Young's modulus (Y) measures resistance to change in length (longitudinal stress/strain). Bulk modulus (B) measures resistance to change in volume (volume stress/strain) under uniform pressure. Shear (rigidity) modulus (G) measures resistance to change in shape (shearing stress/strain).

Stefan's (Stefan-Boltzmann) law: the total energy radiated per unit area per unit time by a black body is proportional to the fourth power of its absolute temperature, E = sigma T^4. Wien's displacement law: the wavelength at which the emission is maximum is inversely proportional to the absolute temperature, lambda_max * T = constant (= 2.9 x 10^-3 m K).

E = sigma T^4 ; lambda_max T = b

The mean free path is the average distance a gas molecule travels between two successive collisions. It increases when the number density of molecules is low and when the molecular diameter is small; that is, it is inversely proportional to the number density and to the square of the molecular diameter.

lambda = 1 / (sqrt(2) pi d^2 n)

The centre of mass is the point at which the entire mass of the system can be considered to be concentrated, and where an applied external force produces the same acceleration as on the whole system. For two particles of masses m1 and m2 at positions x1 and x2, x_cm = (m1 x1 + m2 x2) / (m1 + m2).

x_cm = (m1 x1 + m2 x2) / (m1 + m2)

With depth d, g decreases as g' = g (1 - d/R), assuming uniform density, because only the mass within the smaller inner sphere attracts the body. At the centre (d = R) the value becomes zero, since the mass is symmetrically distributed all around.

g' = g (1 - d/R)

Resolution of a vector is the process of splitting it into two or more components, usually along mutually perpendicular directions. For a vector A at angle theta to the x-axis, the rectangular components are Ax = A cos(theta) along the x-axis and Ay = A sin(theta) along the y-axis, with A = sqrt(Ax^2 + Ay^2).

Ax = A cos(theta) ; Ay = A sin(theta)

A quasi-static process is one carried out infinitely slowly so that the system stays in thermal and mechanical equilibrium with its surroundings at every stage. It is important because only for such (reversible) processes can the state variables (P, V, T) be defined throughout, and the work done can be represented as an area on a P-V diagram.

Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, directed along the line joining them: F = G m1 m2 / r^2, where G is the universal gravitational constant (6.67 x 10^-11 N m^2 / kg^2).

F = G m1 m2 / r^2

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)