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Topic 10 of 15

Electromagnetic Induction

Faraday's discovery that a changing magnetic field induces an electric field — and therefore a current in any nearby conductor — is arguably the most consequential physics result of the 19th century. It is the operating principle behind every generator, transformer, electric guitar pickup, and induction cooktop in existence. Lenz's law tells us the direction of the induced effect, and it always acts to oppose the change that caused it.

Key Concepts

Magnetic Flux

Magnetic flux through a surface:

\Phi_B = BA\cos\theta

, where

\theta

is the angle between

\vec{B}

and the outward normal to the surface. For non-uniform fields:

\Phi_B = \int\vec{B}\cdot d\vec{A}

. Units: weber (Wb = T·m²).

Faraday's Law

The induced EMF in a loop equals the negative rate of change of magnetic flux through it:

\mathcal{E} = -N\,d\Phi_B/dt

, where

N

is the number of turns. EMF is induced whenever

B

A

, or

\theta

changes with time.

Lenz's Law

The induced current flows in a direction such that the magnetic field it creates opposes the change in flux that induced it. This is a consequence of energy conservation — you must do work to maintain the change against the opposing induced effect.

Motional EMF

When a conductor of length

L

moves with velocity

v

perpendicular to a magnetic field

B

, free charges experience a magnetic force that creates an EMF:

\mathcal{E} = BLv

. This is the basis of electric generators.

AC Generators

A coil of

N

turns and area

A

rotating at angular velocity

\omega

in a field

B

produces a sinusoidal EMF:

\mathcal{E}(t) = NBA\omega\sin(\omega t)

. Peak EMF:

\mathcal{E}_0 = NBA\omega

Key Equations

Magnetic flux

\Phi_B = BA\cos\theta

Flux through a flat surface of area A in uniform field B; θ is the angle between B and the surface normal. Unit: weber (Wb).

Faraday's Law

\mathcal{E} = -N\frac{d\Phi_B}{dt}

Induced EMF in a coil of N turns. The negative sign encodes Lenz's law — the induced EMF opposes the flux change.

Motional EMF

\mathcal{E} = BLv

EMF induced in a conductor of length L moving at speed v perpendicular to field B.

Generator peak EMF

\mathcal{E}_0 = NBA\omega

Peak EMF of a generator coil with N turns, area A, rotating at angular velocity ω in field B.

Worked Example

Motional EMF and Current in a Sliding Rod

Problem

A conducting rod of length $L = 0.30$ m slides along frictionless rails in a uniform $B = 0.80$ T field perpendicular to the rail plane, with speed $v = 4.0$ m/s. A resistor $R = 2.0\,\Omega$ connects the rails. Find (a) the induced EMF, (b) the current, and (c) the braking force on the rod.

Solution

(a) Motional EMF from the moving rod:

\mathcal{E} = BLv = (0.80)(0.30)(4.0) = 0.96\text{ V}

(b) Current through the resistor:

I = \frac{\mathcal{E}}{R} = \frac{0.96}{2.0} = 0.48\text{ A}

F = BIL = (0.80)(0.48)(0.30) = 0.115\text{ N}

Answer EMF = 0.96 V; I = 0.48 A; braking force = 0.115 N.

Practice

Exercises

7 problems

1 of 7

A flat loop of area $A = 0.040$ m² sits in a uniform $B = 0.50$ T field. The normal to the loop makes an angle $\theta = 30°$ with $\vec{B}$ . What is the magnetic flux (in mWb) through the loop?

Your answer

mWb

2 of 7

The flux through a single-turn coil drops from $\Phi_1 = 0.080$ Wb to $\Phi_2 = 0.020$ Wb in $\Delta t = 0.040$ s. What is the magnitude of the induced EMF (in V)?

Your answer

3 of 7

A conducting rod of length $L = 0.50$ m moves at $v = 3.0$ m/s perpendicular to a uniform $B = 0.60$ T field. What is the motional EMF (in V)?

Your answer

4 of 7

The rod from Exercise 3 slides along rails connected by a resistor $R = 3.0\,\Omega$ . What current (in A) flows through the resistor?

Your answer

5 of 7

A coil of $N = 200$ turns and area $A = 0.010$ m² sits in a field that increases uniformly from $B = 0$ to $B = 0.50$ T in $\Delta t = 0.10$ s. What is the magnitude of the induced EMF (in V)?

Your answer

6 of 7

The sliding rod from Exercise 3–4 carries the induced current in the $B = 0.60$ T field. What is the magnitude of the braking force (in N) on the rod?

Your answer

7 of 7

A generator coil has $N = 100$ turns, area $A = 0.025$ m², rotates at $\omega = 80$ rad/s in a field $B = 0.40$ T. What is the peak EMF (in V)?

Your answer

Key Takeaways

Faraday's law: any change in magnetic flux through a loop induces an EMF. The flux can change because $B$ , $A$ , or $\theta$ changes.
Lenz's law (the minus sign in Faraday's law): the induced current always opposes the change that created it — energy is not free.
Motional EMF $\mathcal{E} = BLv$ : the rod is effectively a battery, with EMF proportional to all three of $B$ , $L$ , and $v$ .
Generators rotate a coil in a field; the sinusoidal output $\mathcal{E}_0\sin(\omega t)$ is why household current is AC.
Eddy currents are induced in bulk conductors by changing flux; they dissipate energy as heat (used in induction braking and cooking).