop changes.

Lenz’s law: interpreting the minus sign (direction reasoning)

Lenz’s law says the induced current creates a magnetic field that opposes the change in flux, not necessarily the flux itself.

A reliable way to reason:

1. Decide which direction you chose for $\vec A$ (this sets what “positive flux” means).
2. Determine whether $\Phi_B$ is increasing or decreasing (including sign).
3. The induced current must create a magnetic field through the loop that opposes that change.
4. Use the right-hand rule to convert “which way must the induced $\vec B$ point?” into “which way does the induced current flow?”

Common misconception: “The induced field opposes the external field.” Not always. If the external flux is decreasing, the induced field tries to maintain it, meaning the induced field points the same way as the external field.

Ways to change magnetic flux (and what the induced emf depends on)

Since $\Phi_B = BA\cos\theta$ (for a flat loop in uniform field), flux changes can come from:

• Changing $B$ (time-varying magnetic field)
• Changing $A$ (loop area changes, or loop moves into/out of a field region)
• Changing $\theta$ (rotating the loop)

The induced emf depends on the rate of change, $d\Phi_B/dt$. A large flux is not enough; it must be changing.

Worked example 1: flux through a tilted loop

A circular loop has area $A = 0.020\ \text{m}^2$ in a uniform field $B = 0.50\ \text{T}$. The field makes an angle $\theta = 60^\circ$ with the area vector.

Compute the flux.

Step 1: Use the definition

$\Phi_B = BA\cos\theta$

Step 2: Substitute

$\Phi_B = (0.50)(0.020)\cos 60^\circ$

Step 3: Evaluate

$\Phi_B = (0.010)(0.50) = 0.0050\ \text{Wb}$

Notice: if the loop were rotated so $\theta = 90^\circ$, the flux would be zero even though the loop is still “in” the field.

Worked example 2: induced emf from a changing field

A single loop of area $A = 0.030\ \text{m}^2$ sits in a uniform magnetic field perpendicular to the loop (so $\cos\theta = 1$). The field increases uniformly from $0.20\ \text{T}$ to $0.80\ \text{T}$ in $0.10\ \text{s}$. Find the magnitude of the induced emf.

Step 1: Express flux and its change

$\Phi_B = BA$

So the change in flux is

$\Delta\Phi_B = A\Delta B$

Step 2: Compute the change

$\Delta\Phi_B = (0.030)(0.80 - 0.20) = (0.030)(0.60) = 0.018\ \text{Wb}$

Step 3: Apply Faraday’s law (magnitude)

$|\varepsilon| = \left|\frac{\Delta\Phi_B}{\Delta t}\right| = \frac{0.018}{0.10} = 0.18\ \text{V}$

Direction questions would require Lenz’s law and a right-hand rule, but the magnitude comes straight from the flux rate of change.

Exam Focus

• Typical question patterns
  • Compute flux $\Phi_B$ through a loop at an angle, sometimes requiring careful angle interpretation.
  • Use Faraday’s law to find induced emf from changing $B$, changing area, or rotation (often with $N$ turns).
  • Direction of induced current using Lenz’s law: “field into the page increasing” style questions.
• Common mistakes
  • Using the angle between $\vec B$ and the plane of the loop instead of between $\vec B$ and the area vector.
  • Forgetting the factor of $N$ for multi-turn coils.
  • Opposing the external field rather than opposing the change in flux (the subtle but crucial Lenz’s law idea).

Inductance and Inductors

What an inductor is (physical picture)

An inductor is a circuit element designed so that when current flows through it, it produces magnetic flux that is linked with the circuit itself (often via a coil). When the current changes, the magnetic flux changes, and by Faraday’s law that changing flux induces an emf.

The key idea: inductors resist changes in current, not because they “hate current,” but because changing current implies changing magnetic field, which induces an emf that opposes the change.

Inductors matter because they are the magnetic-field analog of capacitors:

• Capacitors store energy in electric fields and resist changes in voltage.
• Inductors store energy in magnetic fields and resist changes in current.

