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Electromagnetic
Induction

Discover how changing magnetic fields generate electric currents: the principle behind generators, transformers, and modern technology.

High School Physics

What Is Electromagnetic Induction?

Electromagnetic induction is the process by which a changing magnetic flux through a conductor produces an electromotive force (EMF), which can drive an electric current. It was discovered independently by Michael Faraday (1831) and Joseph Henry (1832).

Understanding Magnetic Flux \(\Phi_B\)

Magnetic flux measures how much magnetic field passes through a given area. It depends on three factors: field strength, area, and the angle between them.

A B n \(\theta\)
\(\Phi_B = BA\cos\theta\)
Units: Weber (Wb)

Why Does Induction Matter?

Electromagnetic induction is the backbone of modern electrical power. Nearly all electricity used worldwide is generated by rotating coils inside magnetic fields.

Generators
Convert mechanical energy into electrical energy by spinning a coil in a magnetic field.
Transformers
Step voltage up or down using mutual induction between two coils sharing a core.
Wireless Charging
A changing current in one coil induces current in a nearby coil inside your phone.

Faraday's Law of Induction

Faraday's Law quantifies the induced EMF. The magnitude of the EMF equals the rate of change of magnetic flux through the circuit:

Visualizing Faraday's Law

N-turn coil N S v \(\Phi_B\) increasing \(\mathcal{E} = -N\frac{\Delta\Phi_B}{\Delta t}\) G

Lenz's Law: Nature Resists Change

Lenz's Law states that the direction of the induced current is always such that it opposes the change in flux that produced it. This is the physical meaning of the negative sign in Faraday's Law.

Flux increasing N v B induced I Flux decreasing N v B induced I

Faraday's Law vs. Lenz's Law

Faraday's Law Lenz's Law
What it tells us Magnitude of induced EMF Direction of induced current
Expressed as \(\lvert\mathcal{E}\rvert = N \frac{\Delta\Phi_B}{\Delta t}\) The negative sign in \(\mathcal{E} = -N\frac{\Delta\Phi_B}{\Delta t}\)
Physical basis Experimental observation Conservation of energy
Use case Calculate voltage produced Predict current direction

Three Ways to Change Flux

Since \(\Phi_B = BA\cos\theta\), there are exactly three variables you can change to induce an EMF:

Change B
Increase or decrease the magnetic field strength (move a magnet closer or farther).
Change A
Expand or shrink the area of the loop (stretch a wire loop, or slide a rail).
Change \(\theta\)
Rotate the coil relative to the field (this is how generators work).

How to Solve Induction Problems

1
Identify the loop. What conductor forms the closed path? Count the turns \(N\).
2
Calculate initial and final flux. Use \(\Phi_B = BA\cos\theta\) at both time points.
3
Find the change. Compute \(\Delta\Phi_B = \Phi_{B,f} - \Phi_{B,i}\) and the time interval \(\Delta t\).
4
Apply Faraday's Law. \(\mathcal{E} = -N\,\Delta\Phi_B / \Delta t\).
5
Determine direction via Lenz's Law. The induced current opposes the flux change.

Example 1: Simple Coil

A single circular loop of area \(0.05\;\text{m}^2\) sits in a uniform magnetic field that decreases from \(0.8\;\text{T}\) to \(0.2\;\text{T}\) in \(0.1\;\text{s}\). The field is perpendicular to the loop. Find the induced EMF.

1
Given: \(N = 1\), \(A = 0.05\;\text{m}^2\), \(\theta = 0°\), \(B_i = 0.8\;\text{T}\), \(B_f = 0.2\;\text{T}\), \(\Delta t = 0.1\;\text{s}\).
2
Flux change: \(\Delta\Phi_B = (B_f - B_i) \cdot A \cdot \cos 0° = (0.2 - 0.8)(0.05)(1) = -0.03\;\text{Wb}\).
3
Apply Faraday's Law: \(\mathcal{E} = -N\,\frac{\Delta\Phi_B}{\Delta t} = -(1)\frac{-0.03}{0.1} = +0.3\;\text{V}\).

