Unit 12 · Topic 12.4

12.4 Electromagnetic Induction and Faraday’s Law

Magnetic flux measures how much magnetic field passes straight through an area. Faraday's law says a changing flux through a loop induces an emf, and Lenz's law says the induced current opposes the change. Electromagnetic induction is how generators, transformers and wireless chargers work.

Key terms

magnetic flux
electromagnetic induction
Faraday's law
Lenz's law
induced emf
motional emf

Magnetic flux

Magnetic flux counts how much field passes perpendicularly through a surface, like counting field lines that poke through a loop:

$\Phi_B = BA\cos\theta$

A is the area, and θ is the angle between the field and the area vector, an arrow drawn perpendicular to the surface. If the field passes straight through the loop (θ = 0°), the flux is BA. If the field runs along the surface (θ = 90°), the flux is zero. A negative flux means the field points opposite to the area vector. The unit is T·m², also called the weber (Wb).

Flux changes if the field gets stronger or weaker, if the loop's area inside the field changes, or if the loop turns relative to the field.

Faraday's law

A changing magnetic flux through a loop induces an emf around it:

$\varepsilon = -\frac{\Delta\Phi_B}{\Delta t}$

The faster the flux changes, the bigger the emf. If the loop is a conductor with resistance R, the emf drives a current I = ε/R. Only change matters: a huge but steady flux induces nothing.

Lenz's law: which way the current goes

The minus sign in Faraday's law is Lenz's law. The induced current flows in the direction that makes its own magnetic field oppose the change in flux. It doesn't oppose the field itself, only the change. To find the direction:

Find the direction of the outside field through the loop.
Decide whether the flux is increasing or decreasing.
If it's increasing, the induced field points opposite to the outside field. If it's decreasing, the induced field points the same way, to prop it up.
Use the right-hand rule for a loop: point your thumb along the induced field, and your fingers curl in the direction of the induced current.

A rod sliding on rails

A classic example: a conducting rod of length ℓ slides at speed v along two conducting rails connected at one end, in a uniform field B perpendicular to the rails. As it moves, the area of the circuit changes, so the flux changes and an emf appears:

$\varepsilon = B\ell v$

You can also see it from 12.2: the charges in the rod move with it through the field, so the magnetic force pushes them along the rod. The induced current in the rod then feels a magnetic force IℓB that opposes the motion. To keep the rod moving at constant speed, something must push it, and the work done becomes electrical energy, then thermal energy in the resistance. Lenz's law is really energy conservation: if the induced current helped the motion, you'd get energy for free.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
A loop in a growing field
A flat loop of wire with area 0.020 m² and resistance 0.50 Ω sits with its face perpendicular to a uniform magnetic field that points out of the page. The field increases from 0.10 T to 0.50 T in 0.20 s. Find the induced emf, the current, and the current's direction as seen from the front of the page.
Show the solutionHide the solution
1. Step 1: ΔΦ = AΔB = (0.020)(0.50 − 0.10) = 8.0 × 10⁻³ T·m².
2. Step 2: $\lvert \varepsilon \rvert = \dfrac{\Delta\Phi_B}{\Delta t} = \dfrac{8.0 \times 10^{-3}}{0.20} = 0.040$ V.
3. Step 3: I = ε/R = 0.040/0.50 = 0.080 A.
4. Step 4: Direction: the outward flux is increasing, so the induced field inside the loop points into the page. A current that makes an inward field at the center flows clockwise as seen from the front.
Answer: 0.040 V, 0.080 A, clockwise as seen from the front
Example 2Calculator allowed
Rod on rails
A 0.50 m rod slides to the right at a steady 4.0 m/s along rails connected by a 3.0 Ω resistor at the left end. A 0.60 T field points into the page. Find the emf, the current, the magnetic force on the rod, and the power needed to keep it moving.
Show the solutionHide the solution
1. Step 1: ε = Bℓv = (0.60)(0.50)(4.0) = 1.2 V.
2. Step 2: I = ε/R = 1.2/3.0 = 0.40 A.
3. Step 3: Direction: the circuit's area, and the inward flux, are increasing, so the induced field inside points out of the page, which needs a counterclockwise current: up through the rod.
4. Step 4: Force on the rod: F = IℓB = (0.40)(0.50)(0.60) = 0.12 N. By the right-hand rule (current up, field into the page), it points left, against the motion.
5. Step 5: Power to keep it moving: P = Fv = (0.12)(4.0) = 0.48 W. Check: I²R = (0.40)²(3.0) = 0.48 W, so the work done ends up as thermal energy in the resistor.
Answer: 1.2 V; 0.40 A (counterclockwise); 0.12 N to the left; 0.48 W
Example 3
Moving through a uniform field (classic trap)
A square loop moves to the right at constant speed. It starts outside a region of uniform magnetic field, moves fully into it, travels through it, and then leaves. During which parts of the trip is a current induced?
Show the solutionHide the solution
1. Step 1: Entering: more and more of the loop's area is inside the field, so the flux increases. A current is induced.
2. Step 2: Fully inside: the flux is large but not changing, so there's no emf and no current. This is the trap; a strong field alone induces nothing.
3. Step 3: Leaving: the flux decreases, so a current is induced again, in the opposite direction to when it entered.
Answer: Only while entering and while leaving the field, with opposite current directions

Common mistakes

Thinking a strong magnetic field through a loop induces a current. Only a changing flux does.
Saying the induced field always opposes the outside field. It opposes the change: when flux is decreasing, the induced field points the same way as the outside field.
Using the wrong angle in Φ = BA cos θ. θ is measured from the area vector, which is perpendicular to the loop's surface.

On the exam

Expect Lenz's law direction questions with a magnet moving toward or away from a loop, or a field growing or shrinking; walk through the steps in words.
Rod-on-rails problems often combine Bℓv, Ohm's law, the force IℓB and energy conservation.
Graph questions may show flux against time and ask you to sketch the emf, which follows the slope of the flux graph.

Connected topics

Videos

Induced EMF Made Easy: Faraday’s Law & Lenz’s Law Explained | AP Physics 2 - Unit 12 Lesson 7
Allen Tsao The STEM CoachWatch on YouTube (opens in a new tab)
Electromagnetic Induction
Bozeman ScienceWatch on YouTube (opens in a new tab)
Induction - An Introduction: Crash Course Physics #34
CrashCourseWatch on YouTube (opens in a new tab)
Lenz's Law
Flipping PhysicsWatch on YouTube (opens in a new tab)
Faraday's Law Introduction | Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Moving Rod and Loop in Magnetic Field Made Easy | AP Physics 2 - Unit 12 - Lesson 8
Allen Tsao The STEM CoachWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 12.4 Electromagnetic Induction and Faraday’s Law. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A flat loop of area 0.050 m² is in a uniform 0.40 T magnetic field. The field makes an angle of 60° with the loop's area vector (the direction perpendicular to the loop's plane). What is the magnetic flux through the loop?

Question 2 of 4Calculator allowed

The magnetic flux through a single loop of wire falls steadily from 0.020 T·m² to zero in 0.10 s. What is the magnitude of the average induced emf?

Question 3 of 4Calculator allowed

The north end of a bar magnet is pushed toward a loop of wire, along the loop's axis. Viewed from the magnet's side, which way does the induced current flow in the loop?

Question 4 of 4Calculator allowed

A strong bar magnet sits at rest inside a coil of wire that is connected to a sensitive ammeter. What does the ammeter read?

0 of 4 answered