12.4 Electromagnetic Induction and Faraday's Law

Electromagnetic induction is how a changing magnetic flux through a loop produces an emf (voltage) and—if the loop is closed—an induced current. Flux ΦB = B A cosθ measures how much of the B-field goes through an area. Faraday’s law tells you the size: |ε| = |ΔΦB/Δt| (and the signed form ε = −ΔΦB/Δt). Lenz’s law gives direction: the induced current creates its own B-field that opposes the change in flux (the minus sign); use the right-hand rule to relate current direction to that opposing field. Common examples: moving a loop into/out of a uniform B region (flux changes) or a conducting rod sliding on rails (motional emf ε = Bℓv). On the AP exam you’ll be asked to compute flux, rate of change, induced emf, and use Lenz’s law to get current/force directions (Topic 12.4, CED). For a focused review and practice problems, see the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3), the Unit 12 overview (https://library.fiveable.me/ap-physics-2-revised/unit-12), and thousands of practice questions (https://library.fiveable.me/practice/ap-physics-2-revised).

Changing magnetic flux creates an electric current because a changing flux produces an induced emf (an electric potential difference) according to Faraday’s law: |ε| = |ΔΦB/Δt|. Magnetic flux ΦB = B·A·cosθ describes how much B-field passes through a loop. If ΦB changes (B, area, or angle changes), an emf appears around the loop. If the loop is closed and conductive, that emf drives charges and you get an induced current. Lenz’s law gives the current’s direction: the induced current creates its own magnetic field that opposes the change in the original flux (ε = −ΔΦB/Δt), which ensures energy conservation (you must do work to change the flux). A common example is a conducting rod sliding on rails in a uniform B (motional emf ε = Bℓv), which produces a measurable current when the circuit is closed. For AP review, focus on flux, Faraday’s law, Lenz’s law, and motional emf (see the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3)) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Short answer: almost—but not always. Magnetic flux ΦB measures the amount of the B-field passing through a surface, so for a uniform field that’s perpendicular to the area ΦB = B A (more generally ΦB = B A cosθ, CED 12.4.A.2). Key points you need to keep in mind for the AP: - “B times area” only holds when B is uniform across the surface and you use the perpendicular component (use the area vector perpendicular to the surface). - If the field varies across the surface, you must integrate ΦB = ∫ B · dA. - The sign of ΦB comes from whether B is parallel or antiparallel to the area vector (12.4.A.2.i/ii). - Only a change in flux induces an emf (Faraday’s law: |ε| = |ΔΦB/Δt|) and Lenz’s law gives the direction (12.4.A.3–4). If you want practice sketching nonuniform/angled cases or problems tied to the AP CED, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3), the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-12), or the practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Faraday’s law tells you how big the induced emf is: an emf is produced whenever the magnetic flux through a loop changes, and |ε| = |ΔΦB/Δt| (CED 12.4.A.3). Lenz’s law tells you which way that emf (and any induced current) points: the induced current creates a magnetic field that opposes the change in flux, represented by the minus sign ε = −ΔΦB/Δt (CED 12.4.A.4). So think: Faraday = magnitude (how much emf), Lenz = direction (which way current flows to oppose the flux change). Use the area vector and right-hand rule to apply the sign convention and get the actual current direction in problems (CED 12.4.A.2, 12.4.A.4.i). For quick review and AP-style practice, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3), the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-12), and lots of practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Quick recipe you can use on any induction problem: 1) Find how the magnetic flux through the loop changes (ΦB = B A cosθ). Is flux increasing or decreasing? (CED 12.4.A.1–A.3) 2) Use Lenz’s law: the induced current produces a magnetic field that opposes that change (CED 12.4.A.4). So decide which direction the induced B must point (e.g., if external B into page is increasing, induced B must be out of page). 3) Apply the right-hand rule for a current loop: curl your fingers in the direction of the loop current; your thumb shows the loop’s magnetic dipole / area vector (direction of the induced B). To find current direction, pick the curl that makes the thumb point the way the induced B must point. 4) For single moving charges or a rod (motional emf ε = Bℓv), use the RHR for magnetic force: point fingers along v, curl toward B; thumb is force on a positive charge (direction of charge separation/current). Practice this sequence on problems (study guide: https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3). For lots of practice Qs, check Fiveable’s practice bank (https://library.fiveable.me/practice/ap-physics-2-revised).

