Faraday's first law states that whenever the magnetic flux linked with a closed circuit changes, an emf is induced in the circuit, and it lasts as long as the flux is changing. Faraday's second law states that the magnitude of the induced emf is equal to the rate of change of magnetic flux linked with the circuit: e = -N (d(flux)/dt), where N is the number of turns. The negative sign is due to Lenz's law.

e = -N d(flux)/dt

Lenz's law states that the direction of the induced current (or emf) is always such that it opposes the change in magnetic flux that produces it. For example, if a magnet is pushed towards a coil, the induced current opposes its approach. Lenz's law is a consequence of the law of conservation of energy, because work has to be done against the opposing force, and this work appears as electrical energy.

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

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)

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)

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

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

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

The power of a lens is a measure of its ability to converge or diverge light and is defined as the reciprocal of its focal length in metres: P = 1/f (f in metres). Its SI unit is the dioptre (D). A converging (convex) lens has positive power and a diverging (concave) lens has negative power. For two thin lenses in contact, the total power is the sum: P = P1 + P2.

P = 1/f; P = P1 + P2

Huygens' principle states that: (1) every point on a given wavefront acts as a source of new disturbance called secondary wavelets, which spread out in all directions with the speed of the wave; and (2) the new wavefront at a later instant is the forward envelope (tangential surface) of all these secondary wavelets. This principle is used to explain the laws of reflection and refraction and the propagation of light as a wave.

For constructive interference (bright fringe), the path difference between the two interfering waves must be an integral multiple of the wavelength: path difference = n lambda, where n = 0, 1, 2, ... For destructive interference (dark fringe), the path difference must be an odd multiple of half the wavelength: path difference = (2n - 1) lambda/2. Equivalently, the phase difference is 2n pi for constructive and (2n - 1) pi for destructive interference.

constructive: n lambda; destructive: (2n-1) lambda/2

Coherent sources are two sources of light that emit waves of the same frequency (or wavelength) and have a constant phase difference between them. Conditions for sustained (steady) interference: (1) the two sources must be coherent; (2) they must have the same frequency and nearly equal amplitudes; and (3) they must be narrow and close together, and the light should preferably be monochromatic. In practice, coherent sources are obtained from a single source (for example, using two slits).

(1) Interference is due to the superposition of waves from two (or more) different coherent sources, whereas diffraction is due to the superposition of secondary wavelets coming from different parts of the same wavefront. (2) In interference all bright fringes are of equal intensity and equal width, whereas in diffraction the central maximum is the brightest and the intensity of the secondary maxima decreases rapidly on either side.

Polarisation is the phenomenon of restricting the vibrations of the electric field of a light wave to a single plane perpendicular to the direction of propagation; it shows that light is a transverse wave. Brewster's law states that when unpolarised light is incident on a transparent surface at a particular angle called the polarising angle (theta_p), the reflected light is completely plane-polarised, and the refractive index of the medium is n = tan(theta_p).

n = tan(theta_p)

The photoelectric effect is the phenomenon of emission of electrons (called photoelectrons) from the surface of a metal when light of suitable frequency (usually ultraviolet or visible for some metals) falls on it. The emitted electrons carry kinetic energy. The effect occurs only when the frequency of the incident light is greater than a certain minimum value called the threshold frequency, and it provided evidence for the particle (photon) nature of light.