In this explainer, we will learn how to describe the electric current induced in a wire that is placed in a changing magnetic field.
Consider a loop of conducting wire as shown below.
This wire carries no current because there is no potential difference between any two points in the wire causing charge to flow.
It is possible, however, to create a current in this loop—without using a cell or battery.
Imagine that we position a bar magnet near the wire as follows.
Some of the magnetic field lines from the magnet pass through the wire loop as shown below.
Still, if the magnet and the wire remain stationary, there will be no current in the wire.
However, if the magnet is set in motion, then the magnetic field experienced by the loop will change.
That is, when the loop lies in a different part of the magnet’s field, the number of field lines passing through the loop—or the direction of the field lines, or both—will be different. This is what we mean when we say the magnetic field through the wire is changing.
The changing magnetic field through the loop creates a current in the wire. This process is called electromagnetic induction.
Definition: Electromagnetic Induction
When a loop of conducting wire is exposed to a changing magnetic field, a current will be induced in the wire.
There are many ways for a wire to experience a changing magnetic field.
Consider again our bar magnet and wire loop, this time with field lines omitted for clarity. We have seen that if they both are stationary, no current is induced, as shown below.
However, if the magnet is in motion (in any direction) while the wire remains still, the magnetic field through the wire will change over time and current will be induced in the wire as follows.
It is also possible for the magnet to be stationary while the wire moves as shown below.
Once again, current is induced in the wire. Note that the wire could be moving in any direction relative to the magnet and current would still be induced.
As long as the magnetic field source and the wire are moving relative to one another, current is induced in the wire.
Interestingly, the magnitude of current induced depends on how quickly the field through the wire loop changes.
For example, imagine that the wire is stationary, while the magnet moves very slowly toward it, as depicted below.
The magnetic field through the wire will change slowly, and the current induced in the wire will be small.
However, if the magnet instead moves very quickly toward the wire, a greater current will be induced as follows.
This greater induced current is caused by a greater rate of change of the magnetic field through the wire.
There are still other ways of increasing induced current. These include increasing the strength of the magnet being used and adding more loops to the wire.
Example 1: Understanding Electromagnetic Induction
The diagram shows a permanent magnet being moved through a loop of copper wire. This motion induces an electric current of 0.5 A in the wire.
- If the magnet is moved through the loop at half the speed, what will the current in the loop be?
- If the permanent magnet is changed for one that is twice as strong and moves through the loop at the original speed, what will the current in the loop be?
Moving at its initial speed, the bar magnet creates a change in the magnetic field through the copper loop such that a current of 0.5 A is induced.
The magnitude of induced current is directly related to the rate at which the magnetic field through the loop changes, and therefore the magnet’s speed.
A faster-moving magnet will therefore induce more current and a slower-moving magnet less current.
If the magnet moves at half its initial speed, the current in the loop will be less than 0.5 A.
Considering a change in the magnet’s strength, we recall that the current induced in the copper loop is due to the rate at which the magnetic field through the loop changes.
Increasing the magnet’s strength means that the magnetic field through the loop will be changing at a greater rate. A greater current will therefore be induced, so we can say that if the strength of the magnet is doubled and it moves at the same speed as before, then more than 0.5 A of current will be induced in the loop.
Another way to strengthen induction is to replace a single loop of wire with a wire arranged in many identical loops, called a solenoid. Each loop or turn in a solenoid multiplies the induced potential difference in the wire. An example of a solenoid is shown below.
Example 2: Understanding Relative Motion in Electromagnetism
Part (a) of the diagram shows a bar magnet moving at a speed toward a stationary solenoid. This induces an electric potential difference across the two ends of the solenoid. Part (b) of the diagram shows a stationary bar magnet, with a solenoid moving toward it at speed . How is the potential difference induced in the solenoid in part (b) different from that in part (a)?
Recall that a solenoid is a coil of wire, as shown in the above diagrams.
In diagram (a), we see the north pole of the bar magnet approaching the solenoid at a speed .
