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## Physics library

### Course: Physics library>Unit 13

Lesson 4: Magnetic flux and Faraday's law

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 E induced in the loop. The relationship is
E, equals, start fraction, d, \Phi, divided by, d, t, end fraction
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.
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,
E, equals, minus, start fraction, d, \Phi, divided by, d, t, end fraction
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.
E, equals, minus, N, start fraction, d, \Phi, divided by, d, t, end fraction

# 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?
Exercise 1b:
If the magnet is dropped north-pole first, what direction (clockwise or counterclockwise) will the current first flow in the coil?
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.

# 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.

# 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 
Figure 8: Construction of a typical transformer 
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, start subscript, s, end subscript depends on the alternating voltage across the primary coil V, start subscript, p, end subscript and the ratio of the turns in the primary and secondary coils (N, start subscript, s, end subscript, slash, N, start subscript, p, end subscript). Because energy is conserved, the maximum current available increases when the voltage is stepped down.
V, start subscript, s, end subscript, equals, V, start subscript, p, end subscript, start fraction, N, start subscript, s, end subscript, divided by, N, start subscript, p, end subscript, end fraction

1. By Peripitus GFDL or CC BY-SA 4.0-3.0-2.5-2.0-1.0, via Wikimedia Commons
2. OpenStax Physics

## Want to join the conversation?

• There is a direction of the current indicated with an arrow on Figure 1 ("Charge in a moving wire."). Shouldn't the arrow point in the opposite direction? I used the formula V = L v × B and the right-hand rule. Thank you. • I think the figure is very misleading, or maybe just plain wrong, depending on where we are supposed to understand the magnetic field is located. They have TWO black arrows on the wires, indicating the current going in a loop. The current is shown going "southwest" in the wire near us, and "northeast" in the wire far from us. For any wire moving in the direction indicated, the induced current will be SOUTHWEST only. So if ONLY the near wire is "in" the B field, then the diagram is actually OK. But if the entire loop (both parallel wires) is in the B field, then there would be no induced current. The current would be trying to flow southwest in both wires.

Because they only draw B field arrows along a single line which looks like it is BETWEEN the wires, it is a very ambiguous figure.
• Question on inducing current in the coil
I cannot understand why the emf produced in the coil be opposite after the magnet has moved halfway through the coil. Thank you • I don't understand Faraday's experiment : Induction from a magnet moving through a coil . Can some one explain it better for me please
(1 vote) • I'll try to explain it
Consider that the magnet's north pole moves towards the coil. Upper end of the coil acquires north polarity, hence work is done against the force of repulsion to move the magnet.
If the magnet is withdrawn from the coil upper end acquires south polarity, so work is done against the force of attraction.

Now we know that as the magnet moves through the coil magnetic flux linked with the coil changes inducing a current. The direction of the induced current is found from Lenz' law as follows.

The work done in moving the magnet towards the coil is converted into electrical energy, which gets dissipated into heat energy.The current flows in a direction to oppose the motion of the magnet. If in case the induced current promotes the motion of the magnet, it stars moving at a faster rate and the electric energy(induced) and kinetic energy(of the magnet) starts increasing, without any work done. This is not possible (law of conservation of energy) .
So induced current always flows in a direction that oppose the motion of the magnet.This is Lenz's law.

pheww....Hope it helped!
• If the rate of change plays a role in the amount of induced voltage then why different frequencies of AC voltage still get transformed by the same amount by a transformer? Wouldn't 120hz AC came out higher than 60hz AC on the other end of the transformer? (like lets say we connect 10V 60hz AC and 10V 120hz AC to a 1:2 transformer, why both of these voltages end up 20V at the secondary?) • I am not able to understand the difference between potential difference and electro motive force.
(1 vote) • Hello Aryan,

Here is an answer from an electrical engineer's perspective - it may not line up with a physicists answer.

If you have a voltmeter in had you could measure voltage as the difference between any two point in a circuit. The red probe goes to one part of the circuit and the black probe goes to another. It does not matter what is being measured. It could be a battery, solar cell, or coil of wire as shown in this video. We would say the the meter is measuring the potential difference between the two points. This is often called voltage.

Forgive me as I add an analogy that would appear not to belong. Suppose you are riding a bicycle on a level road. You would have a "no-load" speed. As you started to climb a hill you will experience an increased load and start to slow down.

The same thing happens to sources of electrical energy. It will have a certain "no-load" voltage that usually drops when a load is applied. For example, a battery may have a no-load voltage of 13 VDC that drops to 12 VDC when a load is applied.

Regards,

APD
• what is the difference between moving the magnet through the coil from the right side versus the left side? • Why an approaching north pole induces a counter-clockwise current with respect to the bar magnet as shown in ex. 1b.    