The resistance of a metallic conductor increases with a rise in temperature, because increased thermal motion of the atoms causes more frequent collisions of electrons, increasing the resistance. The relation is R(t) = R0 (1 + alpha (delta T)), where R0 is the resistance at the reference temperature, alpha is the temperature coefficient of resistance and delta T is the rise in temperature.

R = R0 (1 + alpha delta T)

A charge q moving with velocity v in a magnetic field B experiences a magnetic force F = q v B sin theta, where theta is the angle between v and B; in vector form F = q(v x B). The force is maximum (F = qvB) when the charge moves perpendicular to the field (theta = 90 deg), and it is zero when the charge moves parallel or antiparallel to the field (theta = 0 or 180 deg). The force is always perpendicular to the velocity, so it does no work.

F = q v B sin theta

The Biot-Savart law states that the magnetic field dB due to a small current element I dl at a point P at distance r is directly proportional to the current I, the length dl and the sine of the angle theta between the element and the line joining it to P, and inversely proportional to the square of the distance r. Mathematically, dB = (mu0/4 pi) (I dl sin theta)/r^2, and its direction is perpendicular to the plane containing dl and r.

dB = (mu0/4 pi)(I dl sin theta)/r^2

Two long parallel wires separated by distance d carrying currents I1 and I2 experience a force per unit length F/L = (mu0 I1 I2)/(2 pi d); the force is attractive if the currents are in the same direction and repulsive if opposite. The ampere is defined as that steady current which, when maintained in two infinitely long parallel wires of negligible cross-section placed 1 metre apart in vacuum, produces a force of 2 x 10^-7 newton per metre of length between them.

F/L = mu0 I1 I2/(2 pi d)

A current loop behaves as a magnetic dipole. The magnetic dipole moment of a coil of N turns each of area A carrying current I is m = N I A, and it is directed perpendicular to the plane of the loop (given by the right-hand rule). Its SI unit is ampere metre squared (A m^2). The torque on it in a field B is m x B.

m = N I A

Diamagnetic substances are weakly repelled by a magnetic field and move from stronger to weaker regions; they have a small negative susceptibility (e.g. bismuth, copper). Paramagnetic substances are weakly attracted by a magnetic field and have a small positive susceptibility (e.g. aluminium, sodium). Ferromagnetic substances are strongly attracted and can be permanently magnetised; they have a large positive susceptibility (e.g. iron, cobalt, nickel).

(1) Magnetic field lines are continuous closed curves that pass from the south to the north pole inside the magnet and from the north to the south pole outside it. (2) The tangent drawn at any point on a field line gives the direction of the magnetic field at that point. (3) Two field lines never intersect each other (as the field can have only one direction at a point). (4) The lines are crowded where the field is strong and spread apart where it is weak.

Magnetic susceptibility (chi) is the ratio of the intensity of magnetisation (M) produced in a material to the magnetising field (H): chi = M/H; it measures how easily a material can be magnetised. Relative permeability (mu_r) is the ratio of the permeability of the material to that of free space. The relation between them is mu_r = 1 + chi.

mu_r = 1 + chi

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