Learn what Faraday's law means and how to use it to determine the induced electro-motive force.

What is electromagnetic induction?

Electromagnetic induction is the process by which a current can be induced to flow due to a changing magnetic field.
In our article on the magnetic force we looked at the force experienced by moving charges in a magnetic field. The force on a current-carrying wire due to the electrons which move within it when a magnetic field is present is a classic example. This process also works in reverse. Either moving a wire through a magnetic field or (equivalently) changing the strength of the magnetic field over time can cause a current to flow.

How is this described?

There are two key laws that describe electromagnetic induction:
1. Faraday's law, due to 19ᵗʰ century physicist Michael Faraday. This relates the rate of change of magnetic flux through a loop to the magnitude of the electro-motive force $\mathcal{E}$ induced in the loop. The relationship is
$\mathcal{E} = \frac{\mathrm{d}\Phi}{\mathrm{d}t}$
The electromotive force or EMF refers to the potential difference across the unloaded loop (i.e. when the resistance in the circuit is high). In practice it is often sufficient to think of EMF as voltage since both voltage and EMF are measured using the same unit, the volt.
The take-away point here is that a wire moving in a field does not necessarily represent an ideal voltage source; the voltage you might measure with a high-impedance voltmeter would only equal the EMF if the load is small.
2. Lenz's law is a consequence of conservation of energy applied to electromagnetic induction. It was formulated by Heinrich Lenz in 1833. While Faraday's law tells us the magnitude of the EMF produced, Lenz's law tells us the direction that current will flow. It states that the direction is always such that it will oppose the change in flux which produced it. This means that any magnetic field produced by an induced current will be in the opposite direction to the change in the original field.
Lenz's law is typically incorporated into Faraday's law with a minus sign, the inclusion of which allows the same coordinate system to be used for both the flux and EMF. The result is sometimes called the Faraday-Lenz law,
$\mathcal{E} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t}$
In practice we often deal with magnetic induction in multiple coils of wire each of which contribute the same EMF. For this reason an additional term $N$ representing the number of turns is often included, i.e.
$\mathcal{E} = -N \frac{\mathrm{d}\Phi}{\mathrm{d}t}$

What is the connection between Faraday's law of induction and the magnetic force?

While the full theoretical underpinning of Faraday's law is quite complex, a conceptual understanding of the direct connection to the magnetic force on a charged particle is relatively straightforward.
Figure 1: Charge in a moving wire.
Figure 1: Charge in a moving wire.
Consider an electron which is free to move within a wire. As shown in figure 1, the wire is placed in a vertical magnetic field and moved perpendicular to the magnetic field at constant velocity. Both ends of the wire are connected, forming a loop. This ensures that any work done in creating a current in the wire is dissipated as heat in the resistance of the wire.
A person pulls the wire with constant velocity through the magnetic field. As they do so, they have to apply a force. The constant magnetic field can’t do work by itself (otherwise its strength would have to change), but it can change the direction of a force. In this case some of the force that the person applies is re-directed, causing an electromotive force on the electron which travels in the wire, establishing a current. Some of the work the person has done pulling the wire ultimately results in energy dissipated as heat within the resistance of the wire.

