Induced Voltage and Magnetic Flux

Chapter contents: Chapter 10: Magnetism

The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in Figure 10.9.1. When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in the coil on the bottom. It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. Closing and opening the switch induces the current. It is the change in magnetic field that creates the current in the following way: a changing magnetic field induces an electric field, which results in the induced voltage. When this induced voltage occurs over a conducting path, as in this example, the induced voltage causes a current to flow. As a shorthand, we call the resulting current induced current; the changing magnetic field does not induce the current directly but through the induced voltage and an application of Ohm's law.

The picture shows Faraday’s apparatus for demonstrating that a magnetic field can produce a current. It consists of a cylinder shaped battery. The positive end of the battery is connected to an open switch. There is a ring shaped iron core consisting of a set of coils one on the top and another at the bottom. The other end of the switch is connected to one end of the top coil. The other end of the top coil is connected back to the battery. Both the ends of the bottom coil are shown connected across a galvanometer box which shows a null deflection. — **Figure 10.9.1.** Faraday’s apparatus for demonstrating that a magnetic field can produce a current. A change in the field produced by the top coil induces an voltage and, hence, a current in the bottom coil. When the switch is opened and closed, the galvanometer registers currents in opposite directions. No current flows through the galvanometer when the switch remains closed or open.

An experiment easily performed and often done in physics labs is illustrated in Figure 10.9.2. A voltage is induced in the coil when a bar magnet is pushed in and out of it. Voltages of opposite signs are produced by motion in opposite directions, and the voltages are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the voltage, and there is no voltage when the magnet is stationary relative to the coil.

The diagram shows five stages of an experiment done by moving a magnet relative to a coil and measuring the e m f produced. The first stage of the experiment shows a wire coil with two loops connected across a galvanometer. The loop is in horizontal plane. A cylindrical rod shaped magnet is moved upward through the loop with the north pole of the magnet facing the loop and the South Pole away from the loop. The magnetic lines of force of the magnet are shown to emerge out from the North Pole and intersect the coil. A current is shown to be induced in the coil in clockwise direction. The galvanometer needle is shown to deflect toward right. The second stage of the experiment shows the next state of the first stage of the experiment. The cylindrical rod shaped magnet is now moved downward away from the loop with the north pole of the magnet facing the loop and South Pole away from the loop. The magnetic lines of force of the magnet are shown to emerge out from the North Pole and intersect the coil. A current is shown to be induced in the coil in anti clockwise direction. The galvanometer needle is shown to deflect toward left. The third stage of the experiment shows a wire coil with two loops connected across a galvanometer. The loop is in horizontal plane. A cylindrical rod shaped magnet is moved upward through the loop with the south pole of the magnet facing the loop and the North Pole away from the loop. The magnetic lines of force of the magnet are shown to merge into the South Pole and intersect the coil. A current is shown to be induced in the coil in anti clockwise direction. The galvanometer needle is shown to deflect toward left. The fourth stage of the experiment shows the next state of the third stage of the experiment. The cylindrical rod shaped magnet is now moved downward away from the loop with the south pole of the magnet facing the loop and the North Pole away from the loop. The magnetic lines of force of the magnet are shown to merge into the South Pole and intersect the coil. A current is shown to be induced in the coil in clockwise direction. The galvanometer needle is shown to deflect toward right. The fifth stage of the experiment shows a wire coil with two loops connected across a galvanometer. The loop is in horizontal plane. A cylindrical rod shaped magnet is held stationary near the loop with the north pole of the magnet facing the loop and south away from the loop. The magnetic lines of force of the magnet are shown to emerge out from the North Pole and intersect the coil. No current is induced in the coil. The galvanometer needle does not deflect. — **Figure 10.9.2.** Movement of a magnet relative to a coil produces voltage as shown. The same voltages are produced if the coil is moved relative to the magnet. The greater the speed, the greater the magnitude of the voltage, and the voltage is zero when there is no motion.

The method of inducing a voltage used in most electric generators is shown in Figure 10.9.3. A coil is rotated in a magnetic field, producing an alternating voltage (and current), which depends on rotation rate and other factors that will be explored in later sections. Note that the generator is remarkably similar in construction to a motor.

