When a coil of wire and a bar magnet are moved in relation to each other, an electric current is produced. This current is produced because the strength of the field at the location of the coil changes.

This current is an induced current and the emf that produces it is an induced emf.

An induced emf can also be produced by changing the area of a coil in a constant magnetic field. The rotation of a coil in a constant magnetic field also produces an induced emf.

The current flows because the coil is a closed circuit. If it were an open circuit, no current would flow, so there would be no induced current. There would, however, be an induced emf.

Producing an induced emf with a magnetic field is called electromagnetic induction.

Moving a conducting rod in a magnetic field causes opposite charges to build up on opposite ends (or sides) of the rod.

This charge difference builds up until the attractive force the charges have for each other is equal in magnitude to the magnetic force.

The induced emf produced is called a motional emf. If the rod stops moving, the magnetic force vanishes, and the emf disappears.

The electric force on a positive charge at the end of the rod is Eq. The electric field magnitude is given by voltage difference between

the ends (the emf ε) divided by the length L.

So, Eq = (ε/L)q. The magnetic force is F = qvB (if the charge moves perpendicular to the field). These two forces are

balanced, so (ε/L)q = qvB.

emf, ε = vBL.

Ex. 1 - The rod in the above figure is moving at a speed of 5.0 m/s in a direction perpendicular to a 0.80-T magnetic field. The rod has a length of 1.6 m and a negligible electrical resistance (like the rails). The bulb has a resistance of 96 Ω. Find (a) the emf produced, (b) the induced current, (c) the electrical power delivered to the bulb, (d) the energy used by the bulb in 60.0 s.

Motional emf arises because a magnetic force acts on the charges in a conductor moving through a magnetic field. But if this emf causes a current, a second magnetic force arises because the current I in the conductor is perpendicular to the magnetic field.

The current-carrying material experiences a magnetic force (F = ILB sin90°) that is opposite the velocity of the material. This force, by itself, would slow the rod until it stopped, which would also stop the current.

To keep the rod moving, a force must be added to balance this opposing force. This added force supplies the energy to produce the electrical energy of the current in the conducting material.

Ex. 2 - An external agent supplies a 0.086-N force that keeps the rod moving at a constant speed of 5.0 m/s. Determine the work done in 60.0 s by the external agent.

This opposing force to the motion of the conducting material through a magnetic field is one consequence of the law of conservation of energy.

Ex. 3 - A conducting rod is free to slide down (frictionlessly) between two vertical copper tracks. A constant magnetic field B is directed perpendicular to the motion of the rod. The only force acting on the rod is its weight. Suppose a resistance R is connected between the tops of the tracks. (a) Does the rod now fall with the acceleration due to gravity? (b) How does energy conservation apply to what happens?

A conductor moving across a magnetic field crosses an area A. The product of the field strength B and the area A, BA is called the magnetic flux: BA = . The magnitude of the induced emf is the change in magnetic flux ∆ = - 0 divided by the time interval ∆t = t - t0 during which the

change occurs. ε = ∆/∆t.

This is almost always written with a minus sign: ε = -∆/∆t as a reminder that the polarity of the induced emf is such that it produces a force opposing the direction of motion.

This formula,

ε = -∆/∆t, canbe applied to all possible ways of generating induced emfs.

If the direction of the magnetic field is not perpendicular to the surface swept out by the moving conductor, we use the following general equation to calculate magnetic flux:

ε = -∆/∆t cos .

The unit of magnetic flux is the tesla•meter2 (T•m2). This unit is called a weber (Wb).

Ex. 4 - A rectangular coil of wire is situated in a constant magnetic field whose magnitude is 0.50 T. The coil has an area of 2.0 m2. Determine the magnetic flux for the three orientations, = 0°, 60.0°, and 90.0°.

The magnetic flux is proportional to the number of field lines that passes through a surface. Thus, one often encounters phrases like. “the flux that passes through a surface bounded by a loop of wire.”

The fact that a change of flux through a loop of wire produces an emf was discovered by Joseph Henry (USA) and Michael Faraday (English). The key word here is “change.” Without a change in flux, there is no emf. A flux that is constant over time creates no emf.

Faraday’s Law of Electromagnetic Induction:

The average emf ε induced in a coil of N loops is:

ε = -N ∆/∆t. The unit is the volt (surprise, surprise).

An emf is generated if the flux changes for any reason. Since = BA cos , B, A, or could change.

Ex. 5 - A coil of wire consists of 20 turns, each of which has an area of 1.5 x 10-3 m2. A magnetic field is perpendicular to the surface of the loops at all times. At time t0 = 0, the magnitude of the magnetic field at the location of the coil is B0 = 0.050 T. At a later time t = 0.10 s, the magnitude of the field has increased to B = 0.060 T. (a) Find the average emf induced in the coil during this time. (b) What would be the value of the average induced emf if the magnitude of the magnetic field decreased from 0.060 T to 0.050 T in 0.10 s?

Ex. 6 - A flat coil of wire has an area of 0.020 m2 and consists of 50 turns. At t0 = 0 the coil is oriented so the normal to its surface is parallel to a constant magnetic field of magnitude 0.18 T. The coil is then rotated through an angle of f = 30.0° in a time of 0.10 s. (a) Determine the average induced emf. (b) What would be the induced emf if the coil were returned to its initial orientation in the same time of 0.10 s?