The negative sign in Faraday’s law (ε = −ΔΦB/Δt) comes from Lenz’s law: the induced emf (and any induced current) always acts to oppose the change in magnetic flux through the loop. Flux ΦB = B·A·cosθ can increase or decrease; the sign of ΔΦB tells you whether flux is rising or falling. The minus sign flips that sign so the induced emf produces a current whose magnetic field fights the change—e.g., if external B into the page increases, the induced current makes a B out of the page to reduce the net change. Practically this sets the direction of the induced current (use the right-hand rule with the area vector) and ensures energy conservation (induced currents require work to change flux). This is exactly what the CED calls out in 12.4.A.4 (Lenz’s law). Review the topic study guide for more examples (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and practice problems at the unit page (https://library.fiveable.me/ap-physics-2-revised/unit-12) or the AP practice bank (https://library.fiveable.me/practice/ap-physics-2-revised).

Magnetic flux sign just tells you whether the B-field points with or against the surface’s area vector. Flux ΦB = B A cosθ—the area vector is perpendicular to the surface (for a loop use the chosen orientation). If θ < 90° then cosθ is positive and ΦB > 0 (B has a component parallel to the area vector). If θ > 90° then cosθ is negative and ΦB < 0 (B is antiparallel to the area vector). Why it matters: Faraday’s law uses ΔΦB—the magnitude tells you how big the induced emf is, and the sign (with the minus in ε = −ΔΦB/Δt) together with Lenz’s law tells you the direction of the induced emf/current (it opposes the change in flux). So a flux that goes from positive to more positive gives an opposite-current sense than a flux that goes from negative to less negative. For review and AP-style practice, see the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3), unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-12), and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

In the rod-on-rails setup a conducting rod completes a rectangular loop whose area changes as the rod slides through a uniform B into/out of the page. That changing area changes the magnetic flux ΦB = B A, so by Faraday’s law an emf is induced: |ε| = |ΔΦB/Δt|. For a rod moving at speed v with rail separation ℓ this becomes the motional emf ε = Bℓv. The emf drives a current around the loop; by Lenz’s law the current’s magnetic field opposes the change in flux, so the magnetic force on the rod opposes its motion (you feel a retarding force). Energy put in by whatever pushes the rod goes into Joule heating in the loop (and work done against the magnetic force). This example directly ties to CED 12.4.A (flux, Faraday, Lenz, and ε = Bℓv) and is a common AP problem—review the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and practice questions on the unit page (https://library.fiveable.me/ap-physics-2-revised/unit-12).

Because of Lenz’s law: the minus sign in Faraday’s law (ε = −ΔΦB/Δt) tells you the induced emf produces a current whose magnetic field opposes the change in flux. Physically this is just energy conservation—the induced current makes a magnetic force that resists whatever change produced the changing flux, so you can’t get free energy. Quick way to use it: 1) Determine how the flux through the loop is changing (increasing/decreasing and which way is “positive” for the area vector). 2) Ask: what direction of B produced by an induced current would oppose that change? 3) Use the right-hand rule (curl fingers in current direction, thumb gives B direction) to pick the current direction. This is exactly what the CED calls for in 12.4.A.4 (Lenz’s law). For more worked examples and practice applying the sign and right-hand rule, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and the AP practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Use the cosθ whenever the magnetic field isn’t perfectly perpendicular to the surface you care about. θ is the angle between the magnetic field vector B and the area vector (the area vector is perpendicular to the surface). Flux = B A cosθ tells you the component of B that actually “goes through” the loop. Quick cases: - B perpendicular to the surface → θ = 0°, cos0 = 1 → ΦB = BA (max flux). - B parallel to the surface → θ = 90°, cos90 = 0 → ΦB = 0 (no flux through). - If B is at some angle, use cosθ to get the perpendicular component: B⊥ = B cosθ. Remember sign: if B points opposite the area vector, cosθ is negative and flux is negative—that matters for Lenz’s law and determining emf direction via ΔΦB/Δt. For more examples and practice, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and tons of practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

EMF (electromotive force) is the induced "driving" potential produced by a changing magnetic flux (Faraday’s law: ε = −ΔΦB/Δt). It’s what appears when a magnetic field through a loop changes or when a conductor moves through a B-field (motional emf ε = Bℓv). Regular voltage (potential difference) usually refers to the conservative electric potential difference between two points produced by charges (ΔV = −∫E·dl for electrostatic fields). Key differences: an induced emf comes from a nonconservative electric field created by changing magnetic flux, so you can get a net emf around a closed loop even if no battery or charge separation exists. In a closed circuit that emf drives current; across two points in an open circuit you may measure a potential difference only if charges separate. For AP topic 12.4, focus on flux, Faraday’s law, and Lenz’s sign (direction of induced current). For extra practice and exam-style problems, see the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and unit resources (https://library.fiveable.me/ap-physics-2-revised/unit-12).