In diagram (b), the magnet is stationary while the solenoid now approaches the magnet’s north pole at the same speed .
The relative motion in both of these instances is the same; the magnet’s north pole and the solenoid approach one another at speed .
Thus, the change in magnetic field through the solenoid’s loops is the same, whether the magnet is in motion or the solenoid.
This changing magnetic field is what induces a potential difference in the solenoid, so we expect the potential difference induced to be the same in both scenarios.
When a wire experiences a changing magnetic field, the direction of the induced current corresponds to the changing field.
Consider a bar magnet moving through a solenoid as shown below.
Current is induced in the solenoid, and this current creates its own magnetic field.
As with any magnetic field, the induced field can be modeled as possessing a north and south pole. Effectively, the solenoid can be replaced with a bar magnet. The fields of these two objects are depicted below.
Supposing this substitution is made, the orientation of the magnet replacing the solenoid must be determined. There are two possibilities for this orientation as shown below.
To determine which orientation is correct, we consider the effect of each one on the moving bar magnet.
In orientation A, the south pole of the stationary magnet would attract the moving magnet’s north pole. The moving magnet would then accelerate, manifesting a kinetic energy that increases continually. This would imply that the total energy of the isolated system of magnets is increasing. The law of conservation of energy prohibits this from happening.
In orientation B, rather than attracting the moving magnet, the stationary magnet repels it. This slows down the moving magnet and eventually stops it.
In light of energy conservation, the stationary magnet must take orientation B. It repels the incoming magnet and slows it down.
This result holds true regardless of the moving magnet’s orientation. Whichever of the magnet’s poles is nearest to the wire it approaches, the wire can be considered as a bar magnet with opposing polarity.
Considering again a wire interacting with a changing magnetic field, we can now make a general statement about the direction of current induced in the wire with respect to the direction of the changing external field.
Definition: Lenz’s Law
When current is created through electromagnetic induction, the direction of the current is such that it generates a magnetic field opposing the change in the original magnetic field.
Example 3: Understanding Lenz’s Law
The diagram shows a bar magnet moving away from a solenoid. This induces an electric current in the solenoid, which creates its own magnetic field in turn. Which end of the solenoid is the north pole of the induced magnetic field?
Since the bar magnet is moving relative to the solenoid, the solenoid experiences a changing magnetic field.
Current is therefore induced in the solenoid, which in turn induces a magnetic field.
We are to consider this field as the field created by a magnet. The magnet’s north pole is either at point A, with the south pole therefore at point B, or at point B, with the south pole at point A.
Let’s consider both of these options in turn.
If point A near the solenoid is a magnetic north pole and point B is a south pole, then the actual magnet and the magnet replacing the solenoid would appear as follows.
Since like magnetic poles repel, we would expect the magnet on the left to be pushed to the left. The magnet already moving that way would accelerate in that direction.
The alternative to the above is for point A near the solenoid to be a south magnetic pole and for point B to be the north magnetic pole. This is shown below.
Opposite magnetic poles attract, so we would expect these two magnets to tend to draw closer together.
This arrangement of magnets would make the moving magnet slow down. Its motion to the left would be opposed by an attraction to the south pole at point A.
Recall from Lenz’s law that induced magnetic fields oppose the changing magnetic fields that cause them. In this example, then, we are looking for the induced magnetic field that works against the motion of the magnet moving to the left.
An induced field that opposes this motion will tend to pull the moving magnet to the right. We have seen that a south magnetic pole at point A has this effect.
Therefore, when the induced magnetic field of the solenoid has a south magnetic pole at A, and a north magnetic pole at B, the induced field and the changing external field interact as described by Lenz’s law.
The north magnetic pole of the induced field is at point B.
A potential difference can be induced across a wire even if the wire does not form a closed loop.
For example, a straight section of wire can move through a magnetic field—even a uniform one—in such a way that a potential difference is established across the ends of the wire.
We can visualize this by picturing the wire as though it points into and out of the screen and the field as pointing from right to left around the wire.