Example 2: Multi-Turn Coil

A coil with \(N = 200\) turns and area \(0.04\;\text{m}^2\) is placed in a magnetic field that increases uniformly from \(0\;\text{T}\) to \(0.5\;\text{T}\) in \(2\;\text{s}\). The field is perpendicular to the coil. Find the induced EMF.

1
Flux change: \(\Delta\Phi_B = (0.5 - 0)(0.04)(1) = 0.02\;\text{Wb}\).
2
Apply Faraday's Law: \(\mathcal{E} = -200 \times \frac{0.02}{2} = -2\;\text{V}\).
3
Magnitude: \(\lvert\mathcal{E}\rvert = 2\;\text{V}\). The negative sign tells us the direction of current opposes the increasing flux.

Example 3: Rotating Coil (Generator)

A 50-turn coil of area \(0.1\;\text{m}^2\) rotates in a \(0.4\;\text{T}\) field. It rotates from \(\theta = 0°\) to \(\theta = 90°\) in \(0.25\;\text{s}\). Find the average induced EMF.

1
Initial flux: \(\Phi_{B,i} = (0.4)(0.1)\cos 0° = 0.04\;\text{Wb}\).
2
Final flux: \(\Phi_{B,f} = (0.4)(0.1)\cos 90° = 0\;\text{Wb}\).
3
EMF: \(\mathcal{E} = -50 \times \frac{0 - 0.04}{0.25} = -50 \times (-0.16) = 8\;\text{V}\).

The AC Waveform from a Generator

When a coil rotates at constant speed in a uniform field, the induced EMF varies sinusoidally: \(\mathcal{E}(t) = \mathcal{E}_0 \sin(\omega t)\).

t \(\mathcal{E}\) \(\mathcal{E}_0\) \(-\mathcal{E}_0\) T/4 T/2 3T/4 T max min

Motional EMF: A Sliding Rail

Sliding bar v ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ B (into page) L R

Common Mistakes to Avoid

✕ Static field means EMF
A constant magnetic field through a stationary loop produces zero EMF. Only changes in flux matter.
✕ Forgetting the angle
Flux is \(BA\cos\theta\), not just \(BA\). If the field is parallel to the loop (\(\theta = 90°\)), flux is zero.
✕ Ignoring N
Forgetting to multiply by the number of turns \(N\). A 100-turn coil produces 100 times the EMF of a single loop.

Summary of Key Formulas

Concept Formula Units
Magnetic flux \(\Phi_B = BA\cos\theta\) Weber (Wb)
Faraday's Law \(\mathcal{E} = -N \frac{\Delta\Phi_B}{\Delta t}\) Volts (V)
Motional EMF \(\mathcal{E} = BLv\) Volts (V)
Generator peak EMF \(\mathcal{E}_0 = NBA\omega\) Volts (V)
Lenz's Law Induced current opposes flux change Direction rule

Question 1 of 5

A magnetic field through a coil increases over time. According to Lenz's Law, the induced current will:

Question 2 of 5

A coil with 50 turns and area \(0.02\;\text{m}^2\) experiences a uniform magnetic field change from \(0.6\;\text{T}\) to \(0\;\text{T}\) in \(0.3\;\text{s}\). The field is perpendicular to the coil. What is the magnitude of the induced EMF?

Question 3 of 5

A wire loop is placed in a constant, uniform magnetic field with the field perpendicular to the loop. The loop is stationary. What is the induced EMF?

Question 4 of 5

A conducting bar of length \(L = 0.5\;\text{m}\) slides at velocity \(v = 4\;\text{m/s}\) along rails in a field of \(B = 0.3\;\text{T}\) (perpendicular, into the page). What is the motional EMF?

Question 5 of 5

Which of the following will NOT induce an EMF in a closed loop of wire?