Self-inductance: defining $L$

When the changing current in a circuit induces an emf in the same circuit, that phenomenon is self-inductance. The proportionality constant is the inductance $L$.

For a coil with $N$ turns, define self-inductance through flux linkage:

$N\Phi_B = LI$

This equation is a definition (for linear materials and typical AP situations): the flux linked is proportional to the current producing it.

From Faraday’s law, if $N\Phi_B = LI$, then

$\varepsilon = -\frac{d(N\Phi_B)}{dt} = -L\frac{dI}{dt}$

Where:

• $L$ is inductance (henry, H), where $1\ \text{H} = 1\ \text{V}\cdot\text{s}/\text{A}$.
• $dI/dt$ is the rate of change of current.

A huge conceptual payoff: this is why inductors oppose changes in current. If you try to increase $I$ quickly, $dI/dt$ is large, so the induced emf magnitude $|\varepsilon|$ is large and acts to oppose that increase.

Where inductance comes from (geometry + magnetic permeability)

Inductance is not a mystical property — it depends on how effectively a given current can create magnetic flux that links the circuit.

A coil with many turns, large area, and tightly packed magnetic field (like inside a long solenoid) has a large flux linkage per current, so it has a large $L$. Introducing a material with higher magnetic permeability (like iron) can also increase inductance by concentrating magnetic fields, though detailed material behavior can be complex; AP problems typically stick to vacuum/air cores using $\mu_0$.

Inductance of a long solenoid (important model)

For an ideal long solenoid of length $\ell$, cross-sectional area $A$, and $N$ turns, the magnetic field inside is approximately uniform:

$B = \mu_0\frac{N}{\ell}I$

Flux through one turn (area perpendicular to field) is

$\Phi_B = BA = \mu_0\frac{N}{\ell}IA$

Flux linkage is

$N\Phi_B = \mu_0\frac{N^2 A}{\ell}I$

Comparing with $N\Phi_B = LI$ gives

$L = \mu_0\frac{N^2 A}{\ell}$

This formula is heavily used because it ties geometry directly to $L$. It also reinforces what “inductance” really means: a measure of how much flux linkage you get per ampere.

Energy stored in an inductor (magnetic energy)

An inductor stores energy in its magnetic field. If you build up current from 0 to $I$, you must do work against the induced emf. That work becomes stored energy:

$U = \frac{1}{2}LI^2$

This is a favorite AP relationship because it mirrors capacitor energy:

$U_C = \frac{1}{2}CV^2$

A common error is treating inductors as dissipating energy like resistors. Ideal inductors do not dissipate; they store and release energy.

Mutual inductance (how one circuit induces emf in another)

If you have two nearby coils, a changing current in coil 1 creates a changing magnetic field, which changes the flux through coil 2 and induces an emf there. This coupling is described by mutual inductance $M$.

Define mutual inductance by the flux linkage in coil 2 due to current in coil 1:

$N_2\Phi_{B2} = MI_1$

Then the induced emf in coil 2 is

$\varepsilon_2 = -M\frac{dI_1}{dt}$

Similarly,

$\varepsilon_1 = -M\frac{dI_2}{dt}$

(For typical AP contexts with fixed geometry, the same $M$ applies both ways.)

This is the physics behind transformers and many forms of wireless power transfer: changing current in one circuit induces voltage in another via changing flux.

Inductors in circuits: the RL time response (conceptual core)

When you put an inductor in series with a resistor and a battery (an RL circuit), the inductor prevents the current from jumping instantly. The reason is the induced emf $\varepsilon_L = -L(dI/dt)$. If the current tried to change abruptly, $dI/dt$ would be enormous, requiring an enormous induced emf.

For a series RL circuit with a constant applied voltage $\mathcal{E}$, Kirchhoff’s loop rule gives

$\mathcal{E} - IR - L\frac{dI}{dt} = 0$

Solving yields a current that approaches a steady value $\mathcal{E}/R$ exponentially with time constant

$\tau = \frac{L}{R}$

The current growth is

$I(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-t/\tau}\right)$

If the battery is removed and the inductor discharges through the resistor, the current decays as

$I(t) = I_0 e^{-t/\tau}$

On AP Physics C: E&M, you’re expected to interpret these equations physically:

• Large $L$ means “more inertia” against current changes, so the response is slower.
• Large $R$ dissipates energy faster and also reduces $\tau$.