When the wire is at rest in this uniform field, no potential difference is induced. However, when the wire is put into motion so that it crosses magnetic field lines, a potential difference is induced across the wire.
The magnitude of the induced potential difference is proportional to the rate at which the wire crosses magnetic field lines.
For a wire and field oriented as shown above, there are many directions in which the wire could move to cross magnetic field lines and therefore induce a potential difference.
Indeed, there are only two ways it could move so that a potential difference is not induced—parallel or antiparallel to the magnetic field.
A potential difference induced in a moving wire can be positive or negative. If the potential difference induced in a wire that moves vertically upward, for example, is positive, then the potential difference induced when the wire moves downward is negative.
Example 4: Understanding Electromagnetic Induction
Parts (a), (b), (c), and (d) in the diagram show a straight piece of copper wire moving through a magnetic field. The magnetic field is uniform, and in each part the wire is moving at the same speed, but in a different direction through the magnetic field. Which of (a), (b), (c), and (d) shows the motion of the wire that would lead to the greatest potential difference being induced in it?
In each of the four diagrams, a copper wire with its axis pointing into and out of the screen moves through a uniform magnetic field.
Such motion is capable of inducing a potential difference across the ends of the wire.
The magnitude of the potential difference induced is proportional to the rate at which the wire crosses magnetic field lines.
We are told that, in all four cases, the speed of the moving wire is the same. Therefore, the only factor affecting how quickly the wire crosses magnetic field lines is its direction of motion.
A wire moving to the left, for example, parallel to the magnetic field, will cross no field lines and thus will not experience any induced current.
A wire moving perpendicularly to the field lines, however, will intersect those lines at the highest rate possible for its speed.
Diagrams (a), and (b) show the wire moving at angles that are not with respect to the magnetic field. Diagram (d) shows the wire moving in the opposite direction to the field, and hence do not cross any field lines. Diagram (c) is different, indicating a wire moving straight upward perpendicularly to the field and therefore crossing the greatest number of field lines possible at its speed.
The greatest potential difference induced in the wire would occur in diagram (c).
Example 5: Understanding Electromagnetic Induction for Circular Motion
Diagram (a) shows a straight piece of copper wire moving along a circular path in a uniform magnetic field. Diagram (b) shows the potential difference across the piece of wire against time as it does so. If point in diagram (a) corresponds to point in diagram (b), what point in diagram (a) does point in diagram (b) correspond to?
Considering these two diagrams, we know that the potential difference induced in the wire moving in diagram (a) is shown graphically in (b).
We are also told that point in diagram (a) corresponds to point in diagram (b). This means that when the wire is moving vertically upward, the potential difference induced is positive and takes on its maximum value.
Note that point in diagram (b) occurs where the potential difference is zero. For our wire moving in a magnetic field, the induced potential difference can only be zero if the wire moves parallel or antiparallel to the surrounding field lines. Therefore, point can only possibly correspond to point or point in diagram (a).
At point , not only is the potential difference across the wire zero, but we also see from diagram (b) that the potential difference is moving from positive to negative.
Looking back at diagram (a) at point , the induced potential difference is moving from negative to positive values (i.e., the wire is transitioning from downward to upward motion). At the top of the circle at point , the induced potential difference is moving from positive to negative.
Therefore, of the two points where the wire is moving horizontally, we choose point as the one corresponding to point on diagram (b).
- When a loop of wire is exposed to a changing magnetic field, a potential difference is induced across the wire.
- The faster the magnetic field changes, the greater the corresponding magnitude of the current induced in the wire.
- For a solenoid—a wire coiled in circular loops—the magnitude of induced current can be increased by adding more loops.
- Current induced in a loop always points in a direction such that the magnetic field it generates opposes the change in the applied magnetic field that originally induced the current, which is a rule known as Lenz’s law.
- Potential difference will be induced across a straight wire moving in a uniform magnetic field if the wire does not move parallel or antiparallel to the field lines.
- The magnitude of such induced potential difference is proportional to the rate at which the wire crosses magnetic field lines.