13.2 Electromagnetic Induction

Electromagnetic induction is how a changing magnetic flux through a loop creates an emf (voltage) and—if the loop’s closed—an induced current. Faraday’s law quantifies it: ε = −dΦB/dt, where ΦB = B·A (or sum for N loops: |εsol| = N|dΦB/dt|). Flux can change if B changes, the loop’s area or orientation changes, or you move the loop. Lenz’s law (the minus sign) tells the induced current’s direction: it produces a B-field that opposes the flux change—use the right-hand rule to get the current sense. Physically, a changing B induces a nonconservative electric field (Maxwell–Faraday: ∮E·dℓ = −dΦB/dt), so charges feel an electric force and move (current) or a motional emf appears in moving conductors. You’ll need these ideas and the equations for AP Topic 13.2 (see the study guide: https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW; unit overview: https://library.fiveable.me/ap-physics-c-e-m/unit-13). For extra practice, use the 1000+ problems at (https://library.fiveable.me/practice/ap-physics-c-e-m).

The negative sign in Faraday’s law, 𝓔 = −dΦB/dt, encodes Lenz’s law: an induced emf (and any induced current) always acts to oppose the change in magnetic flux that produced it. Physically that’s required by energy conservation—the induced current creates its own magnetic field which resists the flux change, so you must do work (or lose energy) to change the flux. Mathematically the minus also comes from the orientation convention: ΦB = ∫B·dA uses an area (normal) vector; if ΦB is increasing, the induced electric field around the loop gives a current whose magnetic field points to reduce that increase, so the sign is negative. In Maxwell’s form, ∮E·dl = −dΦB/dt, the minus simply relates the direction of the line integral of E to the direction of changing flux (use the right-hand rule to connect current direction and flux). For AP review, focus on Lenz’s law for direction questions and the Faraday/Maxwell equations for quantitative work (see the Topic 13.2 study guide on Fiveable: https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW). For extra practice, use the AP practice problems at https://library.fiveable.me/practice/ap-physics-c-e-m.

Short answer: magnetic flux isn’t always just B times A—it’s the dot product of the magnetic field and the area vector: ΦB = ∫ B · dA. What that means in practice: - If B is uniform and the surface is flat, ΦB = B A cosθ (θ is the angle between B and the area normal), so B·A reduces to B·A only when B is perpendicular (cosθ = 1). - If B varies over the surface or the surface is curved, you must integrate: ΦB = ∫ B·dA. - For N turns, flux linkage is NΦB, and Faraday’s law gives emf: ε = −dΦB/dt (and Lenz’s law sets the sign/direction). This is exactly what the CED covers for Topic 13.2 (Faraday’s law, area vector, flux linkage). For worked examples and practice problems, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW), the unit overview (https://library.fiveable.me/ap-physics-c-e-m/unit-13), and lots of practice sets (https://library.fiveable.me/practice/ap-physics-c-e-m).

Faraday’s law gives you the math: a changing magnetic flux through a loop produces an emf. Symbolically E = −dΦB/dt (or ℰ = −d( B·A )/dt), and you use it to calculate the magnitude of the induced emf (including N loops for a solenoid). Lenz’s law tells you the direction: the minus sign in Faraday’s law means the induced current produces a magnetic field that opposes the change in flux. Practically, use Faraday to get |ℰ| and Lenz (with the right-hand rule) to pick the current’s direction so the induced B-field fights the flux change. On the AP exam you’ll be asked to compute magnitudes (13.2.A.1) and justify directions (13.2.A.2)—so show the derivative for ℰ and state how the induced field opposes the change. For a quick review, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Right-hand rule for induction is just a tool to apply Lenz’s law (CED 13.2.A.2): the induced current makes a magnetic field that opposes the change in magnetic flux (Faraday’s law 13.2.A.1). Quick steps you can use on every problem: 1. Decide how the magnetic flux through the loop is changing (is B getting bigger/smaller, or is area/orientation changing?). 2. Use Lenz’s law: if flux into page is increasing, the induced B must point out of the page; if flux into page is decreasing, induced B must point into the page. 3. Use the right-hand rule for loops: point your thumb in the direction of the induced B (the field the induced current must create). Curl your fingers—the finger curl shows the direction of induced current (clockwise or counterclockwise). 4. If asked for emf sign or magnitude, use Faraday: E = −dΦB/dt (and for N turns multiply by N). Example: B into the page and increasing → induced B out → thumb out of page → fingers curl counterclockwise → induced current is CCW. This exactly matches CED skills for Topic 13.2. For more worked examples and practice problems, check the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and the general unit page (https://library.fiveable.me/ap-physics-c-e-m/unit-13).

