Equations are crucial to solving physics questions and there are an enormous number of them. A list of formulae is included in a-level physics exams, so there is no need to remember them all. However, to be able to use these equations they need to be understood, what do the symbols mean and when can the equation be applied? On this formula sheet each equation is listed and then briefly explained. The following formulas are specifically for the AQA physics specification but the topics are similar between different specifications. Where needed, units are shown in square brackets e.g. Length [m].
There is also a formula sheet for AS level physics here.
c - Speed of light in a vacuum = 3 × 108 [m/s]
g - Gravitational acceleration (on Earth) = 9.81 [m/s2]
G - Gravitational constant = 6.67 × 10-11 [Nm2kg-2]
R - Molar gas constant = 8.31 [JK-1mol-1]
k - Boltzmann's constant = 1.38 × 10-23 [J/K]
NA - Avogadro's number = 6.02 × 1023 [mol-1]
ε0 - Permittivity of free space = 8.85 × 10-12 [F/m]
2π radians = 360 degrees
273.15 K = 0 oC
magnitude of angular speed,
Angular velocity [s-1] = Velocity [m/s] ÷ Radial distance from centre of rotation [m]
= 2 × Pi × Frequency of rotation [Hz]
Angular velocity measures the speed of rotation, the change in angle per second. This angle is given in radians.
Centripetal acceleration [m/s2] = Square of velocity [m2/s2] ÷ Radius [m]
= Square of angular velocity [s-2] × Radius [m]
Centripetal acceleration is responsible for keeping an object in circular motion. This acceleration is at right angles to the object's velocity and points towards the centre of rotation. Hence, the speed remains constant but the direction is constantly changing.
Centripetal force [N] = Mass [kg] × Square of velocity [m2/s2] ÷ Radius [m]
= Mass [kg] × Square of angular velocity [s-2] × Radius [m]
By Newton's 2nd law, the centripetal force is equal to the centripetal acceleration multiplied by the mass of the object in circular motion.
Simple harmonic motion
Acceleration [m/s2] = -1 × Square of angular velocity [s-2] × Displacement [m]
Simple harmonic motion (SHM) happens when the acceleration of a body is proportional to its displacement and in the opposite direction to the displacement.
The displacement is measured from the equilibrium position (x = 0).
Displacement [m] = Amplitude [m] × Cosine(Angular velocity [s-1] × Time elapsed since object was at the amplitude position [s])
The amplitude is the maximum displacement of the object from equilibrium.
Velocity [m/s] = (+1 or -1) × Angular velocity [s-1] × Square root(Square of amplitude [m2] - Square of displacement [m2])
The plus or minus symbol indicates a choice of direction for the velocity. Remember that a velocity is a speed in a specific direction.
Maximum speed [m/s] = Angular velocity [s-1] × Amplitude [m]
The maximum speed is achieved when the object passes through the equilibrium position. This corresponds to all of the potential energy having been converted to kinetic energy.
Maximum acceleration [m/s2] = Square of angular velocity [s-2] × Amplitude [m]
The maximum acceleration corresponds to the object being at maximum displacement (x = A). This corresponds to all of the kinetic energy having been converted to potential energy.
for a mass-spring system,
Time period of mass on a spring [s] = 2 × Pi × Square root(Mass [kg] ÷ Spring constant [N/m])
This equation applies to an oscillating mass attached to a spring.
for a simple pendulum,
Time period of a simple pendulum [s] = 2 × Pi × Square root(Length of pendulum [m] ÷ Gravitational acceleration)
A simple pendulum is a point mass hanging from a rod of negligible mass.
energy to change temperature,
Heat energy [J] = Mass [kg] × Specific heat capacity [Jkg-1K-1] × Change in temperature [K]
The heat energy can be added or removed, depending on whether the substance is being heated or cooled.
Specific heat capacity is a property of the material being heated.
energy to change state,
Heat energy [J] = Mass [kg] × Specific latent heat [J/kg]
Specific latent heat is the energy per unit mass to change the state of a specific material. For example, the latent heat of fusion of water = 334,000 J/kg, this is the energy needed to melt ice into water (or freeze water into ice).
Pressure [Pa] × Volume [m3] = Number of moles of gas × Molar gas constant × Temperature [K]
= Number of gas molecules × Boltzmann's constant × Temperature [K]
This equation is only true for an ideal gas, a simplification of the highly complex nature of real life gases.
kinetic theory model,
Pressure [Pa] × Volume [m3] = 1/3 × Number of gas molecules × Mass of one gas molecule [kg] × Square of the root-mean-square speed [m2/s2]
Kinetic theory describes the properties of a gas (pressure, volume etc.) in terms of the random motion of a number of individual gas molecules.
