AQA A-Level Physics: Gravitational and Electric Fields Complete Guide
AQA A-Level Physics: Gravitational and Electric Fields Complete Guide
The fields topics of AQA A-Level Physics are the part of the course where mathematical patterns recur most aggressively. Newton's law of gravitation and Coulomb's law for electric force are the same equation with different constants and different signs. Gravitational and electric field strength share the same definition. Gravitational and electric potential share the same definition. Even orbits and charged-particle motion in uniform fields reuse the centripetal-force algebra from Further Mechanics. If you treat sections 3.7.1 to 3.7.4 as one unified topic rather than four separate ones, you will save weeks of revision time and almost always know what the equation in front of you is supposed to mean.
This guide covers the full content of 3.7.1 (gravitational fields), 3.7.2 (gravitational potential and orbits), 3.7.3 (electric fields), 3.7.4 (electric potential, capacitance and capacitor circuits) and Required Practical 9 (capacitor discharge). It is structured around the analogy between gravitational and electric fields and finishes with the capacitor sub-topic that sits inside Section 3.7.4 but stands a little apart.
Gravitational Fields and Field Strength
A gravitational field is a region in which a mass experiences a force. The gravitational field strength g at a point is the force per unit mass on a small test mass placed there:
g = F / m
so g has units of N kg⁻¹, or equivalently m s⁻². Near the Earth's surface, g ≈ 9.81 N kg⁻¹. Field strength is a vector, pointing in the direction of the force on a positive (i.e. any) test mass — that is, towards the source mass.
There are two field geometries you must know:
- A radial field around a point mass (or a uniform sphere outside its surface): field strength varies as g = GM / r² and the field lines point inward. The Earth, Sun and Moon all produce essentially radial fields outside their surfaces.
- A uniform field in a small region close to a much larger body: the field lines are parallel and evenly spaced, and g is constant. The space inside a typical school laboratory is a uniform-field region of the Earth's gravitational field.
Newton's Law of Gravitation
The force of gravitational attraction between two point masses m₁ and m₂ separated by distance r is given by Newton's law of gravitation:
F = G m₁ m₂ / r²
where G is the universal gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻². The force always acts along the line joining the two masses and is always attractive. The inverse-square form is the source of nearly every nontrivial calculation in the topic — at twice the separation, the force is one quarter; at ten times, one hundredth.
Combining F = GMm / r² with g = F / m gives g = GM / r² for the field strength of a point mass or uniform sphere. At the surface of a planet of mass M and radius R, g_surface = GM / R². Near, but above, the surface, g falls off as 1 / r² where r is measured from the centre, not from the surface — a common slip.
Gravitational Potential and Orbits
The gravitational potential V at a point is the work done per unit mass to bring a small test mass from infinity to that point:
V = −GM / r
The minus sign is essential: gravitational potential is always negative, because work must be done against the attractive field to move the test mass back out to infinity, where V is defined to be zero. Potential is a scalar — it has magnitude but no direction.
The gravitational potential energy of a mass m at a point of potential V is simply E_p = mV. Note that the familiar mgh formula is only the change in potential energy for small height changes near the Earth's surface, where g is effectively constant — for orbital-scale calculations you must use E_p = −GMm / r.
A useful synoptic equation is the escape velocity from a planet of mass M and radius R, found by setting the kinetic energy at the surface equal to the magnitude of the potential energy:
v_esc = √(2GM / R)
For Earth this gives about 11.2 km s⁻¹. Notice that escape velocity does not depend on the mass of the escaping object.
Orbits and Kepler's Third Law
For a satellite in circular orbit at radius r around a body of mass M, the gravitational force provides the centripetal force:
GMm / r² = mv² / r
Cancelling m and rearranging gives the orbital speed v = √(GM / r) and, via v = 2πr / T, the orbital period:
T² = (4π² / GM) r³
This is Kepler's third law: the square of the orbital period is proportional to the cube of the orbital radius. Plotting log T against log r for the planets of the Solar System gives a straight line of gradient 3/2 — a common exam graph.
