Summary

1. The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as, $${\mathrm{\Phi }}_{\mathrm{B}}=\mathbf{B}\cdot \mathbf{A}=BA\cos \theta$$ where θ is the angle between B and A.
2. Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it, $$\epsilon =-N\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{B}}}{\mathrm{d}t}$$ Here ${\mathrm{\Phi }}_{{\rm B}}$ is the flux linked with one turn of the coil. If the circuit is closed, a current $I=\epsilon /R$ is set up in it, where R is the resistance of the circuit.
3. Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it. The negative sign in the expression for Faraday’s law indicates this fact.
4. When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is $$\epsilon =B[v$$
5. Changing magnetic fields can set up current loops in nearby metal (any conductor) bodies. They dissipate electrical energy as heat. Such currents are eddy currents.
6. Inductance is the ratio of the flux-linkage to current. It is equal to $N\mathrm{\Phi }/I$.
7. A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1). This relation is given by, $${\epsilon }_1=-M_{12}\frac{\mathrm{d}{I}_2}{\mathrm{d}t}$$ The quantity $M_{12}$ is called mutual inductance of coil 1 with respect to coil 2. One can similarly define $M_{21}$ . There exists a general equality, $$M_{12}=M_{21}$$
8. When a current in a coil changes, it induces a back emf in the same coil. The self-induced emf is given by, $$\epsilon =-L\frac{\mathrm{d}I}{\mathrm{d}t}$$ L is the self-inductance of the coil. It is a measure of the inertia of the coil against the change of current through it.
9. The self-inductance of a long solenoid, the core of which consists of a magnetic material of relative permeability ${\mu }_r$ , is given by $$L={\mu }_r{\mu }_0{n}^{2}Al$$ where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length.
10. In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction. If coil of N turn and area A is rotated at ν revolutions per second in a uniform magnetic field B, then the motional emf produced is $$\epsilon =\mathrm{NBA}(2\pi v)\sin (2\pi vt)$$ where we have assumed that at time t = 0 s, the coil is perpendicular to the field.