In this explainer, we will learn how to analyze the current induced in conducting loops by changing magnetic fields and the magnetic field due to a current loop.

When we talk about the magnetic field that a conducting loop is within changing, we could mean either of two things:

- The conducting loop is moving from a region with one magnetic field magnitude and direction to a region where the magnetic field has a different magnitude or direction.
- The conducting loop is at rest within a magnetic field that changes magnitude or direction.

Let us first consider a conducting loop that moves from a region with zero magnetic field to a region with a nonzero uniform magnetic field. This is shown in the following figure.

When part of the loop enters the magnetic field, the free electrons in the part of the loop that enters the field are acted on by a force that has a direction perpendicular to both the magnetic field and the motion of the loop. This is shown in the following figure.

Only the free electrons in the part of the loop within the magnetic field are affected by the force. These electrons are pushed toward one end of the wire. This concentration of free electrons at one end of the wire induces a current in the loop by repelling free electrons at the end of the part of the loop where free electrons are concentrated and attracting free electrons at the opposite end of the part of the loop to where free electrons are concentrated. This is shown in the following figure.

We can see from this that there will only be an induced current if two things are true:

- The loop is moving. If the loop does not move, no force acts on the free electrons.
- Part of the loop is within the field and part of the loop is not within the field. If the loop is completely within the magnetic field, the force on the free electrons is the same for all parts of the loop. A current is only produced in the loop if the net force in one part of the loop is greater than in other parts of the loop.

The direction of the current induced in a loop is determined by the right-hand rule. This is shown in the following figure.

In addition to the current induced in a loop, it can be useful to consider the potential difference induced in a loop. For a straight conductor, a potential difference is a difference in potential of the ends of the conductor. A loop however has no ends, so a different idea of potential difference must be considered.

A potential difference, , can be expressed as where is the work done by the potential difference on a charge, , across the potential difference.

For a conducting loop, βacross the potential differenceβ means βaround the loop.β When the magnetic field through the loop changes, work is done on free electrons in the loop to move them around the loop against the resistance of the material that the loop is made of. We can therefore define the potential difference induced in a loop by a change of magnetic field. This is often referred to as the emf induced in the loop, .

The value of depends on the rate at which the magnetic field through a loop changes. This is known as the rate of change of magnetic flux linkage.

The magnetic flux, , in a region of space is given by where is the magnetic field strength in the region and is the area perpendicular to the direction of the magnetic field that the magnetic field lines pass through.

The rate of change of magnetic flux linkage can be represented by the rate at which a loop passes through the lines of a magnetic field. This is shown in the following figure showing the positions at one-second time intervals of two identical loops entering two identical magnetic fields.

We see that the loop with the greater speed has a greater rate of change of magnetic flux linkage.

We could also compare two loops with equal speeds that enter magnetic fields of different strengths. This is shown in the following figure.

As the rate of change of magnetic flux linkage can be represented by the expression where is the magnetic field strength of a magnetic field perpendicular to the direction of the velocity of the loop, is the change in the area of the loop that passes into the magnetic field, and is the time interval in which this area enters the field.

The magnitude of the induced emf, , for a loop is given by

This formula can be made to apply to a coil made of many loops by multiplying the induced emf by the number of loops in the coil, :

Let us now look at an example involving a conducting coil being moved into a uniform magnetic field.

### Example 1: Determining the emf Induced in a Conducting Coil That Enters a Uniform Magnetic Field

A conducting coil has a radius and 35 turns. The coil is moved so that half of its area is within a uniform magnetic field of strength 0.16 T, directed out of the plane of the diagram shown along the axis of the coil. The coil moves from its old position to its new position in 0.24 s.

- What is the magnitude of the electromotive force induced in the coil? Give your answer to two decimal places.
- Is the current through the coil clockwise or counterclockwise?

### Answer

**Part 1**

The magnitude of the emf, , induced across the coil is given by the formula where is the number of turns of the coil, is the magnetic field strength, and is the rate of change of the area of the coil entering the magnetic field.

The value of is stated to be 35, and the value of is stated to be 0.16 T.

The area of the coil, , is given by where is stated to be 13 cm, which equals 0.13 m. Half of the area of the coil enters the magnetic field. The time interval in which this happens is stated to be 0.24 s.