You should be able to use Kepler's third law to:
- Find the orbital radius of a geostationary satellite (T = 86 400 s, r ≈ 4.2 × 10⁷ m, height above surface ≈ 3.6 × 10⁷ m above the equator).
- Find the mass of the central body if T and r are known (as Cavendish-style problems).
- Compare orbital periods of two satellites with different radii.
Electric Fields and Coulomb's Law
An electric field is a region in which a charged object experiences an electric force. The electric field strength E at a point is the force per unit positive charge:
E = F / Q
in N C⁻¹, or equivalently V m⁻¹. Field strength is a vector, pointing in the direction of the force on a positive test charge.
Coulomb's law for the force between two point charges Q₁ and Q₂ separated by distance r is the electric analogue of Newton's law of gravitation:
F = Q₁ Q₂ / (4πε₀ r²)
where ε₀ is the permittivity of free space, 8.85 × 10⁻¹² F m⁻¹. The constant 1 / (4πε₀) ≈ 8.99 × 10⁹ N m² C⁻². Unlike gravity, the force can be attractive or repulsive: like charges repel, unlike charges attract. The field strength of a point charge Q is:
E = Q / (4πε₀ r²)
For a uniform electric field between two parallel plates at potential difference V and separation d, the field is uniform and given by E = V / d. This is the geometry of the parallel-plate capacitor and turns up constantly in cathode-ray and electron-deflection questions.
Electric Potential and Work Done
The electric potential V at a point is the work done per unit positive charge to bring a small positive test charge from infinity to that point:
V = Q / (4πε₀ r)
For a positive source charge V is positive (the field repels the test charge, so work is done on the test charge to bring it in — when you let go it accelerates back out). For a negative source charge V is negative, exactly mirroring the gravitational case.
The work done in moving a charge q from a point at potential V₁ to a point at potential V₂ is W = q(V₂ − V₁) = qΔV. Conservation of energy then gives the kinetic energy gained by an electron accelerated through a potential difference V:
(1/2) m_e v² = eV
so v = √(2eV / m_e). For V = 1000 V this gives v ≈ 1.9 × 10⁷ m s⁻¹ — a useful sanity check for electron-gun problems.
The Gravitational-Electric Analogy
| Quantity | Gravitational | Electric |
|---|---|---|
| Field strength (vector) | g = F / m | E = F / Q |
| Force law | F = −GMm / r² | F = Q₁Q₂ / (4πε₀r²) |
| Field of a point source | g = GM / r² | E = Q / (4πε₀r²) |
| Potential (scalar) | V = −GM / r | V = Q / (4πε₀r) |
| Potential energy | E_p = mV | E_p = qV |
| Constant of proportionality | G ≈ 6.67 × 10⁻¹¹ | 1 / (4πε₀) ≈ 8.99 × 10⁹ |
| Sign | Always attractive (V < 0) | Either sign |
Examiners frequently ask candidates to "compare and contrast" gravitational and electric fields. The mark scheme will almost always credit the inverse-square form, the field-from-potential relationship E = −dV / dr (or g = −dV / dr), the always-attractive nature of gravity vs the dual sign of electric forces, and the much larger relative strength of the electric force between fundamental particles.
Capacitance Fundamentals
A capacitor is a device that stores charge and electrical potential energy. The defining relationship is:
Q = CV
where Q is the charge on one plate, V is the potential difference across the capacitor, and C is the capacitance in farads (F). One farad is one coulomb per volt — a very large unit, so most practical capacitors are measured in microfarads (μF) or smaller.
The energy stored in a capacitor is the area under the Q-against-V graph:
E = (1/2) QV = (1/2) CV² = Q² / (2C)
All three forms are useful in different exam contexts. The factor of one-half is because the potential difference rises linearly from zero to V as the capacitor charges, so the average potential difference across which charge is delivered is V/2, not V.