The gas molecules have a wide range of speeds. Root-mean-square speed is the average speed of the molecules. To calculate it, the molecular speeds are squared, a mean is taken and then square rooted.
kinetic energy of gas molecule,
Average kinetic energy of a molecule [J] = 1/2 × Mass of a gas molecule [kg] × Square of root-mean-square speed [m2/s2]
= 3/2 × Boltzmann's constant × Temperature [K]
= 3/2 × Molar gas constant × Temperature [K] ÷ Avogadro's number
The average kinetic energy is directly proportional to the temperature of the gas. In a warmer gas, the gas molecules will, on average, move faster.
force between two masses,
Gravitational force between two masses [N] = Gravitational constant × First mass [kg] × Second mass [kg] ÷ Square of the distance between the masses [m2]
This is the magnitude of the gravitational force. The direction is always towards the other mass, as gravity is only an attractive force.
All distances are measured from the centre of mass of the masses. The centre of mass is a theoretical position where all the mass of an object can be assumed to be located.
gravitational field strength,
Gravitational field strength [N/kg] = Gravitational force [N] ÷ Mass that the force is acting on [kg]
magnitude of gravitational field strength in a radial field,
Gravitational field strength [N/kg] = Gravitational constant × Mass [kg] ÷ Square of the radial distance [m2]
A radial field has field lines that move radially outwards from a central point. Hence, the separation of the field lines (a measure of field strength) increases with distance from the central point. A spherical mass will produce a radial field.
The gravitational field strength on Earth is usually assumed to be a constant value within a few kilometres of the Earth's surface (9.81 m/s2).
Change in work done [J] = Mass [kg] × Change in gravitational potential [J/kg]
This is the work done to move a mass within a gravitational field.
Gravitational potential [J/kg] = -1 × Gravitational constant × Mass [kg] ÷ Radial distance [m]
This is the gravitational potential for a radial field.
Gravitational field strength [N/kg] = -1 × Change in gravitational potential [J/kg] ÷ Change in radius [m]
Electric fields and capacitors
force between two point charges,
Electrical force between two charges [N] = (1 ÷ (4 × Pi × Permittivity of free space)) × First charge [C] × Second charge [C] ÷ Square of the distance between the charges [m2]
The direction of the force is decided by the signs of the two charges. Like charges repel and opposite charges attract.
Notice the similar appearance of this equation to the gravitational force equation. The equations for gravitational and electrical fields follow a very similar pattern.
force on a charge,
Electrical force [N] = Electric field strength [N/C] × Charge [C]
field strength for a uniform field,
Electrical field strength [N/C] = Voltage [V] ÷ Distance separating the plates [m]
A uniform field is a constant strength at every point within the field. A uniform field is well approximated by applying a voltage across two parallel conducting plates.
Change in work done [J] = Charge [C] × Change in electric potential [V]
This is the work done to move a charge within an electrical field.
field strength for a radial field,
Electric field strength [N/C] = (1 ÷ (4 × Pi × Permittivity of free space)) × Charge [C] ÷ Square of radial distance [m2]
Electrical potential [V] = (1 ÷ (4 × Pi × Permittivity of free space)) × Charge [C] ÷ Radial distance [m]
This is the electric potential for a radial field.
Electrical field strength [N/C] = Change in electrical potential [V] ÷ Change in radial distance [m]
Capacitance [F] = Charge [C] ÷ Voltage [V]
Capacitance is a measure of the ability of an object to store electrical charge.
Capacitance [F] = Area of plates [m2] × Permittivity of free space × Relative permittivity of dielectric ÷ Separation of the plates [m]
This is the capacitance of two parallel metal plates separated by a dielectric, known as a parallel plate capacitor.
capacitor energy stored,
Energy [J] = 1/2 × Charge [C] × Voltage [V]
= 1/2 × Capacitance [F] × Square of the voltage [V2]
= 1/2 × Square of the charge [C2] ÷ Capacitance [F]
Charge [C] = Initial charge [C] × (1 - Exponential(-1 × Time elapsed [s] ÷ Time constant [s]))
Initially, the electrical charge builds up quickly on the capacitor plates. This rate then slows down as the incoming electrons are repelled by the electrons on the plate. Eventually, the charge reaches a constant value, ie. the capacitor is fully charged.
decay of charge,
Charge [C] = Initial charge [C] × Exponential(-1 × Time elapsed [s] ÷ Time constant [s]))
Time constant [s] = Resistance [Ω] × Capacitance [C]
The time constant is the time for the charge to increase/drop by about 37% when a capacitor is charging/decaying.
force on a current,
Force [N] = Magnetic field strength [T] × Current [A] × Length [m]
This equation is only valid when the magnetic field is perpendicular to the current. The force is then perpendicular to both the magnetic field and the current.
