Electricity and magnetism

Lecture 13

1. Induced EMF.

2. Faraday’s law of induction. Lenz’s law.

3.  EMF induced in a moving conductor

4. A changing magnetic flux produces an electric field

Induced EMF.

Scientists in 1820-1821 began to wonder: if electric currents produce a magnetic field, is it possible that a magnetic field can produce an electric current?  

Ten years later the American J. Henry and the Englishman M. Faraday independently found that it was possible.

An Electromotive Force or EMF is said to be induced when the flux of the magnetic field changes in a conductor or coil.

This change in flux can be obtained in two different ways; that is by statically or by dynamically induced emf. They are explained below

In his attempt to produce an electric current from a magnetic field, Faraday used an apparatus like shown in fig.1. A coil of wire X was connected to a battery.

A second coil Y wrapped on the same iron ring. This second circuit Y contained a galvanometer to detect any current but contained no battery.

[pic 1]

A constant current in X produced a constant magnetic field which produced no current in Y. Only when the current in X was starting or stopping was a current produced in Y.

Faraday concluded that although a constant magnetic field produces no current in conductor, a changing magnetic field (B flux) can produce an electric current. Such current is called an induced current. When the magnetic field through coil Y changes, a current occurs in Y as if there were a source of emf in circuit Y.

We therefore say that

A changing magnetic field induces an emf.

This is statical way of induced emf. The statical way involves the notion a self-induction. The self-induction the production of an electromotive force in a circuit when the magnetic flux linked with the circuit changes as a result of a change in current in the same circuit. 

When the current in the circuit changes, the magnetic flux changes proportionally,through the surface bounded by this circuit. A change in this magnetic flux, by virtue of the law of electromagnetic induction, leads to the excitation of an inductive EMF in this circuit. 

The magnitude of the self-induction EMF is proportional to the rate of change in the strength of the (alternating) current I:

[pic 2]

The proportionality coefficient L is called the self-induction coefficient or the inductance of the circuit (coil).

The above property of the coil exists only for the changing current which is the alternating current and not for the direct or steady current. Self-inductance is always opposing the changing current and is measured in Henry (SI unit).

Faraday did further experiments on electromagnetic induction as this phenomenon is called. For example, fig.2 shows that if a magnet is moved quickly into a coil of wire, a current is induced in the opposite direction (B through the coil decreases). Furthermore, if the magnet is held steady and the coil of wire is moved toward or way from the magnet, again an emf is induced and a current flows. Motion or change is required to induce an emf. It doesn’t matter whether the magnet or the coil moves. It is their relative motion that counts. This is dynamical  way induced emf.

Faraday’s law of induction. Lenz’s law.[pic 3]

Faraday’s law or the law of electromagnetic induction is the observation or results of the experiments conducted by Faraday.

Faraday investigated quantitatively what factors influence the magnitude of the emf induced. He found first of all that the more rapidly the magnetic field changes, the greater the induced emf. He also found that the induced emf depends on the area of the circuit loop.

Thus we say that the emf is proportional to the rate of change of the magnetic flux ФВ, passing through the circuit or loop of area of A.

Magnetic flux for a uniform magnetic field is defined in the same way we did for electric flux namely as

[pic 5][pic 4]

Fig.3

Here  is the component of the magnetic field   perpendicular to the face of the loop and θ is the angle between  and the vector  (representing the area) whose direction is perpendicular to the face of the loop. These quantities are shown in Fig.3 for a square loop of side l whose area is A = l2. If the area is of some other shape or  is not uniform the magnetic flux can be written[pic 6][pic 7][pic 8][pic 9][pic 10]

[pic 11]

 As we saw the lines of   can be drawn such that the number of lines per unit area is proportional to the field strength. Then the flux ФВ can be thought of as being proportional to the total number of lines passing through the area enclosed by the loop. This illustrated in Fig.4. For θ = 90o no magnetic field lines pass through the loop and ФВ = 0, whereas ФВ is a maximum when θ = 0o. The unit of magnetic flux is the tesla- meter, this is called a weber: 1Wb = 1T ∙ m2. [pic 12]

 [pic 13]