For a parallel-plate capacitor in vacuum, C = ε₀ A / d, where A is the area of one plate and d is the separation. Inserting a dielectric of relative permittivity ε_r multiplies C by ε_r and so increases stored charge at the same voltage. The dielectric works because its molecules polarise in the field, partially cancelling it and reducing the potential difference between the plates for a given charge.
Capacitor Charge and Discharge: The RC Time Constant
When a capacitor discharges through a resistor R, the charge on the plates decays exponentially:
Q = Q₀ e^(−t / RC)
with V = V₀ e^(−t / RC) and I = I₀ e^(−t / RC) following the same exponential. The product RC is the time constant, τ, of the circuit, with units of seconds (since the farad is C V⁻¹ and the ohm is V A⁻¹, the product is C / A = s).
The time constant has a clear physical meaning: in one time constant, the charge falls to 1/e ≈ 37% of its initial value. After two time constants the charge is at 13%, and after five time constants it is at less than 1% — essentially fully discharged.
The half-life of a discharging capacitor is t_½ = RC ln 2 ≈ 0.693 RC. This is the time for the charge (or voltage, or current) to halve. Examiners often ask candidates to demonstrate the link between t_½ and τ from the exponential decay equation.
For charging through a resistor from a battery of EMF V₀, the equations invert:
- Q = Q_max (1 − e^(−t / RC))
- V = V_max (1 − e^(−t / RC))
- I = I₀ e^(−t / RC)
Note that the current decays in the same way whether the capacitor is charging or discharging — the current at any moment is just the rate at which the capacitor is approaching equilibrium with the supply.
Required Practical 9: Capacitor Discharge
Required Practical 9 asks you to investigate the discharge of a capacitor and determine its capacitance from a graph of ln V (or ln I) against time. The standard procedure is:
- Charge a capacitor of known capacitance to a known voltage through a switch.
- Switch the supply out and start a stopwatch as the capacitor discharges through a known resistor.
- Record the voltage across the capacitor (or the current through the resistor) at regular time intervals using a high-input-impedance voltmeter (or a microammeter).
- Plot ln V (or ln I) against t. The straight-line graph has gradient −1/RC, from which τ = RC and hence C can be calculated.
Three common error sources, all of which appear in mark schemes:
- The voltmeter (if its impedance is not very high) draws current through the resistor in parallel with R, decreasing the effective resistance and the time constant.
- The capacitor's own dielectric is not perfectly insulating; "leakage" through it gives extra discharge paths.
- For long time constants, recording must continue for at least 3-5 time constants to get a good straight line of ln V against t.
How to Study This Topic
The "compare and contrast gravitational and electric fields" question appears on roughly one paper in three. Building your revision around the analogy table is the single most efficient use of revision time.
- Build the analogy table from memory weekly. Add a third column for "magnetic fields" once you've started Section 3.7.5.
- Memorise the central equations on each side of the analogy: g = GM/r², V = −GM/r, E_p = mV, F = GMm/r²; E = Q/(4πε₀r²), V = Q/(4πε₀r), E_p = qV, F = Q₁Q₂/(4πε₀r²).
- Drill orbital-mechanics problems based on Kepler's third law — geostationary satellites and Earth-orbiting satellites are the most common contexts.
- Memorise the three forms of the capacitor energy equation and the role of the time constant τ = RC, including the t_½ = RC ln 2 derivation.
- Practise Required Practical 9 with a fake dataset, including the linearisation step from V = V₀ e^(−t/RC) to ln V = ln V₀ − t/RC.
Related LearningBro Courses
LearningBro's AQA A-Level Physics: Gravity, Electric Fields and Capacitors course covers all of the above in eight lessons:
- Gravitational Fields — g, Newton's law of gravitation, radial and uniform fields.
- Gravitational Potential and Orbits — V = −GM/r, escape velocity, Kepler's third law.
