20.3 Electromagnetic Induction

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field produces an electric current. This single idea is the foundation for electric generators, transformers, and wireless chargers. The two laws that govern it, Faraday's law and Lenz's law, tell you how much emf gets induced and which direction the resulting current flows.

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Production of Currents by Magnetic Fields

A static magnetic field won't induce a current. The field has to be changing relative to the conductor. Faraday's law of induction states that a changing magnetic flux through a loop induces an electromotive force (emf) in that loop, and the induced emf is proportional to the rate at which the flux changes.

Magnetic flux ($\Phi_B$) measures how much magnetic field passes through a given area. It's calculated as:

$\Phi_B = BA\cos\theta$

where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the line perpendicular (normal) to the surface.

• When the field is perfectly perpendicular to the loop's surface ($\theta = 0°$), $\cos\theta = 1$ and flux is maximized.
• When the field runs parallel to the surface ($\theta = 90°$), $\cos\theta = 0$ and no flux passes through.
• A stronger field, a larger loop, or a more favorable angle all increase the flux.

The induced emf drives a current whenever the loop forms a closed circuit. The magnitude of that current depends on the loop's resistance, just as with any other voltage source.

Calculation of Induced EMF and Current

Faraday's law in equation form:

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

where $N$ is the number of turns in the coil and $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux. For a single loop, $N = 1$. The negative sign reflects Lenz's law (covered in the next section) and indicates that the emf opposes the change in flux.

Once you know the induced emf, you find the induced current using Ohm's law:

$I = \frac{\mathcal{E}}{R}$

where $R$ is the resistance of the loop.

Three ways to change the flux (and therefore induce an emf):

• Change $B$: Rapidly increasing or decreasing the magnetic field strength. For example, pushing a stronger magnet toward a coil induces a larger emf than pushing a weak one.
• Change $A$: Altering the area of the loop that's exposed to the field. In a generator, a rotating coil effectively changes the cross-sectional area intercepting the field.
• Change $\theta$: Rotating the loop relative to the field changes $\cos\theta$. A coil spinning in a uniform magnetic field produces a continuously changing emf, which is exactly how AC generators work.

The faster any of these quantities change, the larger the induced emf. And more turns ($N$) multiply the effect, which is the principle behind transformers.

Direction of Induced Currents

Lenz's law tells you which way the induced current flows: it always flows in the direction that opposes the change in flux that caused it. This is a direct consequence of conservation of energy. If the induced current helped the change instead of opposing it, you'd get runaway energy from nothing.

Here's how to apply Lenz's law step by step:

1. Determine whether the flux through the loop is increasing or decreasing. Consider the direction of the external magnetic field and how it's changing.
2. Find the direction of the induced magnetic field. If the flux is increasing, the induced current must create a magnetic field that points opposite to the external field (to fight the increase). If the flux is decreasing, the induced current creates a field in the same direction as the external field (to fight the decrease).
3. Use the right-hand rule to find the current direction. Curl the fingers of your right hand in the direction the current would need to flow so that your thumb points in the direction of the induced magnetic field you identified in step 2.

Electromagnetic Energy and Forces

Electromagnetic induction is fundamentally an energy conversion process. When you move a magnet toward a coil, you're converting mechanical energy (the work you do pushing the magnet) into electrical energy (the induced current). Generators at power plants do this on a massive scale, spinning coils through magnetic fields to produce the current that powers the grid.

The Lorentz force ($\vec{F} = q\vec{v} \times \vec{B}$) is the underlying mechanism. When a conductor moves through a magnetic field, the free charges inside it experience this force, which pushes them along the wire and creates the current. The opposition described by Lenz's law shows up as a real mechanical force: generators require continuous input of energy (from turbines, engines, etc.) precisely because the induced current creates a magnetic force that resists the motion causing it.