Electromagnetic Induction and Faraday’s Law

A changing magnetic field through a circuit can produce a voltage across it. This voltage is called an induced electromotive force (emf), and its production is called electromagnetic induction. An induced current flows if the circuit is closed.

For example, moving a magnet relative to a coil can change the magnetic field through the coil and induce an emf. A changing magnetic field can also induce an emf in a stationary coil. In each case, the emf is produced by a change in the coil’s magnetic flux linkage. If the flux linkage stays constant, no emf is induced.

Magnetic flux and flux linkage

Magnetic flux measures how much magnetic field passes through an area. For a flat area in a uniform magnetic field, it is given by:

Here $$\Phi$$ is magnetic flux in webers, $$B$$ is magnetic flux density, $$A$$ is the area, and $$\theta$$ is the angle between the magnetic field and the normal to the area. The normal is a line perpendicular to the area.

When the field is perpendicular to the area, it is parallel to the normal, so $$\theta = 0$$ and the flux is $$BA$$. When the field is parallel to the area, $$\theta = 90^\circ$$ and the flux is zero.

For a coil with $$N$$ turns and the same flux $$\Phi$$ through each turn, the flux linkage is $$N\Phi$$.

Faraday’s law

Faraday’s law states that the magnitude of the induced emf equals the rate of change of magnetic flux linkage. For a coil of $$N$$ turns with the same flux through each turn, the magnitude of the average induced emf over a time interval is:

Here $$\varepsilon$$ is the induced emf, $$\Delta\Phi$$ is the change in flux through each turn, and $$\Delta t$$ is the time taken. The expression also gives the induced emf when the flux changes at a uniform rate.

A faster change in flux linkage produces a greater induced emf. For example, moving a magnet faster relative to a coil can produce a larger emf because the flux linkage changes more rapidly.

Lenz’s Law

When the magnetic flux linkage of a coil changes, an emf is induced across it. Magnetic flux linkage describes how much magnetic field passes through the turns of the coil. Lenz’s law determines the direction of the induced emf: its effect opposes the change in flux linkage that produces it.

If the circuit is closed, the induced emf drives a current. This current produces its own magnetic field, which opposes the change in flux linkage. The induced field does not necessarily oppose the original field:

• If the original flux linkage is increasing, the induced field acts against the original field, opposing the increase.
• If the original flux linkage is decreasing, the induced field acts in the same direction as the original field, opposing the decrease.

For example, pushing a magnet towards a closed coil increases the flux linkage through it. The induced current produces a field that opposes the increase. Pulling the magnet away decreases the flux linkage, so the induced emf and current reverse direction and the induced field opposes the decrease.

Lenz’s law and energy conservation

The opposition is consistent with conservation of energy. When a magnet is moved towards or away from a closed coil, the induced current produces a magnetic effect that opposes the motion. Work must be done to continue moving the magnet; this supplies the energy transferred to the circuit. If the induced effect assisted the change instead, the change could grow without an external energy supply.

Emf Induced in a Moving Conductor

A straight conductor moving through a magnetic field can develop a potential difference between its ends. The charges in the conductor move through the field with it, so they experience a magnetic force.

For a conductor at right angles to both the magnetic field and its direction of motion, this force acts along the conductor. Charges separate: one end becomes negatively charged and the other positively charged. The resulting potential difference is the induced emf. It can exist even when the conductor is not part of a complete circuit.

As charge builds up at the ends, it creates an electric field inside the conductor. The electric force opposes further charge separation. The separation stops increasing when the electric and magnetic forces balance.

Magnitude of the induced emf

For a conductor of length $$l$$ moving at speed $$v$$ through a uniform magnetic field of flux density $$B$$, the induced emf is:

This relationship applies when the motion is perpendicular to the field and the conductor is perpendicular to both the field and the motion. Here $$\varepsilon$$ is in volts, $$B$$ is in teslas, $$l$$ is in metres and $$v$$ is in metres per second.

A stronger field, a longer conductor or a greater speed produces a larger induced emf, provided the stated orientations remain the same.

Polarity of the induced emf

Reversing the direction of motion reverses the magnetic force on the charges. The charges therefore separate in the opposite direction, so the polarity of the induced emf reverses. Reversing the magnetic field direction, while keeping the motion unchanged, also reverses the polarity.

Emf Induced in a Rotating Coil

As a coil rotates in a magnetic field, its orientation to the field changes. This changes the magnetic flux through the coil and induces an emf. If the coil rotates continuously, the induced emf repeatedly changes direction: it is an alternating emf.

Consider a coil of $$N$$ turns, each with area $$A$$, rotating at constant angular speed $$\omega$$ in a uniform magnetic field of flux density $$B$$. The axis of rotation is perpendicular to the field. A normal to the coil is a line perpendicular to its plane.

Flux linkage of the rotating coil

If the coil’s normal is aligned with the field at $$t=0$$, the angle between the normal and the field after time $$t$$ is $$\omega t$$. The flux linkage—the magnetic flux through one turn multiplied by the number of turns—is then:

The flux linkage is initially at its maximum. As the coil turns, it falls to zero, reaches a minimum when the normal points opposite to the field, and then rises again.

Induced emf of the rotating coil

The induced emf depends on the rate of change of flux linkage. Using the terminal orientation for which $$\varepsilon=-\dfrac{d(N\Phi)}{dt}$$ gives:

The opposite choice of terminal orientation gives the opposite sign. The maximum magnitude is:

The emf is zero when the flux linkage is at a maximum or minimum, because the flux linkage is momentarily not changing. The emf has maximum magnitude when the flux linkage passes through zero, where it changes most rapidly.

During the next half-turn, the flux linkage changes in the opposite direction, so the emf reverses. This repeated reversal produces the alternating emf.