We can substitute these values into the formula to obtain

To two decimal places, is 0.62 V.

**Part 2**

The direction of the current can be determined using the right-hand rule. Using the directions of motion and magnetic field for the part of a conducting loop in the coil that first starts to enter the magnetic field, the direction of current in the loop is shown in the following figure.

We see that the current direction is clockwise.

We have now explained the basic principles of the induction of emf by moving a conducting loop from a region in which the magnetic field is zero to a region in which the magnetic field is nonzero.

Let us now consider the induction of emf in a static loop by a changing magnetic field.

One situation in which this occurs is when a bar magnet is moved toward or away from a conducting loop.

The following figure shows a simplified representation of the magnetic field lines from the north pole of a bar magnet passing through two loops at different distances from the pole of the magnet. The field lines shown are a two-dimensional cross section of the magnetic field around the magnetβs north pole.

We can see that, for the loop further from the pole of the magnet, the field lines are more divergent, and so the density of the field lines decreases further from the magnetβs pole. The magnitude of the magnetic field must then decrease further from the pole.

The right-hand rule can be applied to this system. This is shown in the following figure.

It is important to notice that only the component of the magnetic field perpendicular to the direction of the motion of the magnet induces current in the loop. This is shown in the following figure.

It might not be obvious, but it is important to understand that a current would be induced in exactly the same way if the bar magnet was stationary and the conducting loop was moved toward the pole of the magnet.

This symmetry also applies for uniform magnetic fields. If a uniform magnetic field that could move was to move into a region containing a conducting loop, an emf would be induced in the loop. It is irrelevant whether it is the conducting loop or the magnetic field that is moving.

Let us now look at an example involving a conducting coil being moved within a nonuniform magnetic field.

### Example 2: Determining the Change in the Magnetic Field Strength That Induces an emf in a Conducting Coil

A conducting coil with four turns has a diameter . The coil moves 1.5 cm at a velocity parallel to the axis of a stationary bar magnet, as shown in the diagram. An emf of magnitude 3.6 mV is induced in the coil while it moves past the magnet. Find the change in the magnetic field strength between the points where the coil started to move and where it stopped moving. Give your answer in scientific notation to one decimal place.

### Answer

The magnitude of the emf, , induced across a coil entering a magnetic field is given by the formula where is the number of turns of the coil, is the magnetic field strength, and is the rate of change of the area of the coil entering the magnetic field.

In this case, however, the area of the coil that is within the magnetic field does not change. Instead, the magnetic field strength changes. We can therefore express the equation for as

In this form of the equation, is constant and a change in , , is a term in the equation.

We wish to determine the magnitude of .

To determine , the equation must be rearranged to make the subject, as follows:

The emf induced is stated to be 3.6 mV, or 0.0036 V.

The value of is stated to be 4.

The area of the coil, , is given by

The value of is not given, but this is equal to half of the diameter, , of the coil. The value of is stated to be 25 cm. This gives a value of of 12.5 cm, or 0.125 m.

The area of the coil is then given by

The coil moves a distance of 1.5 cm, or 0.015 m.

The speed of the coil is stated to be 7.5 cm/s, or 0.075 m/s.

The time, , for which the coil moves is given by

We can now use the equation

Substituting the values of and we have determined,

To one decimal place, is T.

The formula for the magnitude of is not the complete version of the formula. The emf induced has a sign as well as a magnitude.

### Formula: Emf Induced in a Current Loop due to a Changing Magnetic Field

The emf, , induced across a coil entering a magnetic field is given by the formula where is the number of turns of the coil, is the magnetic field strength, and is the rate of change of the area of the coil perpendicular to the magnetic field that is entering the magnetic field.

The formula can also have the form which applies where a constant area of loop is within a changing magnetic field.

The negative sign in the full version of the formula indicates that the emf is considered to have a direction that is opposite to the direction of a quantity that corresponds to a term in the formula. If the negative sign of the induced emf is included in the formula, it is known as Lenzβs law.

We can explain this as follows.

- The emf induced in a loop by a changing magnetic field produces a current.
- The induced current then produces a second magnetic field.
- The direction of the second magnetic field opposes that of the magnetic field that induces the current.

