Consider a conducting rod of length l moving with velocity v perpendicular to a uniform magnetic field B. In time dt the rod sweeps an area dA = l (v dt), so the change in flux is d(flux) = B dA = B l v dt. By Faraday's law, the induced emf e = d(flux)/dt = B l v. Alternatively, the free electrons in the rod experience a force qvB that pushes them to one end, setting up an emf e = B v l across the rod.

e = B l v

Self-inductance is the property of a coil by virtue of which it opposes any change in the current flowing through it, by inducing an opposing emf (back emf). It is defined as the flux linkage per unit current (N flux = L I) or from the induced emf e = -L (dI/dt), where L is the self-inductance. Its SI unit is the henry (H).

e = -L dI/dt

Mutual inductance is the property by which a change of current in one coil (primary) induces an emf in a neighbouring coil (secondary) due to the change in flux linkage. It is defined by e2 = -M (dI1/dt), where M is the mutual inductance, whose SI unit is the henry. It depends on the number of turns of the coils, their geometry (area and length), the distance and orientation between them, and the permeability of the core material.

e2 = -M dI1/dt

Given N = 500, initial flux = 0.01 Wb, final flux = 0.05 Wb, so change in flux = 0.05 - 0.01 = 0.04 Wb, and time = 0.1 s. The induced emf (magnitude) = N (change in flux)/time = 500 x 0.04/0.1 = 500 x 0.4 = 200 V.

e = N (change in flux)/time

Eddy currents are the circulating induced currents produced in the body of a conductor when the magnetic flux linked with it changes. They flow in closed loops within the conductor and generally cause heating and energy loss. Applications: they are used in induction furnaces (to melt metals by the heat produced), in electromagnetic braking of trains, in electric (induction) meters, and in induction cooktops. Laminating the cores of transformers reduces energy loss due to eddy currents.

The RMS (root mean square) value of an alternating current is that value of steady direct current which produces the same heating effect in a given resistance in the same time as the alternating current does. For a sinusoidal current of peak value I0, the RMS value is Irms = I0/sqrt(2) = 0.707 I0. Similarly Vrms = V0/sqrt(2). AC meters read RMS values.

Irms = I0/sqrt(2)

Inductive reactance is the opposition offered by an inductor to the flow of alternating current, given by XL = omega L = 2 pi f L; it increases with frequency. Capacitive reactance is the opposition offered by a capacitor to alternating current, given by XC = 1/(omega C) = 1/(2 pi f C); it decreases with frequency. Both are measured in ohm. For direct current (f = 0), XL = 0 and XC is infinite.

XL = omega L; XC = 1/(omega C)

In a series LCR circuit connected to an AC source, the total opposition to current is called impedance Z, given by Z = sqrt(R^2 + (XL - XC)^2), where XL is the inductive reactance and XC is the capacitive reactance. The phase angle phi between the applied voltage and the current is given by tan(phi) = (XL - XC)/R. The current is I = V/Z.

Z = sqrt(R^2 + (XL - XC)^2)

Resonance in a series LCR circuit occurs when the inductive reactance equals the capacitive reactance (XL = XC). At resonance the impedance is minimum (Z = R), the current is maximum, and the circuit behaves as purely resistive. The condition XL = XC gives omega L = 1/(omega C), so omega^2 = 1/(LC), and the resonant frequency f = 1/(2 pi sqrt(LC)). Such a circuit is used for tuning radios and televisions.

f = 1/(2 pi sqrt(LC))

The average power in an AC circuit is P = Vrms Irms cos(phi), where cos(phi) is called the power factor and phi is the phase difference between voltage and current. Thus the power factor is the ratio of true power to apparent power (cos phi = R/Z). Wattless current is the component of the AC current (Irms sin phi) that is 90 degrees out of phase with the voltage; it consumes no average power, so it is called the idle or wattless current.

