46 Induced Emf and Magnetic Flux

In the previous chapter, we learned that electric currents produce magnetic fields. Michael Faraday showed that the reverse process is also possible: a changing magnetic field can produce an electromotive force (emf) and, when a closed conducting path is available, an electric current.

Faraday’s original apparatus is shown in Figure 46.1. A battery and switch are connected to the upper coil wrapped around an iron ring. When the switch is closed, current begins to flow through the upper coil and creates a magnetic field in the iron ring. The changing magnetic field passes through the lower coil, which is connected to a galvanometer that detects electric current.

Each time the switch is closed, the galvanometer briefly indicates a current in one direction. When the switch is opened, it briefly indicates a current in the opposite direction. However, if the switch remains either open or closed, the galvanometer returns to zero. A steady magnetic field does not continue to induce an emf. It is the change in the magnetic field that produces the emf.

A simpler experiment is shown in Figure 46.2. When a bar magnet is moved into or out of a coil, an emf is induced in the wire. Moving the magnet in opposite directions produces emfs with opposite signs. Reversing the magnet’s poles also reverses the direction of the induced emf.

The result is the same if the coil moves while the magnet remains still. What matters is the relative motion between the magnet and the coil. Moving the magnet or coil more quickly produces a larger emf because the magnetic conditions change more rapidly. When the magnet and coil are stationary relative to each other, no emf is induced.

Most electric generators use a related method, illustrated in Figure 46.3. A coil rotates inside a magnetic field. As the coil turns, its orientation relative to the field continually changes, producing an alternating emf. Mechanical work used to rotate the coil is therefore converted into electrical energy.

A generator is similar in construction to an electric motor, but the direction of energy conversion is reversed. A motor converts electrical energy into mechanical motion, whereas a generator converts mechanical motion into electrical energy.

Magnetic Flux

The previous experiments show that an emf is produced whenever the magnetic conditions through a loop change. To describe this quantitatively, we use a quantity called magnetic flux, represented by the Greek letter [latex]\Phi[/latex].

For a uniform magnetic field passing through a flat area, magnetic flux is defined as

where

• [latex]B[/latex] is the magnetic field strength,
• [latex]A[/latex] is the area of the loop or surface, and
• [latex]\theta[/latex] is the angle between the magnetic field and the line perpendicular to the surface.

The angle is measured from the magnetic field to the normal, an imaginary line perpendicular to the surface. This is important because magnetic flux depends only on the component of the magnetic field that passes directly through the surface.

The perpendicular component of the magnetic field is

Therefore, magnetic flux can also be written as

The SI unit of magnetic flux is the tesla–meter squared,

which is also called the weber (Wb).

Magnetic flux is greatest when the magnetic field is perpendicular to the surface. In that case, the field is parallel to the normal, so [latex]\theta=0^\circ[/latex] and [latex]\cos\theta=1[/latex]. The flux is then

If the magnetic field lies parallel to the surface, it is perpendicular to the normal. Then [latex]\theta=90^\circ[/latex], [latex]\cos\theta=0[/latex], and the magnetic flux is zero even though the magnetic field itself is not zero.

A change in magnetic flux can be produced in several ways:

• changing the magnetic field strength [latex]B[/latex],
• changing the area [latex]A[/latex] through which the field passes, or
• changing the angle [latex]\theta[/latex] between the field and the surface normal.

Any change in magnetic flux can induce an emf. This process is called electromagnetic induction.

In Faraday’s apparatus, opening and closing the switch changes the magnetic field strength. Moving a magnet relative to a coil changes the field passing through the coil. Rotating a generator coil changes the angle between the coil and the magnetic field. Although these situations appear different, they all produce an emf by changing the magnetic flux.

Electromagnetic induction is also important in healthcare technology. Changing magnetic fields induce signals in the receiver coils of MRI systems, allow wireless charging of some implanted and portable medical devices, and are used in sensors that monitor motion, position, and physiological activity.