Coulomb's law states that the force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, acting along the line joining them. Mathematically, F = k q1 q2 / r^2, where k = 1/(4 pi epsilon0) = 9 x 10^9 N m^2 C^-2 in free space.

F = k q1 q2 / r^2

The electric field intensity at a point is the force experienced by a unit positive test charge placed at that point. It is a vector quantity given by E = F/q0, where q0 is the small positive test charge. Its SI unit is newton per coulomb (N/C) or equivalently volt per metre (V/m).

E = F/q0

When a dipole of moment p is placed in a uniform electric field E at an angle theta, the two charges experience equal and opposite forces qE, forming a couple. The magnitude of the torque = force x perpendicular distance = qE x (2a sin theta) = (q x 2a) E sin theta = pE sin theta. In vector form, torque = p x E. The torque tends to align the dipole with the field.

torque = pE sin theta = p x E

The electric potential at a point is the work done in bringing a unit positive charge from infinity to that point against the electric field. It is a scalar quantity and its SI unit is the volt (V), where 1 volt = 1 joule per coulomb. The potential due to a point charge Q at distance r is V = kQ/r = Q/(4 pi epsilon0 r).

V = kQ/r

The electric field is the negative gradient of the electric potential: E = -dV/dr. This means the field points in the direction in which the potential decreases most rapidly. The negative sign indicates that the electric field is directed from a region of higher potential to a region of lower potential, that is, potential decreases along the direction of the field.

E = -dV/dr

The capacitance of a conductor is the ratio of the charge given to it to the resulting rise in its potential: C = Q/V. It is a measure of the ability of the conductor to store charge. Its SI unit is the farad (F), where 1 farad = 1 coulomb per volt. Capacitance depends on the size and shape of the conductor and the surrounding medium.

C = Q/V

The energy stored in a charged capacitor is the work done in charging it. It can be written in three equivalent forms: U = (1/2) C V^2 = (1/2) Q V = Q^2/(2C), where C is the capacitance, Q the charge and V the potential difference. This energy is stored in the electric field between the plates.

U = (1/2) C V^2 = (1/2) QV = Q^2/(2C)

When a dielectric of dielectric constant K is fully inserted between the plates, the capacitance increases K times: C = K C0, where C0 is the capacitance with air. This is because the dielectric gets polarised and sets up an internal field opposite to the applied field, reducing the net field and hence the potential difference for the same charge; since C = Q/V, a smaller V means a larger C.

C = K C0

Ohm's law states that, at constant temperature, the current flowing through a conductor is directly proportional to the potential difference across its ends, so V = IR, where R is a constant called the resistance. Resistance is the opposition offered by a conductor to the flow of current and is defined as R = V/I. Its SI unit is the ohm.

V = IR

Resistivity (specific resistance) is the resistance of a conductor of unit length and unit area of cross-section; it depends on the material and temperature but not on its dimensions. The resistance of a wire is R = rho L/A, so it is directly proportional to its length L and inversely proportional to its area of cross-section A. The SI unit of resistivity is the ohm metre.

R = rho L/A

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

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 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