The figure shows a schematic diagram of an electric generator. It consists of a rotating rectangular coil placed between the two poles of a permanent magnet shown as two rectangular blocks curved on side facing the coil. The magnetic field B is shown pointing from the North to the South Pole. The two ends of this coil are connected to the two small rings. The two conducting carbon brushes are kept pressed separately on both the rings. The coil is attached to an axle with a handle at the other end. The axle may be mechanically rotated from outside to rotate the coil inside the magnetic field. Outer ends of the two brushes are connected to the galvanometer. A current is shown to flow in the coil in anti clockwise direction and the galvanometer shows a deflection. — **Figure 10.9.3.** Rotation of a coil in a magnetic field produces a voltage. This is the basic construction of a generator, where work done to turn the coil is converted to electric energy. Note the generator is very similar in construction to a motor.

So we see that changing the magnitude or direction of a magnetic field produces a voltage. Experiments revealed that there is a crucial quantity called the magnetic flux,

Φ

, given by

Φ = B_{⊥} A,

(10.9.1)

where $B$ is the magnetic field strength over an area $A$ , at an angle

θ

with the perpendicular to the area as shown in Figure 10.9.4. Any change in magnetic flux $Φ$ induces a voltage. This process is defined to be electromagnetic induction. Units of magnetic flux $Φ$ are

T \cdot m^{2}

Figure shows a flat square shaped surface A. The magnetic field B is shown to act on the surface at an angle theta with the normal to the surface A. The cosine component of magnetic field B cos theta is shown to act parallel to the normal to the surface. — **Figure 10.9.4.** Magnetic flux $Φ$ is related to the magnetic field and the area $A$ over which it exists. Only the portion of the magnetic field that is perpendicular to the area ( $B_{⊥}$ ) contributes to the flux. The flux $Φ = B_{⊥} A$ is related to induction; any change in $Φ$ induces a voltage.

All induction, including the examples given so far, arises from some change in magnetic flux $Φ$ . For example, Faraday changed $B$ and hence $Φ$ when opening and closing the switch in his apparatus (shown in Figure 10.9.1). This is also true for the bar magnet and coil shown in Figure 10.9.2. When rotating the coil of a generator, the angle

θ

and, hence, $Φ$ is changed. Just how great a voltage and what direction it takes depend on the change in $Φ$ and how rapidly the change is made, as examined in the next section.

Section Summary

The crucial quantity in induction is magnetic flux $Φ$ , defined to be $Φ = B_{⊥} A$ , where $B_{⊥}$ is the magnetic field strength perpendicular to the area $A$ .
Units of magnetic flux $Φ$ are $T \cdot m^{2}$ .
Any change in magnetic flux $Φ$ induces a voltage—the process is defined to be electromagnetic induction.

Conceptual Questions

Exercise 11

How do the multiple-loop coils and iron ring in the version of Faraday’s apparatus shown in Figure 10.9.1 enhance the observation of induced voltage?

Exercise 12

When a magnet is thrust into a coil as in Figure 10.9.2(a), what is the direction of the force exerted by the coil on the magnet? Draw a diagram showing the direction of the current induced in the coil and the magnetic field it produces, to justify your response. How does the magnitude of the force depend on the resistance of the galvanometer?

Exercise 13

Explain how magnetic flux can be zero when the magnetic field is not zero.

Exercise 14

Is a voltage induced in the coil in Figure 10.E.1 when it is stretched? If so, state why and give the direction of the induced current.

The first part of the figure shows a circular coil of wire held in a magnetic field. The magnetic field points into the paper. The coil is held using both the hands to stretch it. The second part of the figure shows the same circular coil of wire stretched in the magnetic field. — **Figure 10.E.1.** A circular coil of wire is stretched in a magnetic field.

induced current

the current created by a changing magnetic field through voltage induced over a conducting path

magnetic flux

the amount of magnetic field going through a particular area, calculated with

Φ = B_{⊥} A

, where

B_{⊥}

is the magnetic field strength perpendicular to the area

A

electromagnetic induction

the process of inducing a voltage with a change in magnetic flux