First, use Faraday + Lenz: a changing magnetic flux through the loop induces an emf that creates a current whose magnetic field opposes the change in flux (CED 12.4.A.3–4). Steps you can always follow: 1. Pick an area vector for the loop (perpendicular to the loop). 2. Determine the sign of the magnetic flux and how it’s changing (is |ΦB| increasing or decreasing? Is the B-field pointing parallel or anti-parallel to the area vector?) (CED 12.4.A.2). 3. By Lenz’s law, the induced B-field points to oppose that change: if flux into the page is increasing, the induced B must point out of the page; if flux out is decreasing, induced B points out to try to restore it. 4. Use the right-hand rule: curl your fingers in the direction of current; your thumb gives the induced B direction. Match the curl that gives the required induced B (CED 12.4.A.4.ii). Quick example: B into the page is increasing → induced B out of the page → current is counterclockwise (viewed from your side). Practice these on Fiveable’s topic study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and the AP unit page (https://library.fiveable.me/ap-physics-2-revised/unit-12).

You need electromagnetic induction because it explains how changing magnetic flux makes useful voltages and currents—the exact ideas AP names in 12.4 (magnetic flux, Faraday’s law, Lenz’s law, motional emf). Real-life examples: electric generators (mechanical → electrical power), transformers (voltage step-up/step-down for the grid), induction cooktops and wireless chargers (induced currents heat or power devices), regenerative braking and many sensors (moving magnets change flux to generate signals). Those all depend on |ε| = |ΔΦB/Δt| and on Lenz’s law to get the direction of induced current. On the exam, Unit 12 is worth ~12–15% of MCQs, so being able to calculate flux (ΦB = BA cosθ), use ε = −ΔΦB/Δt, and apply the right-hand rule is directly testable. For targeted review, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3), the full unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-12), and do practice problems (https://library.fiveable.me/practice/ap-physics-2-revised) to see these concepts in real exam-style questions.

You were supposed to see that changing magnetic flux through the coil makes a measurable emf/current. Practically: move a bar magnet toward or into a coil and the galvanometer needle (or LED/buzzer) deflects; pull it away and the deflection reverses. Faster motion → bigger deflection (|ε| = |ΔΦB/Δt|). More turns or larger coil area → larger signal (ΦB = B A cosθ). When the coil is completely inside a uniform B and B isn’t changing, the reading goes to zero (no change in flux). Use Lenz’s law to predict direction: induced current creates a B-field that opposes the change in flux. Classic demos: dropping a magnet through a copper tube or through a coil gives short pulses; pushing/pulling a magnet through a multi-turn coil gives larger, opposite pulses. For AP stuff, be ready to describe observations, connect them to ΦB, Faraday’s law, and Lenz’s law (CED 12.4.A). Review the topic study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Use flux ΦB = B A cosθ and Faraday’s law ε = −dΦB/dt. If B, A, or θ change with time, just take the time derivative: ε = −d/dt(BA cosθ) = −[A cosθ (dB/dt) + B cosθ (dA/dt) − B A sinθ (dθ/dt)] So: - If B(t) changes: ε = −A cosθ · dB/dt. - If area changes (loop shrinking/expanding): ε = −B cosθ · dA/dt. - If angle changes (loop rotating): ε = +B A sinθ · dθ/dt (note sign from derivative). For a moving rod of length ℓ sliding at speed v through uniform B: motional emf ε = Bℓv (use when flux change comes from motion). Use Lenz’s law to pick the sign/direction: induced current produces B that opposes the change in flux. On the AP exam you may be asked to derive symbolic expressions (FRQ) or compute numerically (MCQ); show the derivative step and argue direction with Lenz’s law. For extra practice, check the Topic 12.4 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-4/4-electromagnetic-induction-and-faradays-law/study-guide/UqgM4DyzPPfjroi3) and hit the AP practice set (https://library.fiveable.me/practice/ap-physics-2-revised).