Motional emf, eddy currents, inductance and the AC generator

3.1 Motional emf

When a conductor moves through a magnetic field, the free electrons inside it feel the magnetic force and shift along the rod, which sets up an emf across its ends. For a rod of length l moving with speed v perpendicular to a field B, the emf is B l v, and the end towards which the electrons move is the lower potential. If the rod slides on a pair of rails that form a closed circuit with resistance R, the current is B l v / R. The flux through the loop changes because the area changes, so the same result follows from Faraday's law, and this agreement is a useful check in an exam.

The moving rod needs a force to keep it moving at constant speed. The rod feels a magnetic force that opposes its motion, so an external agent must work at the rate F v = (B squared l squared v squared)/R. This mechanical power equals the electrical power dissipated in the resistor, which is another example of energy conservation in induction.

For a rod of length l rotating with angular speed omega about one end, in a field perpendicular to the plane of rotation, the different parts of the rod have different speeds. Adding the contributions gives an emf of half B omega l squared. For a coil of N turns and area A rotating with angular speed omega in a uniform field, the flux is N B A cos omega t, so the emf is N B A omega sin omega t. The emf alternates, its peak value is N B A omega, and it is zero when the coil plane is perpendicular to the field and maximum when the plane is parallel to the field.

3.2 Eddy currents

When a solid conductor sits in a changing magnetic field, or moves through a field, the induced currents circulate in the body of the conductor instead of following a wire. These eddy currents produce their own fields that oppose the change, and their energy is dissipated as heat. The effect has both good and bad uses. It can be used to brake a moving train or a galvanometer coil smoothly, to heat a metal in an induction furnace, or to damp the oscillation of a balance. It is a nuisance in a transformer, where the changing flux in the iron core would set up large eddy currents, so the core is made of thin insulated sheets called laminations that break up the current paths and reduce the loss.

3.3 Self and mutual inductance

When the current in a coil changes, the flux through the coil itself changes, and an emf is induced that opposes the change. This is self induction: the induced emf is minus L times di/dt, with L the self inductance of the coil. Take a long solenoid of N turns, length l and area A, the flux per turn is mu0 n I A, the total flux linkage is N phi, and the self inductance is mu0 N squared A / l, or mu0 n squared A l. The unit is the henry, the inductance for which a current changing at one ampere per second gives an emf of one volt.

When two coils are placed close together, a changing current in one produces a changing flux in the other and an emf is induced in the second coil. The mutual inductance M measures this coupling: emf2 = -M di1/dt. For two coaxial solenoids of length l and area A, with N1 turns in one and N2 in the other, M = mu0 N1 N2 A / l. Mutual inductance depends on the geometry of the two coils, their separation and relative orientation, and on the material of the core; it is largest when the coils are wound on top of each other and share the whole flux.

An inductor stores energy in its magnetic field. The energy is half L I squared, and for a solenoid this works out to the energy density B squared / (2 mu0) spread through the volume of the field. That expression is general: wherever a magnetic field exists, the field carries energy with density B squared / (2 mu0).

3.4 The AC generator

A generator turns mechanical energy into electrical energy by the same motional emf. In its simplest form a rectangular coil of N turns rotates in a uniform magnetic field about an axis perpendicular to the field. The flux linkage is N B A cos omega t, and the emf is N B A omega sin omega t, so the output alternates with the rotation. Slip rings and brushes connect the rotating coil to the external circuit. In an actual generator the coil is the armature, the field is produced by electromagnets, and the emf is increased by using many turns and a strong field. The direction of the emf reverses every half rotation, which is the origin of alternating current, and the peak value is N B A omega, so faster rotation and a stronger field both increase the output.

3.5 Inductance in practice

A coil has self inductance for two reasons. The flux through each turn depends on the current, and the geometry decides how much flux a given current produces. That is why a solenoid with more turns per unit length has a large inductance, and why a coil on an iron core has a much larger inductance than the same coil in air. Mutual inductance works the same way for two coils, and it is the principle of the transformer: a changing current in the primary produces a changing flux in the shared core, and that flux links the secondary. The two coils need a common magnetic circuit to couple well, which is why a transformer core is a closed loop of laminated soft iron.