Induction in a Motionless Loop

Figure 8.4.1: A single loop of wire in the presence of an impressed spatially-uniform but time-varying magnetic field.

In this section, we consider the problem depicted in Figure 8.4.1, which is a single motionless loop of wire in the presence of a spatially-uniform but time-varying magnetic field. A small gap is introduced in the loop, allowing us to measure the induced potential

. Additionally, a resistance

is connected across

in order to allow a current to flow. This problem was considered in Section 8.3 as an introduction to Faraday’s Law; in this section, we shall actually work the problem and calculate some values. This is intended to serve as an example of the application of Faraday’s Law, a demonstration of transformer emf, and will serve as a first step toward an understanding of transformers as devices.

In the present problem, the loop is centered in the

plane. The magnetic flux density is

; i.e., time-varying magnitude

and a constant direction

. Because this magnetic field is spatially uniform (i.e., the same everywhere), we will find that only the area of the loop is important, and not it’s specific shape. For this reason, it will not be necessary to specify the radius of the loop or even require that it be a circular loop. Our task is to find expressions for

and

To begin, remember that Faraday’s Law is a calculation of electric potential and not current. So, the approach is to first find

, and then find the current

that flows through the gap resistance in response.

The sign convention for

is arbitrary; here, we have selected “+” and “−” terminals as indicated in Figure 8.4.1.¹ Following the standard convention for the reference direction of current through a passive device,

should be directed as shown in Figure 8.4.1 . It is worth repeating that these conventions for the signs of

and

are merely references; for example, we may well find that

is negative, which means that current flows in a clockwise direction in the loop.

We now invoke Faraday’s Law:

The number of windings

in the loop is 1, and

is the magnetic flux through the loop. Thus:

where

is any open surface that intersects all of the magnetic field lines that pass through the loop. The simplest such surface is simply the planar surface defined by the perimeter of the loop. Then

, where

is the differential surface element and

is the normal to the plane of the loop. Which of the two possible normals to the loop? This is determined by the right-hand rule of Stokes’ Theorem. From the “−” terminal, we point the thumb of the right hand in the direction that leads to the “+” terminal by traversing the perimeter of the loop. When we do this, the curled fingers of the right hand intersect

in the same direction as

. To maintain the generality of results derived below, we shall not make the substitution

; nevertheless we see this is the case for a loop parallel to the

plane with the polarity of

indicated in Figure 8.4.1.

Taking this all into account, we have

Since the magnetic field is uniform,

may be extracted from the integral. Furthermore, the shape and the orientation of the loop are time-invariant, so the remaining integral may be extracted from the time derivative operation. This leaves:

The integral in this expression is simply the area of the loop, which is a constant; let the symbol

represent this area. We obtain

which is the expression we seek. Note that the quantity

is the projected area of the loop. The projected area is equal to

when the the magnetic field lines are perpendicular to the loop (i.e.,

), and decreases to zero as

. Summarizing:

The magnitude of the transformer emf induced by a spatially-uniform magnetic field is equal to the projected area times the time rate of change of the magnetic flux density, with a change of sign. (Equation 8.4.5).

A few observations about this result:

As promised earlier, we have found that the shape of the loop is irrelevant; i.e., a square loop having the same area and planar orientation would result in the same $V_{T}$ . This is because the magnetic field is spatially uniform, and because it is the magnetic flux ( $\Phi$ ) and not the magnetic field or shape of the loop alone that determines the induced potential.
The induced potential is proportional to $A$ ; i.e., $V_{T}$ can be increased by increasing the area of the loop.
The peak magnitude of the induced potential is maximized when the plane of the loop is perpendicular to the magnetic field lines.
The induced potential goes to zero when the plane of the loop is parallel to the magnetic field lines. Said another way, there is no induction unless magnetic field lines pass through the loop.
The induced potential is proportional to the rate of change of $\mathbf{B}$ . If $\mathbf{B}$ is constant in time, then there is no induction.

Finally, the current in the loop is simply

Again, electromagnetic induction induces potential, and the current flows only in response to the induced potential as determined by Ohm’s Law. In particular, if the resistor is removed, then

and

, but

is unchanged.

One final comment is that even though the current

is not a direct result of electromagnetic induction, we can use

as a check of the result using Lenz’s Law (Section 8.2). We’ll demonstrate this in the example below.

Exercise

#1. Induction in a motionless circular loop by a linearly-increasing magnetic field.

Let the loop be planar in the

plane and circular with radius

cm. Let the magnetic field be

where

i.e.,

begins at zero and increases linearly to

at time

, after which it remains constant at

. Let

= 0.2 T,

= 3 s, and let the loop be closed by a resistor

. What current

flows in the loop?

Solution: Adopting the sign conventions of Figure 8.4.1, we first note that

; this is determined by the right-hand rule with respect to the indicated polarity of

. Thus, Equation 8.4.5 becomes

Note

since

; i.e., because the plane of the loop is perpendicular to the magnetic field lines. Since the loop is circular,

. Also

Putting this all together:

and

before and after this time interval, since

is constant during those times. Subsequently the induced current is

and

before and after this time interval. We have found that the induced current is a constant clockwise flow that exists only while

is increasing.

Finally, let’s see if the result is consistent with Lenz’s Law. The current induced while

is changing gives rise to an induced magnetic field

. From the right-hand rule that relates the direction of

to the direction of

(Section 7.5), the direction of

is generally

inside the loop. In other words, the magnetic field associated with the induced current opposes the increasing impressed magnetic field that induced the current, in accordance with Lenz’s Law.