P = Vrms Irms cos(phi)

A transformer works on the principle of mutual induction: an alternating current in the primary coil produces a changing flux that induces an emf in the secondary coil. For an ideal transformer, Vs/Vp = Ns/Np = Ip/Is, where V is voltage, N is number of turns and I is current. A step-up transformer has more turns in the secondary (Ns > Np) and increases the voltage; a step-down transformer has fewer turns in the secondary (Ns < Np) and decreases the voltage.

Vs/Vp = Ns/Np = Ip/Is

For an ideal transformer, Ns/Np = Vs/Vp. Given Vp = 2200 V, Vs = 220 V, Np = 5000. So Ns = Np x (Vs/Vp) = 5000 x (220/2200) = 5000 x 0.1 = 500 turns. Since the secondary has fewer turns than the primary, it is a step-down transformer, as expected.

Ns/Np = Vs/Vp

Displacement current is the current that arises due to a changing electric field (or changing electric flux) between the plates of a capacitor, even though no charge actually flows across the gap. It is given by Id = epsilon0 (d(electric flux)/dt). Maxwell introduced it to make Ampere's law consistent while charging a capacitor: the total current (conduction current plus displacement current) is continuous, so the modified Ampere-Maxwell law is the integral of B.dl = mu0 (I + Id).

Id = epsilon0 d(electric flux)/dt

(1) Electromagnetic waves are transverse in nature, with the electric field E and magnetic field B oscillating perpendicular to each other and to the direction of propagation. (2) They do not require a material medium and can travel through vacuum. (3) They travel through vacuum with the speed of light, c = 3 x 10^8 m/s. (4) They carry energy and momentum, and the ratio of the amplitudes of E and B equals c (E0/B0 = c).

c = E0/B0

In order of increasing frequency (decreasing wavelength): (1) Radio waves - used in radio and television communication. (2) Microwaves - used in radar and microwave ovens. (3) Infrared - used in remote controls and thermal imaging. (4) Visible light - used for vision and photography. (5) Ultraviolet - used to sterilise water and in detecting forgery. (6) X-rays - used in medical imaging and detecting fractures. (7) Gamma rays - used in cancer treatment (radiotherapy).

The speed of electromagnetic waves in vacuum is given by c = 1/sqrt(mu0 epsilon0), where mu0 is the permeability and epsilon0 the permittivity of free space. Substituting mu0 = 4 pi x 10^-7 and epsilon0 = 8.85 x 10^-12 gives c = 3 x 10^8 m/s, which equals the measured speed of light, showing that light is an electromagnetic wave. In a medium the speed is v = 1/sqrt(mu epsilon).

c = 1/sqrt(mu0 epsilon0)

The mirror formula relates the object distance u, image distance v and focal length f of a spherical mirror: 1/v + 1/u = 1/f. The linear magnification is m = -v/u = height of image/height of object. The focal length f = R/2, where R is the radius of curvature. The New Cartesian sign convention is used for the distances.

1/v + 1/u = 1/f

Using the sign convention, u = -20 cm and f = -15 cm. From the mirror formula 1/v = 1/f - 1/u = 1/(-15) - 1/(-20) = -1/15 + 1/20 = (-4 + 3)/60 = -1/60, so v = -60 cm. The magnification m = -v/u = -(-60)/(-20) = -3. The image is real (v negative), inverted (m negative) and magnified three times, formed 60 cm in front of the mirror.

1/v + 1/u = 1/f

The laws of refraction are: (1) The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane. (2) For a given pair of media and a given colour of light, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, called the refractive index: sin i/sin r = n (Snell's law). Refraction occurs because light travels at different speeds in different media.

sin i/sin r = n

Total internal reflection is the phenomenon in which a ray of light travelling from a denser to a rarer medium is completely reflected back into the denser medium when the angle of incidence exceeds a certain angle called the critical angle. Conditions: (1) light must travel from a denser to a rarer medium; (2) the angle of incidence must be greater than the critical angle C, where sin C = 1/n. Applications: optical fibres, sparkling of diamonds, and mirages.

sin C = 1/n