The length is only the length of conductor (carrying the current) that lies within the magnetic field.
force on a moving charge,
Force [N] = Magnetic field strength [T] × Charge [C] × Velocity [m/s]
This equation is only valid when the magnetic field is perpendicular to the velocity of the charge. The force is then perpendicular to both the magnetic field and the velocity.
Magnetic flux [Wb] = Magnetic field strength [T] × Area [m2]
Magnetic flux can be visualised as the number of magnetic field lines passing through a surface area. Therefore, a stronger magnetic field corresponds to an increased density of field lines.
magnetic flux linkage,
Magnetic flux linkage [Wb] = Number of turns in the coil × Flux through each coil [Wb]
= Magnetic field strength [T] × Cross-sectional area of the coil [m2] × Number of turns in the coil × Cosine(Angle between the normal to the coil's cross sectional area and the direction of the magnetic field)
The magnetic flux linkage is the total magnetic flux through a coil of wire.
magnitude of induced emf,
Emf [V] = Number of turns in the coil × Change in magnetic flux through each coil [Wb] ÷ Change in time [s]
This states that the emf induced by moving a coil of wire through a magnetic field is equal to the rate of change of magnetic flux linkage.
The direction of the induced emf is such that the magnetic field it induces opposes the motion of the coil.
emf induced in a rotating coil,
Emf [V] = Magnetic field strength [T] × Cross-sectional area of the coil [m2] × Number of turns in the coil × Angular velocity [s-1] × Sine(Angular velocity [s-1] × Time [s])
Root mean square current [A] = Peak current [A] ÷ Square root(2)
Root mean square voltage [V] = Peak voltage [V] ÷ Square root(2)
An alternating current (AC) periodically changes the direction of flow. This is in contrast to direct current (DC) that only flows in one direction. The root mean square (RMS) values of an AC supply use the same amount of power as a DC supply with equal values.
Number of turns in the secondary coil ÷ Number of turns in the primary coil = Voltage across the secondary coil [V] ÷ Voltage across the primary coil [V]
A transformer is used to step up (increase) or step down (decrease) the voltage of an AC supply. The alternating current in the primary coil causes a changing magnetic field in the metal core. This in turn induces a changing emf in the secondary coil.
Efficiency = (Current through the secondary coil [A] × Voltage across the secondary coil [V]) ÷ (Current through the primary coil [A] × Voltage across the primary coil [V])
Efficiency is simply the ratio of the power produced (in the secondary coil) to the power input (in the primary coil). Laminating the metal core will increase efficiency by reducing the formation of eddy currents.
inverse square law for γ radiation,
Intensity of radiation [W/m2] = Constant ÷ Square of the distance from the radiation source [m2]
The intensity of gamma radiation is inversely proportional to the square of the distance e.g. doubling the distance from the radiation source decreases the intensity by 1/4. The constant is determined by the radiation source used.
Change in the number of atoms ÷ Change in time [s] = -1 × Decay constant of the radionuclide [s-1] × Number of atoms
The decay constant is the probability per second that a nucleus will decay.
Number of atoms at current time = Number of atoms initially × Exponential(-1 × Decay constant of the radionuclide [s-1] × Time elapsed [s])
Radioactive decay is an exponential decay, meaning the number of radioactive atoms falls rapidly at first and then slowly tend towards zero.
Activity of a sample [Bq] = Decay constant of the radionuclide [s-1] × Number of atoms of this radionuclide in the sample
The activity of a sample is the number of decays per second.
Half life of a radioactive nuclide [s] = Natural logarithm(2) ÷ Decay constant of the radionuclide [s-1]
The half life is the time for half of the sample to decay e.g. after three half lives there will only be 1/8 of the original sample left that hasn't decayed.
Nuclear radius [m] = Radius of a nucleon (approximately 1.3 fm) × Number of nucleons to the power of 1/3
This relation has been derived from experimental data. The femtometre is a convenient unit for measuring the size of nuclei (1 fm = 1 × 10-15 m).
Change in energy [J] = Change in mass [kg] × Square of the speed of light in a vacuum [m2/s2]
© 2017 Sam Brind