- Electric Fields and Coulomb's Law — Coulomb's inverse-square force and parallel-plate fields.
- Electric Potential — V = Q/(4πε₀r) and the work done on a charge.
- Capacitance Fundamentals — Q = CV and parallel-plate geometry.
- Capacitor Charge and Discharge — exponential decay, RC time constant, half-life.
- Energy Stored in Capacitors and Applications — E = (1/2)CV² and capacitor uses.
- Required Practical 9: Capacitor Discharge — method, ln-graph linearisation, error analysis.
Common Exam Pitfalls
This unit lives or dies by careful sign conventions and a willingness to compare gravitational and electric quantities side by side. The biggest mark-droppers are predictable.
- Forgetting the minus sign on potential. Gravitational potential V = -G M / r is always negative because gravity is attractive and we set V = 0 at infinity. A "find the work done to move a satellite from orbit A to orbit B" question requires Δ V to be computed as V_B - V_A with the negative signs intact. Candidates who quietly drop the minus signs get a magnitude right but a direction wrong.
- Field strength versus potential. g and E are vectors with units of N kg^-1 and N C^-1 (equivalently V m^-1); V_g and V_e are scalars in J kg^-1 and J C^-1. The relation between them, g = -dV/dr and E = -dV/dr, is asked routinely on 4-mark "explain" questions.
- Inverse-square versus inverse-cube confusion. Force and field strength are inverse-square in r; potential is inverse-first in r; potential energy U = -G M m / r is also inverse-first. Sketching graphs from memory without re-deriving them leads to wrongly-curved lines.
- Capacitor energy formulas used carelessly. E = (1/2) Q V = (1/2) C V^2 = Q^2 / (2 C). All three are equivalent — but only at the same instant. If a charged capacitor is connected to an identical uncharged capacitor in parallel, charge is conserved but energy is not: half of the initial energy is dissipated as heat in the connecting wires or radiated electromagnetically. This is the classic "where did the energy go" question.
- Time constant arithmetic. τ = R C in seconds when R is in ohms and C is in farads. Microfarads and kilo-ohms cancel to give τ in milliseconds, which is convenient — but only if you remember to do it consistently. The half-life of capacitor discharge is t_(1/2) = τ ln 2, not τ / 2.
- Kepler's third law base. T^2 / r^3 is constant for orbits about the same central body. Comparing the orbital period of a moon of Jupiter to a moon of Saturn using a single constant is wrong; the constants differ because G M is different.
- Geostationary parameters. A geostationary orbit has T = 24 hours (more precisely 23 h 56 min for sidereal alignment), sits over the equator and orbits prograde. A "polar orbit at 24 hours" is not geostationary and a "geostationary orbit over London" is impossible.
How This Topic Connects to Other A-Level Physics
The deliberate parallels between gravity, electrostatics, and capacitors are the synoptic backbone of Paper 2.
- Further mechanics. Orbital motion is uniform circular motion driven by an inverse-square gravitational force. The centripetal equation m v^2 / r = G M m / r^2 produces Kepler's third law in two algebraic steps.
- Magnetic fields and induction. The charged-particle deflector that you used in electric-field questions reappears in cyclotrons and mass spectrometers, but now with B v q in place of E q. The same dynamical machinery handles both.
- Nuclear physics. Alpha-particle scattering by a gold nucleus is just an inverse-square Coulomb scattering problem with the geometry worked out. The closest-approach distance is found by setting kinetic energy equal to electric potential energy, exactly as in the field-and-potential lessons of this unit.
- Engineering Physics (Option C). Flywheel energy storage is the rotational analogue of capacitor energy storage, with (1/2) I omega^2 in place of (1/2) C V^2. The two are introduced in different units but obey identical pattern of reasoning.
Treat this unit as the template of A-Level Physics: many subsequent units repeat the same field-potential-energy logic. Mastering the algebra here pays back across the rest of the course.