Worked example 1: induced emf from a changing current (self-inductance)

An inductor has $L = 0.40\ \text{H}$. The current decreases from $3.0\ \text{A}$ to $1.0\ \text{A}$ in $0.20\ \text{s}$. Find the magnitude of the induced emf.

Step 1: Use the inductor emf relation (magnitude)

$|\varepsilon| = L\left|\frac{\Delta I}{\Delta t}\right|$

Step 2: Compute $\Delta I/\Delta t$

$\left|\frac{\Delta I}{\Delta t}\right| = \left|\frac{1.0 - 3.0}{0.20}\right| = \left|\frac{-2.0}{0.20}\right| = 10\ \text{A/s}$

Step 3: Multiply by $L$

$|\varepsilon| = (0.40)(10) = 4.0\ \text{V}$

Direction/sign: because the current is decreasing, the induced emf acts to keep current flowing in the original direction (it “fights” the decrease).

Worked example 2: long-solenoid inductance and stored energy

A long air-core solenoid has $N = 800$ turns, length $\ell = 0.50\ \text{m}$, and cross-sectional area $A = 2.0\times 10^{-4}\ \text{m}^2$. Find $L$ and the energy stored when $I = 1.5\ \text{A}$.

Step 1: Use the solenoid inductance formula

$L = \mu_0\frac{N^2 A}{\ell}$

Step 2: Substitute values (with $\mu_0 = 4\pi\times 10^{-7}\ \text{T}\cdot\text{m/A}$)

$L = (4\pi\times 10^{-7})\frac{(800)^2(2.0\times 10^{-4})}{0.50}$

Compute piecewise:

• $(800)^2 = 6.4\times 10^5$
• $(6.4\times 10^5)(2.0\times 10^{-4}) = 128$
• $128/0.50 = 256$

So

$L = (4\pi\times 10^{-7})(256)$

$L \approx 3.2\times 10^{-4}\ \text{H}$

Step 3: Energy stored

$U = \frac{1}{2}LI^2$

$U = \frac{1}{2}(3.2\times 10^{-4})(1.5)^2$

$(1.5)^2 = 2.25$, so

$U \approx 0.5\times 3.2\times 10^{-4}\times 2.25 \approx 3.6\times 10^{-4}\ \text{J}$

This is a small energy because $L$ is small; increasing turns or area increases $L$ dramatically (note the $N^2$ dependence).

What typically goes wrong with inductance

Students often memorize $\varepsilon = -L(dI/dt)$ without internalizing that it’s just Faraday’s law applied to a circuit whose own current is changing its own flux. If you keep the flux linkage picture in mind, the sign and the “opposes change” behavior stop feeling like magic.

Another common mistake is confusing the induced emf across an inductor with a potential difference that is always present. In DC steady state (after a long time with constant current), $dI/dt = 0$, so the ideal inductor’s emf is zero — it behaves like a wire (though real inductors have some resistance).

Exam Focus

• Typical question patterns
  • Use $\varepsilon = -L(dI/dt)$ to find induced emf given how current changes (or infer $dI/dt$ from emf).
  • Compute inductance of an ideal solenoid from geometry, then use it for energy $U = (1/2)LI^2$.
  • RL transients: interpret or compute time constant $\tau = L/R$ and current growth/decay expressions.
• Common mistakes
  • Dropping the minus sign conceptually and getting Lenz’s law direction backwards (especially when current is decreasing).
  • Treating an inductor like a resistor (thinking it “uses up” energy rather than storing and returning it).
  • Forgetting that at steady DC, $dI/dt = 0$ so the induced emf across an ideal inductor is zero (many students incorrectly keep a nonzero inductor voltage forever).