Kirchhoff's junction (current) law states that the algebraic sum of currents meeting at a junction is zero, that is, the total current entering a junction equals the total current leaving it; it is based on conservation of charge. Kirchhoff's loop (voltage) law states that the algebraic sum of the changes in potential around any closed loop of a circuit is zero; it is based on conservation of energy.

sum(I) at junction = 0; sum(V) around loop = 0

A Wheatstone bridge has four resistances P, Q, R and S arranged in a quadrilateral, with a galvanometer between the junctions of P-Q and R-S, and a cell across the other diagonal. When the bridge is balanced, no current flows through the galvanometer, so the junction connected by the galvanometer are at the same potential. Applying Kirchhoff's laws, the potential drops give P/Q = R/S. This is the balance condition; if three resistances are known, the fourth can be found.

P/Q = R/S

Given emf E = 2 V, internal resistance r = 0.5 ohm, external resistance R = 4.5 ohm. Current I = E/(R + r) = 2/(4.5 + 0.5) = 2/5 = 0.4 A. Terminal potential difference V = I R = 0.4 x 4.5 = 1.8 V (equivalently V = E - I r = 2 - 0.4 x 0.5 = 1.8 V).

I = E/(R + r); V = E - I r

When n identical cells, each of emf E and internal resistance r, are connected in series (all in the same direction), the effective emf is n E and the total internal resistance is n r. The current through an external resistance R is I = n E/(R + n r). Series grouping is advantageous when the external resistance is much larger than the internal resistance.

I = nE/(R + nr)

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 force on a current-carrying conductor is F = B I L sin theta. Here B = 0.2 T, I = 4 A, L = 0.5 m and theta = 90 deg (so sin theta = 1). F = 0.2 x 4 x 0.5 x 1 = 0.4 N. The direction of the force is given by Fleming's left-hand rule.

F = B I L 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

Consider a circular loop of radius R carrying current I. By the Biot-Savart law, each element I dl is perpendicular to the line joining it to the centre (theta = 90 deg), so dB = (mu0/4 pi)(I dl)/R^2. All elements produce fields in the same direction (perpendicular to the plane of the loop) at the centre, so they simply add. Integrating dl around the loop gives the circumference 2 pi R: B = (mu0/4 pi)(I/R^2)(2 pi R) = mu0 I/(2R). For N turns, B = mu0 N I/(2R).

B = mu0 I/(2R)

Ampere's circuital law states that the line integral of the magnetic field B around any closed loop is equal to mu0 times the total current enclosed by the loop: the integral of B.dl = mu0 I(enclosed). Applying it to a long solenoid with n turns per unit length carrying current I, the magnetic field inside is uniform and given by B = mu0 n I, directed along the axis; the field outside an ideal long solenoid is nearly zero.

B = mu0 n I

When a charged particle of charge q and mass m enters a magnetic field B perpendicularly with speed v, the magnetic force qvB provides the centripetal force m v^2/r. Equating: q v B = m v^2/r, so the radius r = m v/(q B). The time period T = 2 pi r/v = 2 pi m/(q B), which is independent of the speed and radius. This principle is used in the cyclotron.

r = mv/(qB); T = 2 pi m/(qB)

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 moving coil galvanometer works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. The deflection of the coil is directly proportional to the current passing through it (I is proportional to the deflection). To convert a galvanometer into an ammeter, a small resistance (shunt) is connected in parallel with it, so most of the current bypasses the galvanometer. To convert it into a voltmeter, a high resistance is connected in series with it, so that only a small current flows and it can measure potential difference.

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.

The three elements of the earth's magnetism are: (1) Magnetic declination - the angle between the geographic meridian and the magnetic meridian at a place. (2) Magnetic dip or inclination - the angle made by the earth's total magnetic field with the horizontal at a place. (3) Horizontal component of the earth's field (BH) - the component of the earth's total magnetic field in the horizontal direction, given by BH = B cos(dip). These three completely specify the earth's field at a place.

BH = B cos(dip)

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.