This process is shown in the following sequence of figures.

We see here the magnetic field of the bar magnet inducing a current, both looking across and looking through the area of the loop.

Next, we will see the effect of this induced current.

We see here, looking first through and then across the area of the loop, the magnetic field induced by the current in the loop.

The field lines of the magnetic field induced by the current point to the right side of the loop. This means that the right side of the loop acts as a north pole of a magnet.

We see then that the magnetic field induced in the loop acts to repel the bar magnet.

This opposite direction of the induced magnetic field to the inducing magnetic field is the meaning of the negative sign in the formula for induced emf.

It is interesting to ask whether the magnetic field of a loop induced by a current induced in a loop then will induce another current in the loop. The answer is yes, and this phenomenon is called the self-induction of a loop. This explainer does not consider this phenomenon, however.

The examples that have been considered have both involved motion, either of a loop or of a magnetic field source. A magnetic field, though, can change without either the motion of the source of the field or the motion of a loop.

Suppose that a conducting coil is initially at rest within a larger conducting coil, as shown in the following figure.

Each coil is really just a single wire that happens to be formed into a series of loops. If the wire that forms the large coil has a potential difference connected across it, there will be a current in the large coil.

The current in the large coil will induce a magnetic field in the large coil. The small coil will be within the magnetic field of the large coil. Initially, there was no magnetic field through the small coil, and so the induction of the magnetic field of the large coil will change the strength of the magnetic field that the small coil is within. A current will then be induced in the small coil.

Let us look at an example involving one coil inducing a current in another.

### Example 3: Determining the emf Induced in a Coil by a Second Coil

A coil with 6 turns and a radius of 2.1 cm is placed with the center of its base at the center of the base of a coil with four turns and a radius of 7.2 cm, as shown in the diagram. The top of the smaller coil is below the top of the larger coil. The larger coil is connected to a variable voltage source and a current is induced in the larger coil that produces a uniform magnetic field inside it, which changes from zero magnitude to a magnitude of 360 mT in a time of 23 ms.

- What is the magnitude of the electromotive force induced in the smaller coil? Give your answer to two decimal places.
- Is the current through the smaller coil clockwise or counterclockwise?

### Answer

**Part 1**

The emf, , induced across the small coil entering the magnetic field is given by the formula where is the number of turns of the coil, is the area of the coil, and is the rate of change of the magnetic field that the small coil is within. So,

The number of turns in the coil is stated to be 6.

The area of the coil, , is given by where is 2.1 cm, or 0.021 m.

The area of the coil is then given by

The magnetic field changes from 0 T to 360 mT. The change in the magnetic field is then given by

The time, , in which the magnetic field strength changes is stated to be 23 ms, or 0.023 s.

Substituting the values obtained into we obtain

To two decimal places, is 0.13 V.

**Part 2**

The current in the large coil induces a magnetic field. At the center of the large coil, this field acts vertically downward. The magnetic field of the large coil must induce a current in the small coil that will induce a magnetic field at the center of the small coil that will act vertically upward. This is shown in the following figure.

For the small coil to produce an upward magnetic field, the current in the small coil must be clockwise.

It is interesting to ask whether the magnetic field induced in the small loop would induce a current in the large loop. The answer is yes, and this phenomenon is called the mutual induction of two coils. This explainer does not consider this phenomenon, however.

Let us now summarize what has been learned in this explainer.

### Key Points

- A conducting loop that has a changing magnetic field through it has an emf induced in the loop and hence a current induced through the loop.
- It makes no difference whether a loop moves through a region in which the magnetic field varies in space through the region or whether the magnetic field that the loop is within varies. Either of these situations results in induced emf in the loop.
- The emf, , induced across a coil entering a magnetic field is given by the formula
where is the number of turns of the coil, is the magnetic field strength, and is the rate of change of the area of the coil perpendicular to the magnetic field that is entering the magnetic field.

The formula can also have the form which applies where a constant area of loop is within a changing magnetic field. - The current induced in a loop will induce a magnetic field around the loop.
- The magnetic field due to an induced current in a loop will have the opposite direction to the external magnetic field that induced the current.