13.2 Electromagnetic Induction

Electromagnetic induction means a changing magnetic flux through a loop creates an induced emf. Faraday's law, $\mathcal{E} = \dfrac{d\Phi_B}{dt}$, gives the size of that emf, and Lenz's law (the negative sign) tells you its direction: the induced current always opposes the flux change that created it.

Why This Matters for the AP Physics C: E&M Exam

Electromagnetic Induction makes up a meaningful part of the exam, and this topic is the conceptual core of the whole unit. Almost every induction question starts here, whether you are calculating an induced emf, deciding which way the current flows, or reasoning about an induced electric field.

On both the multiple-choice and free-response sections, you will be asked to identify and analyze how variables relate. A common setup gives you a changing magnetic flux and asks for the induced current, the direction of that current, or how the result changes if you double the field, area, or rate of change. You also need to connect math to physical meaning, so practice explaining why an emf appears, not just plugging into a formula. Lab-style reasoning about loops and solenoids in changing fields shows up too, since these setups make abstract flux ideas concrete.

Key Takeaways

• Faraday's law: an induced emf equals the negative rate of change of magnetic flux, $\mathcal{E} = -\dfrac{d\Phi_B}{dt} = -\dfrac{d(\vec{B}\cdot\vec{A})}{dt}$.
• Flux can change three ways: the field changes, the area changes, or the angle between $\vec{B}$ and $\vec{A}$ changes.
• For a coil or solenoid with $N$ turns, multiply the single-loop emf by $N$: $|\mathcal{E}_{\mathrm{sol}}| = N\left|\dfrac{d\Phi_B}{dt}\right|$.
• Lenz's law sets direction: the induced current makes a magnetic field that opposes the flux change.
• A changing magnetic flux also creates a nonconservative electric field, written in integral form as $\oint \vec{E}\cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$ (Maxwell's third equation).
• Induced current depends on resistance through $I = \mathcal{E}/R$, so emf and current are separate steps.

Faraday's Law of Induction

Faraday's law gives the induced emf produced by changing magnetic flux:

$\mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{d(\vec{B} \cdot \vec{A})}{dt}$

Magnetic flux is defined as:

$\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$

where $\theta$ is the angle between the magnetic field vector and the area vector (the vector perpendicular to the surface). The negative sign in Faraday's law represents Lenz's law, which is covered below.

Any time the magnetic flux through a surface changes, for any reason, an emf is induced. This is the foundational principle of electromagnetic induction.

Special Cases of Faraday's Law

Faraday's law simplifies in two commonly tested scenarios:

1. Constant area, changing magnetic field: If the area of the surface is constant, the induced emf equals the area multiplied by the rate of change in the component of the magnetic field perpendicular to the surface:

$|\mathcal{E}| = A\left|\frac{dB_\perp}{dt}\right|$

1. Constant magnetic field, changing area: If the magnetic field is constant, the induced emf equals the magnetic field multiplied by the rate of change in the area perpendicular to the magnetic field:

$|\mathcal{E}| = B\left|\frac{dA_\perp}{dt}\right|$

Here $B_\perp$ is the component of $\vec{B}$ perpendicular to the surface, and $A_\perp$ is the area perpendicular to the field.

Multiple Turns: Coils and Solenoids

For a long solenoid with $N$ loops, each loop experiences the same changing magnetic flux, so the total induced emf across the solenoid is the emf in one loop multiplied by the number of loops:

$|\mathcal{E}_{\mathrm{sol}}| = N\left|\frac{d\Phi_B}{dt}\right|$

A long solenoid acts like many identical loops connected in series. The number of turns multiplies the emf; the resulting current still depends on the circuit resistance through $I = \mathcal{E}/R$.

Lenz's Law

Lenz's law tells you the direction of the induced emf and current:

• If the external flux into the page is increasing, the induced current produces a field out of the page (to oppose the increase).
• If the external flux into the page is decreasing, the induced current produces a field into the page (to oppose the decrease).

This is the physical meaning of the negative sign in Faraday's law. The induced effects always work against the change that caused them, so nature resists changes in magnetic flux.

Using the Right-Hand Rule with Induction

To determine the direction of the induced current:

1. First, use Lenz's law to determine the direction of the magnetic field that the induced current must create (it opposes the change in flux).
2. Then, use the right-hand rule for a current loop: curl the fingers of your right hand in the direction the current flows, and your thumb points in the direction of the magnetic field produced by that current. Choose the current direction that gives a field opposing the flux change.

For forces on current-carrying conductors in a magnetic field, use the right-hand rule for $\vec{F} = I\vec{L} \times \vec{B}$: point your fingers in the direction of the current (along $\vec{L}$), curl them toward $\vec{B}$, and your thumb points in the direction of the magnetic force on a positive current element.

Faraday's Law as Maxwell's Third Equation

A changing magnetic flux does not just create an emf in a physical wire. It produces a nonconservative electric field in space, even where no conductor is present. In integral form, Faraday's law is expressed as:

$\mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}$

This means the induced emf can be understood as the circulation of the electric field around a closed loop. This is Maxwell's third equation, and it tells you something important: changing magnetic fields create electric fields. This is fundamentally different from the electrostatic fields created by charges, because these induced electric fields form closed loops and are nonconservative.