Lenz’s law says the induced emf (and any induced current) always acts to oppose the change in magnetic flux that produced it. Think of it like a “reaction”: if the magnetic flux through a loop tries to increase, the induced current creates its own magnetic field that points the opposite way to reduce that increase; if flux tries to decrease, the induced field boosts it. Mathematically Faraday’s law shows this: ε = −dΦB/dt—the minus sign is Lenz’s law built in. Use the right-hand rule to get the induced field direction from the current direction (and the right-hand curl to find current given an induced B). Physically this arises from conservation of energy: the induced current resists change so you must do work to change the flux (that work becomes Joule heat or stored energy). For AP E&M, practice applying Faraday’s law + Lenz’s sign rule and the right-hand rule (see the Topic 13.2 study guide on Fiveable (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and try practice problems at (https://library.fiveable.me/practice/ap-physics-c-e-m)).

Use Faraday’s law (general form) any time you need the induced emf from a changing magnetic flux: E = −dΦB/dt = −d(B·A)/dt (or the loop integral form E = ∮E·dℓ = −dΦB/dt). That covers cases where B changes, A changes (motional emf), the loop rotates, or the shape/orientation matters (CED 13.2.A.1 and 13.2.A.1.i–ii). Use the “solenoid equation” (emf for a long solenoid with N turns) when the changing flux comes from many identical loops stacked into a long solenoid—then total emf = N·(emf per loop) = −N dΦB/dt (CED 13.2.A.1.iii). Practically: if you have a long solenoid with n turns per length and a small coil/loop inside it, substitute Binside = μ0 n I and multiply flux by the number of turns (or use N when you’re given total turns). If you’re unsure on the exam, start with Faraday’s law and then include N (or n and loop area) when the problem states multiple turns/solenoid (see the Topic 13.2 study guide for worked examples: https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW). For extra practice, try AP-style problems at (https://library.fiveable.me/practice/ap-physics-c-e-m).

Magnetic flux is ΦB = B · A (more precisely B·A·cosθ). If B is constant but the loop’s area A changes, ΦB still changes because ΦB depends on A. By Faraday’s law, an emf is induced whenever dΦB/dt ≠ 0: ε = −dΦB/dt = −B (dA/dt) (if B and the loop’s orientation are constant). So changing the loop’s size (sliding part of the loop, stretching it, or pulling it out of a uniform-B region) changes A over time, gives a nonzero dA/dt, and produces an emf. Lenz’s law then tells you the induced current’s direction: it will create a magnetic field that opposes the change in flux (i.e., opposes the increase or decrease of ΦB). This is exactly what the CED expects you to use (Faraday’s law 13.2.A.1 and Lenz’s law 13.2.A.2). For a quick review and practice problems on this topic, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and the AP practice question bank (https://library.fiveable.me/practice/ap-physics-c-e-m).

Use Faraday’s law: emf = −dΦB/dt = −d( B · A )/dt. If both the magnetic field and the area (or orientation) are changing, the total induced emf is the sum of the contributions from each change: emf = −[ A (dB/dt) + B (dA/dt) ] (for B and A perpendicular; more generally use the dot product and take its time derivative). If you have N loops, multiply that result by N (flux linkage). Direction: use Lenz’s law—the induced current produces a magnetic field that opposes the net change of flux (i.e., opposes the combined effect of dB/dt and dA/dt). For AP-style problems, set up ΦB(t) explicitly, differentiate, and use signs to get direction. For practice and worked examples, see the Topic 13.2 study guide on Fiveable (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and more practice at (https://library.fiveable.me/practice/ap-physics-c-e-m).

You need Maxwell’s equations in AP Physics C because they’re the compact set of laws that explain everything you study in Topic 13.2: Faraday’s law (Maxwell’s third equation) is exactly the statement that a changing magnetic flux produces an induced emf (E = ∮E·dℓ = −dΦB/dt). Knowing Maxwell’s equations ties Faraday’s law to Lenz’s law, motional emf, and how induced electric fields behave in circuits and solenoids—all tested on the exam. They also show (conceptually) that E and B fields satisfy wave equations and predict the constant speed c = 1/√(ε0μ0) in free space (you don’t have to derive that on the AP). For targeted review, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and practice problems at (https://library.fiveable.me/practice/ap-physics-c-e-m) to drill Faraday/Lenz setups and induced-emf calculations.

