Faraday's Law — Electromagnetic Induction
Faraday's Law — Electromagnetic Induction
In 1831, Michael Faraday discovered one of the most important principles in all of physics: a changing magnetic field creates an electric field, and therefore an electromotive force that can drive a current. This discovery — electromagnetic induction — is the basis for electric generators, transformers, induction cooktops, wireless charging, and essentially all electrical power generation on Earth.
Faraday's law states that the induced EMF in a loop equals the negative rate of change of magnetic flux through the loop. Mathematically, the induced EMF epsilon equals negative d-phi-B over d-t, where phi-B is the magnetic flux defined as the integral of B dot dA over the surface bounded by the loop. For a flat loop of area A in a uniform field, the flux simplifies to phi equals B A cosine theta, where theta is the angle between the field and the normal to the loop.
There are three ways to change the flux and induce an EMF: change the magnetic field strength B, change the area A of the loop, or change the angle theta between the field and the loop. Electric generators exploit the third method — a coil rotates in a magnetic field, continuously changing theta and producing alternating current. Transformers use the first method — an alternating current in one coil creates a changing B that induces EMF in a nearby coil.
Lenz's law provides the direction of the induced current: it always flows in a direction that opposes the change in flux that caused it. This is a consequence of energy conservation — if the induced current enhanced the change, you would get runaway energy creation from nothing.
The mathematics of Faraday's law brings together calculus (rates of change), trigonometry (the cosine angular dependence), and algebra. These are the same tools you have been developing throughout this course, now applied to one of the most consequential discoveries in the history of science.
The induced EMF in a loop is equal to the negative rate of change of magnetic flux:
$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$
For $N$ loops: $\mathcal{E} = -N\frac{d\Phi_B}{dt}$
$$\Phi_B = BA\cos\theta$$
where $B$ is the field strength, $A$ is the loop area, and $\theta$ is the angle between $\vec{B}$ and the normal to the loop. Units: weber (Wb) $= $ T·m².
The direction of the induced current is such that its own magnetic field opposes the change in flux that produced it.
A circular loop of radius 10 cm is in a magnetic field that increases uniformly from 0.2 T to 0.8 T in 0.5 seconds. Find the induced EMF.
Step 1: Area: $A = \pi r^2 = \pi(0.10)^2 = 0.0314$ m²
Step 2: Change in flux: $\Delta\Phi = \Delta B \cdot A = (0.8 - 0.2)(0.0314) = 0.01885$ Wb
Step 3: EMF: $|\mathcal{E}| = \Delta\Phi / \Delta t = 0.01885 / 0.5 = 0.0377$ V $= 37.7$ mV
A generator coil has 100 turns with area 0.04 m² rotating at 60 Hz in a 0.5 T field. Find the peak EMF.
Step 1: Angular frequency: $\omega = 2\pi f = 2\pi(60) = 377$ rad/s
Step 2: Peak EMF for a rotating coil:
$$\mathcal{E}_{\max} = NAB\omega = (100)(0.04)(0.5)(377) = 754 \text{ V}$$
A rectangular loop (20 cm × 30 cm) is pulled at 5 m/s out of a 0.4 T field region. If the loop has resistance 2 Ω, find the induced current.
Step 1: As the loop is pulled out, the rate of flux change:
$$\frac{d\Phi}{dt} = B \cdot w \cdot v = (0.4)(0.20)(5) = 0.4 \text{ V}$$
where $w = 0.20$ m is the width of the side crossing the field boundary.
Step 2: Induced current: $I = \mathcal{E}/R = 0.4/2 = 0.2$ A
Practice Problems
Show Answer Key
1. $|\mathcal{E}| = N|\Delta\Phi/\Delta t| = 50 \times (0.6 \times 0.03)/0.02 = 45$ V
2. $\mathcal{E}_{\max} = NAB\omega = 200(0.1)(0.8)(2\pi \times 50) = 5{,}027$ V
3. $I = 0.3/5 = 0.06$ A. Direction opposes the flux change (Lenz's law).
4. $|\mathcal{E}| = NA(dB/dt) = 500(0.02)(0.04) = 0.4$ V
5. If current enhanced the change, induced B would increase flux further, creating more EMF, more current — violating energy conservation.
6. $\mathcal{E} = BLv = (0.6)(0.25)(3) = 0.45$ V