Faraday's experiment : Induction from a magnet moving through a coil

The key experiment which lead Michael Faraday to determine Faraday's law was quite simple. It can be quite easily replicated with little more than household materials. Faraday used a cardboard tube with insulated wire wrapped around it to form a coil. A voltmeter was connected across the coil and the induced EMF read as a magnet was passed through the coil. The setup is shown in Figure 2.
Figure 2: Faraday's experiment: a magnet is passed through a coil.
Figure 2: Faraday's experiment: a magnet is passed through a coil.
The observations were as follows:
1. Magnet at rest in or near the coil: No voltage observed.
2. Magnet moving toward the coil: Some voltage measured, rising to a peak as the magnet nears the center of the coil.
3. Magnet passes through the middle of the coil: Measured voltage rapidly changes sign.
4. Magnet passes out and away from the coil: Voltage measured in the opposite direction to the earlier case of the magnet moving into the coil.
An example of the EMF measured is plotted against magnet position in Figure 3.
These observations are consistent with Faraday's law. Although the stationary magnet might produce a large magnetic field, no EMF can be induced because the flux through the coil is not changing. When the magnet moves closer to the coil the flux rapidly increases until the magnet is inside the coil. As it passes through the coil the magnetic flux through the coil begins to decrease. Consequently, the induced EMF is reversed.
Exercise 1a:
A small 10 mm diameter permanent magnet produces a field of 100 mT. The field drops away rapidly with distance and is negligible more than 1 mm from the surface. If this magnet moves at a speed of 1 m/s through a 100-turn coil of length 1 mm and diameter just larger than the magnet, what is the EMF induced?
We can use Faraday's law of induction to find the induced EMF. This requires us to know the change in flux through the coil and how quickly the change happens.
We can start by separately considering the cases when the magnet is outside and inside the coil. Since we are told the field decays rapidly, we can assume the flux is zero when the magnet is outside the coil. Because the coil is a tight fit around the magnet we can assume that the field is always orthogonal to the coil and the flux is
$\Phi = BA$
Because the magnet is known to be moving at 1000 mm/s, we know that it will be inside the 1 mm long coil for just 1/1000 s (1 ms). So applying Faraday's law,
\begin{aligned}\mathcal{E} &= -N \frac{\mathrm{d}\Phi}{\mathrm{d}t} \\ &= -(100~\mathrm{turns})\frac{(100\cdot 10^{-3}~\mathrm{T})\pi (5\cdot 10^{-3}~\mathrm{m})^2}{1\cdot 10^{-3}~\mathrm{s}}\\&\simeq 0.78~\mathrm{V}\end{aligned}
Exercise 1b:
If the magnet is dropped north-pole first, what direction (clockwise or counterclockwise) will the current first flow in the coil?
According to Lenz's law, the field produced by the induced current must oppose the changing flux which produced it. Applying the right-hand-grip rule we see that the current in the coil must initially be counter-clockwise when viewed from above to achieve this as shown in Figure 4.
Figure 4: Direction of induced current due to Lenz's law.
Figure 4: Direction of induced current due to Lenz's law.
Exercise 1c:
Suppose the ends of the coil are electrically connected to each other, ensuring that any current generated is dissipated as heat in the resistance of the wires. What effect would you expect this to have on the falling magnet? Hint: consider conservation of energy.
The energy lost to heat must have come from the falling magnet. Therefore, the velocity of the falling magnet must decrease as it travels through the coil. This would be consistent with the effect of a repulsive force from the two opposing magnetic fields. Interestingly, this effect can occur in any conductor moving through a magnetic field. This effect has many applications in engineering where it is known as eddy-current braking and employed on trains and amusement park rides.

Induction in parallel wires

If a pair of wires are set parallel to one another it is possible for a changing current in one of the wires to induce an EMF pulse in the neighboring wire. This can be a problem when the current flowing in neighboring wires represents digital data. Ultimately this effect can limit the rate at which data can be reliably sent in this manner.
Exercise 2:
Figure 5 shows a pair of parallel wires. One is connected to a battery via a switch and current meter while its neighbor forms a loop with just a current meter in series. Suppose the switch is briefly switched on then off. Qualitatively speaking, what will happen to the current measured in the neighbor?
Figure 6: Current pulses due to induction between parallel wires.
Figure 6: Current pulses due to induction between parallel wires.
When the switch is closed current begins to flow down through the left wire. This causes a magnetic field to build up around the wire and along its length according to the right-hand-grip rule. Some of this field comes out of the page and intersects the right wire. The resulting change in flux causes a current to be briefly induced until the magnetic field stops changing. This current consists of a pulse which is maximized just when the switch is thrown because at this time the field is increasing at the maximum rate. The current does not depend on the strength of the magnetic field, just the rate that it is changing.
By Lenz's law, the direction of the current pulse in the right wire is opposite to that in the left wire. When the switch is opened again the process is repeated in reverse. Figure 6 show how the current in the two wires would change as a result of the switch opening and closing.
Figure 6: Current pulses due to induction between parallel wires.
Figure 6: Current pulses due to induction between parallel wires.

What is a transformer?

In the simplest form, a transformer is simply a pair of coils wound on the same core. The core is often shaped as a square loop with primary and secondary coils wound on opposite sides. The construction of a transformer allows the magnetic flux generated by a current changing in one coil to induce a current in the neighboring coil.
Figure 8: Construction of a typical transformer [2]
Figure 8: Construction of a typical transformer [2]
Large transformers are a key component of the electrical distribution system. They are especially useful because the number of turns on each coil does not need to be the same. Because the EMF induced depends on the number of turns, transformers allows the voltage of an alternating current to be drastically stepped up or down. This is crucial as it allows high voltages to be used to efficiently distribute power over long distance with much safer lower voltages made available to consumers.
For a transformer with no losses, the alternating voltage generated across a secondary coil $V_s$ depends on the alternating voltage across the primary coil $V_p$ and the ratio of the turns in the primary and secondary coils ($N_s/N_p$). Because energy is conserved, the maximum current available increases when the voltage is stepped down.
$V_s = V_p \frac{N_s}{N_p}$