Quick checklist you can use on any problem: 1) Identify how magnetic flux is changing (is B getting stronger/weaker, into or out of the page? or is the loop area changing?). Use ΦB = B·A and Faraday’s law (ε = −dΦB/dt) to tell if the induced emf is nonzero. 2) Use Lenz’s law: the induced magnetic field from the induced current will oppose the change in flux. If flux into the page is increasing, the induced B must come out of the page; if flux out is decreasing, the induced B must come out to replace it, etc. (That minus sign in Faraday’s law encodes Lenz’s law.) 3) Use the right-hand rule to get current direction: point your thumb in the induced-B direction, your curled fingers show the direction of induced current (counterclockwise vs clockwise as seen from your viewpoint). Practice this sequence on a few diagrams—the AP C exam expects you to use Lenz’s law + the right-hand rule (Topic 13.2). For more worked examples, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and try practice problems at (https://library.fiveable.me/practice/ap-physics-c-e-m).

Generators are just real-world applications of Faraday’s law and Lenz’s law. Faraday: E = −dΦB/dt, so any device that makes the magnetic flux through a coil change will produce an induced emf. In a typical AC generator a loop (or many turns, N) rotates in a (nearly) constant B-field, so ΦB = B A cosθ and θ = ωt. That gives an emf E(t) = N B A ω sin(ωt)—an alternating voltage whose frequency is f = ω/2π. Lenz’s law tells you the direction of the induced current: it creates a magnetic field opposing the change in flux (this is why you feel mechanical resistance when you turn a generator). Motional emf (v × B) on moving conductors is the same idea for simple rod-on-rails setups. For AP exam work, you should be able to derive and use E = −dΦB/dt, apply Lenz’s rule, and handle N-turn coils (13.2.A.1–2). Review Topic 13.2 on Fiveable (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Think of a changing magnetic flux as the cause, and an induced electric field as the effect—Faraday’s law gives the exact link. In integral form (the AP CED expects this): ∮ E · dℓ = −dΦB/dt. That means if the magnetic flux through any loop changes with time, the line integral of E around that loop is nonzero—there must be an electric field circulating around the region. Physically: a time-changing B-field produces a nonconservative electric field whose field lines form closed loops (often circular around the changing B). Lenz’s law sets the direction: the induced E (and any induced current in a conductor) works to oppose the change in flux. For moving conductors you can also get a motional emf from v × B, but that’s a different viewpoint that still matches Faraday’s law. You should be ready to use ∮E·dℓ = −dΦB/dt and Lenz’s law on the exam. For a quick refresher and practice problems, see the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and the AP Physics C practice set (https://library.fiveable.me/practice/ap-physics-c-e-m).

Short recipe for rod-through-field problems: 1) Draw B, v, and rod length ℓ. Identify area swept per time: dA/dt = ℓ v (if rod moves perpendicular to B and along rails). 2) Use Faraday: ε = −dΦB/dt. For constant B this gives motional emf ε = B ℓ v (magnitude). That’s the AP C 13.2.A idea: emf from changing flux. 3) If the rod closes a circuit with resistance R, get current I = ε/R. Use Lenz’s law/right-hand rule to pick the current direction so its B opposes the flux change (CED 13.2.A.2). 4) Magnetic force on charges (and rod): F = I ℓ B (direction opposes motion). If you need acceleration, apply Newton’s 2nd law: ma = Fnet (include applied forces). Power/energy: electrical power P = Iε = I^2R and mechanical power lost = F v. On the exam, explicitly cite Faraday’s law and Lenz’s law in your setup. For more worked examples and practice, see the Topic 13.2 study guide (Fiveable) here: (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and thousands of practice problems at (https://library.fiveable.me/practice/ap-physics-c-e-m).

Maxwell’s equations show that changing electric and magnetic fields produce each other (Faraday’s law and the Maxwell–Ampère law), which leads to wave equations for E and B. Those wave equations have solutions that are electromagnetic waves traveling through free space at a single, constant speed c, where c = 1 / sqrt(ε0 μ0). So the speed of light is not just an empirical number—it’s the natural propagation speed of E and B fields predicted by Maxwell’s laws. In those waves E and B are transverse, perpendicular to each other and to the direction of travel, and their magnitudes are linked by the vacuum impedance (√(μ0/ε0)). You don’t need to derive this for AP—the CED notes that Maxwell’s equations can be used to show EM fields obey wave equations and that c = 1/√(ε0μ0) (see Topic 13.2). For review, check the Topic 13.2 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/2-electromagnetic-induction/study-guide/b2D8zUgtWmPcdNoW) and more unit resources (https://library.fiveable.me/ap-physics-c-e-m/unit-13).