Electromagnetic Waves and the Speed of Light

Maxwell's equations imply that electric and magnetic fields can propagate as electromagnetic waves in free space at speed:

$c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$

You are not expected to derive this expression for AP Physics C: E&M, but you should recognize it as an important consequence of Maxwell's equations.

Brief Note: Forces on Conductors from Induced Currents

If an induced emf drives a current in a closed circuit within an external magnetic field, that current can experience magnetic forces. For example, as a conducting loop enters or exits a region of magnetic field, the changing flux induces a current, and the magnetic force on that current opposes the loop's motion (consistent with Lenz's law). The primary focus of this topic, though, is calculating the induced emf from changing magnetic flux and determining its direction using Lenz's law. Forces on induced currents are developed more fully in the next topic.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

Work induction problems in two clear steps. First find the magnitude of the emf using Faraday's law, choosing the special case that matches the setup (constant area, constant field, or changing angle). Then, if the loop is closed, find the current with $I = \mathcal{E}/R$. Keep these separate so you do not accidentally mix emf and current.

MCQ

Many questions test functional dependence: how does the induced emf change if you double the field, the area, or the rate of change? Read the relationship straight from the equation. For a coil, remember the factor of $N$. For direction questions, apply Lenz's law first, then the right-hand rule.

Free Response

You may be asked to derive a symbolic expression for the emf, sketch how flux or emf changes over time, or justify the direction of an induced current using a physical law. Write a clear logical path: state Faraday's law, show how you take the derivative, and explain your reasoning for direction with Lenz's law. A correct final number with no supporting steps usually loses points.

Common Trap

When the magnetic field points at an angle to the loop, only the perpendicular component drives flux. Use $\Phi_B = BA\cos\theta$ and be careful about whether $\theta$ is measured from the surface or from the area vector.

Practice Problem 1: Induced EMF from a Changing Magnetic Field

Solution

Since the area is constant and the magnetic field is changing, use the constant-area special case of Faraday's law:

$|\mathcal{E}| = A\left|\frac{dB}{dt}\right|$

The area of the circular loop is:

$A = \pi r^2 = \pi (0.08 \text{ m})^2 = 6.4 \times 10^{-3}\pi \text{ m}^2 \approx 0.0201 \text{ m}^2$

So the induced emf is:

$|\mathcal{E}| = (0.0201 \text{ m}^2)(0.3 \text{ T/s}) = 6.03 \times 10^{-3} \text{ V} \approx 6.0 \text{ mV}$

The induced current is:

$I = \frac{|\mathcal{E}|}{R} = \frac{6.03 \times 10^{-3} \text{ V}}{2 \; \Omega} = 3.0 \times 10^{-3} \text{ A} = 3.0 \text{ mA}$

Direction: The magnetic field points into the page and is increasing. By Lenz's law, the induced current must create a magnetic field opposing this increase, that is, out of the page. Using the right-hand rule, the induced current flows counterclockwise as viewed from above.

Practice Problem 2: Multi-Turn Coil with Changing Flux

Solution

For a solenoid with $N$ turns, the total induced emf is:

$|\mathcal{E}_{\mathrm{sol}}| = N\left|\frac{d\Phi_B}{dt}\right|$

Since the area is constant and the field changes uniformly:

$\left|\frac{d\Phi_B}{dt}\right| = A\left|\frac{\Delta B}{\Delta t}\right| = (4.0 \times 10^{-4} \text{ m}^2) \cdot \frac{|0.10 - 0.50|}{0.02 \text{ s}}$

$\left|\frac{d\Phi_B}{dt}\right| = (4.0 \times 10^{-4})(20) = 8.0 \times 10^{-3} \text{ V}$

The total emf is:

$|\mathcal{E}_{\mathrm{sol}}| = 200 \times 8.0 \times 10^{-3} \text{ V} = 1.6 \text{ V}$

The field is decreasing, so by Lenz's law, the induced current flows in a direction to maintain the original flux. It creates a magnetic field in the same direction as the original field inside the solenoid.

Common Misconceptions

• A steady magnetic field induces an emf. Only a changing flux induces an emf. A strong but constant field through a stationary loop gives zero induced emf because $\frac{d\Phi_B}{dt} = 0$.
• The negative sign means a negative voltage. The negative sign encodes direction through Lenz's law. It is not telling you the emf is a negative number; it tells you the induced effect opposes the change.
• Lenz's law means the induced current opposes the existing field. The induced current opposes the change in flux, not the field itself. If the flux is decreasing, the induced current actually supports the original field to slow the decrease.
• More turns always means more current. Adding turns multiplies the emf by $N$, but the current still depends on resistance through $I = \mathcal{E}/R$. Do not skip the resistance step.
• Induced electric fields behave like fields from charges. Fields from static charges are conservative and point from high to low potential. Induced electric fields from changing flux form closed loops and are nonconservative.
• Flux uses the full field even at an angle. Only the component of $\vec{B}$ perpendicular to the loop contributes. Use $\Phi_B = BA\cos\theta$ with the correct angle.

Related AP Physics C: E&M Guides

• 13.3 Induced Currents and Magnetic Forces
• 13.4 Inductance
• 13.5 Circuits with Resistors and Inductors (LR Circuits)
• 13.6 Circuits with Capacitors and Inductors (LC Circuits)
• 13.1 Magnetic Flux