College Physics for AP® Courses

# 23.1Induced Emf and Magnetic Flux

College Physics for AP® Courses23.1 Induced Emf and Magnetic Flux

### Learning Objectives

By the end of this section, you will be able to:

• Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation.
• Describe methods to produce an electromotive force (emf) with a magnetic field or a magnet and a loop of wire.

The information presented in this section supports the following AP® learning objectives and science practices:

• 4.E.2.1 The student is able to construct an explanation of the function of a simple electromagnetic device in which an induced emf is produced by a changing magnetic flux through an area defined by a current loop (i.e., a simple microphone or generator) or of the effect on behavior of a device in which an induced emf is produced by a constant magnetic field through a changing area. (S.P. 6.4)

The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in Figure 23.3. 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. (You can also observe this in a physics lab.) 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. More basic than the current that flows is the emfthat causes it. The current is a result of an emf induced by a changing magnetic field, whether or not there is a path for current to flow.

Figure 23.3 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 emf 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 23.4. An emf is induced in the coil when a bar magnet is pushed in and out of it. Emfs of opposite signs are produced by motion in opposite directions, and the emfs 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 emf, and there is no emf when the magnet is stationary relative to the coil.

Figure 23.4 Movement of a magnet relative to a coil produces emfs as shown. The same emfs are produced if the coil is moved relative to the magnet. The greater the speed, the greater the magnitude of the emf, and the emf is zero when there is no motion.

The method of inducing an emf used in most electric generators is shown in Figure 23.5. A coil is rotated in a magnetic field, producing an alternating current emf, 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 (another symmetry).

Figure 23.5 Rotation of a coil in a magnetic field produces an emf. 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 an emf. Experiments revealed that there is a crucial quantity called the magnetic flux, $ΦΦ size 12{Φ} {}$ , given by

$Φ=BAcosθ,Φ=BAcosθ, size 12{Φ= ital "BA""cos"θ} {}$
23.1

where $BB size 12{B} {}$ is the magnetic field strength over an area $AA size 12{A} {}$, at an angle $θ θ$ with the perpendicular to the area as shown in Figure 23.6. Any change in magnetic flux $ΦΦ size 12{Φ} {}$ induces an emf. This process is defined to be electromagnetic induction. Units of magnetic flux $ΦΦ size 12{Φ} {}$ are $T⋅m2T⋅m2 size 12{T cdot m rSup { size 8{2} } } {}$. As seen in Figure 23.6, $Bcosθ=BBcosθ=B size 12{B"cos"θ=B rSub { size 8{ ortho } } } {}$, which is the component of $BB size 12{B} {}$ perpendicular to the area $AA size 12{A} {}$. Thus magnetic flux is $Φ=B⊥AΦ=B⊥A size 12{Φ=B rSub { size 8{ ortho } } A} {}$, the product of the area and the component of the magnetic field perpendicular to it.

Figure 23.6 Magnetic flux $ΦΦ size 12{Φ} {}$ is related to the magnetic field and the area over which it exists. The flux $Φ=BAcosθΦ=BAcosθ size 12{Φ= ital "BA""cos"θ} {}$ is related to induction; any change in $ΦΦ size 12{Φ} {}$ induces an emf.

All induction, including the examples given so far, arises from some change in magnetic flux $ΦΦ size 12{Φ} {}$. For example, Faraday changed $BB size 12{B} {}$ and hence $ΦΦ size 12{Φ} {}$ when opening and closing the switch in his apparatus (shown in Figure 23.3). This is also true for the bar magnet and coil shown in Figure 23.4. When rotating the coil of a generator, the angle $θθ size 12{θ} {}$ and, hence, $ΦΦ size 12{Φ} {}$ is changed. Just how great an emf and what direction it takes depend on the change in $ΦΦ size 12{Φ} {}$ and how rapidly the change is made, as examined in the next section.