AQA A-Level Physics: Fields and their consequences (7408) – Revision Notes

For full explanations and worked examples, start with the study guide. These notes condense section 3.7 Fields and their consequences of the AQA AS and A-level Physics specification (7407/7408, version 1.4, AS and A-level exams June 2016 onwards): 3.7.1 Fields to 3.7.5 Magnetic fields. All of 3.7 is A-level only and is assessed in Paper 2. It is core content, not an option; the options are sections 3.9 to 3.13, one of which you study for Paper 3 Section B.

Links: course hub, printable checklist, practice questions, free diagnostics.

3.7.1 Fields: definitions

• Force field: a region in which a body experiences a non-contact force.
• Field strength is a vector; decide its direction by inspection.
• Sources: mass (gravitational), static charge (electric), moving charge (magnetic).
• Gravity vs electrostatics. Same: inverse-square laws, field lines, potential, equipotential surfaces. Different: masses always attract; charges can attract or repel.

Formula table

Quantity	Gravitational (3.7.2)	Electric (3.7.3)
Force between points	F = Gm₁m₂/r²	F = (1/4πε₀) Q₁Q₂/r²
Field strength (definition)	g = F/m	E = F/Q
Radial field strength	g = GM/r²	E = (1/4πε₀) Q/r²
Uniform field	—	E = V/d
Radial potential	V = −GM/r	V = (1/4πε₀) Q/r
Work done	ΔW = mΔV	ΔW = QΔV
Field from potential	g = −ΔV/Δr	E = ΔV/Δr (magnitude)
Potential difference from graph	area under g–r	area under E–r

Capacitors (3.7.4)	Magnetism (3.7.5)
C = Q/V	F = BIl (B ⊥ I)
C = Aε₀εᵣ/d	F = BQv (B ⊥ v)
E = ½QV = ½CV² = ½Q²/C	Φ = BA; NΦ = BAN cos θ
Time constant RC; T½ = 0.69RC	ε = NΔΦ/Δt
Q = Q₀e^(−t/RC) (discharge)	ε = BANω sin ωt
Q = Q₀(1 − e^(−t/RC)) (charge)	I_rms = I₀/√2; V_rms = V₀/√2
—	Nₛ/Nₚ = Vₛ/Vₚ; efficiency = IₛVₛ/IₚVₚ

Orbit results to derive, not just quote: v = √(GM/r), T² = (4π²/GM)r³, total energy = −GMm/2r, escape velocity v = √(2GM/R).

3.7.2 Gravitational fields: key points

• Potential V: work done per unit mass bringing a mass from infinity. Zero at infinity, negative everywhere else because the force is attractive.
• Equipotential surfaces: no work done moving along them; perpendicular to field lines.
• Graphs: g ∝ 1/r² outside the mass; V ∝ −1/r, rising towards zero. g is minus the gradient of V–r.
• Orbits: T² ∝ r³. A log T against log r plot has gradient 1.5.
• Satellite energy: Ek = GMm/2r, Ep = −GMm/r, total = −GMm/2r.
• Synchronous orbit: period equals the rotation period of the body orbited.
• Geostationary orbit: Earth-synchronous (24 h), equatorial plane, west to east, radius about 4.2 × 10⁷ m. Fixed dishes can point at it.
• Low orbits: short period, close to the surface; good for observation, but a satellite passes overhead quickly.

Method: deriving T² ∝ r³

1. Set gravitational force equal to centripetal force: GMm/r² = mv²/r.
2. Substitute v = 2πr/T.
3. Rearrange: T² = (4π²/GM) r³. Since 4π²/GM is constant for one central mass, T² ∝ r³.

3.7.3 Electric fields: key points

• Coulomb’s law is for point charges in a vacuum; air counts as a vacuum. A charged sphere acts as if its charge is at the centre.
• Gravitational force between subatomic particles is negligible compared with the electrostatic force (by a factor of order 10³⁹ for a proton and electron).
• Field lines: positive to negative. Uniform between parallel plates; radial around a point charge.
• E = V/d comes from work done: Fd = QΔV.
• A charge entering a uniform field at right angles follows a parabola: constant velocity along the plates, constant acceleration across.
• Absolute potential: work done per unit positive charge from infinity. Positive near a positive charge, negative near a negative one.

Method: deflection between plates

1. E = V/d, then F = QE and a = F/m.
2. Time between plates: t = plate length ÷ horizontal speed.
3. Sideways displacement y = ½at²; sideways velocity = at.
4. Check the direction: positive charges move towards the negative plate.

3.7.4 Capacitance: key points

• Relative permittivity = dielectric constant = εᵣ.
• Dielectric action: polar molecules rotate to line up with the field. Their own field opposes the applied field, so p.d. falls for a given charge and C rises.
• Energy: area under the charge–p.d. graph (straight line through the origin) = ½QV.
• Graphs:

	Charging	Discharging
Q and V	rise to a maximum, curving off	fall exponentially to zero
I	falls exponentially to zero	falls exponentially to zero (reversed direction)

• Gradient of Q–t = current. Area under I–t = charge.
• After one RC on discharge: 37% left. After one RC on charge: 63% reached.
• Required practical 9: plot ln V against t for discharge. Gradient = −1/RC.

Method: using the exponential

1. Find RC (check units: Ω × F = s).
2. Substitute into the right equation for charge or discharge.
3. To find time, take natural logs: t = −RC ln(Q/Q₀) for discharge.

3.7.5 Magnetic fields: key points

• Tesla: 1 T gives a force of 1 N on 1 m of wire carrying 1 A perpendicular to the field.
• Fleming’s left-hand rule for the motor effect; current = direction of positive charge flow.
• Charges in B: force ⊥ velocity, so no work done, constant speed, circular path, r = mv/BQ.
• Cyclotron: time per orbit 2πm/BQ does not depend on speed, so a fixed-frequency alternating p.d. can keep accelerating particles.
• Required practical 10: top pan balance; F = Δm × g on the magnet equals the force on the wire.
• Flux linkage NΦ = BAN cos θ, θ between field and the normal to the coil.
• Required practical 11: search coil and oscilloscope; flux linkage (and peak emf) varies with cos θ.
• Faraday’s law: emf = rate of change of flux linkage. Lenz’s law: the induced current opposes the change producing it.
• Straight conductor: ε = Blv.
• Rotating coil: ε = BANω sin ωt; emf is maximum when the coil plane is parallel to B (flux linkage zero).
• AC: the rms, peak and peak-to-peak relations here are for sinusoidal waveforms only.
• Oscilloscope: voltage = divisions × Y-gain; time = divisions × time-base. Usable as a dc or ac voltmeter and for measuring time intervals and frequencies.
• Transformer losses: eddy currents (reduce with a laminated core), coil resistance, magnetising the core, flux leakage.
• Transmission: high voltage → low current → low I²R loss.

Must-know distinctions

• Potential vs potential difference: V is per unit mass or charge relative to infinity; ΔV is between two points.
• Flux vs flux linkage: Φ = BA; flux linkage multiplies by N.
• Time constant vs half-time: RC vs 0.69RC.
• Peak vs rms vs peak-to-peak: V₀, V₀/√2, 2V₀.
• Left-hand rule (force on a current) vs Lenz’s law (direction of induced current).

Quick self-test

1. Calculate the gravitational force between two 50 kg masses 2.0 m apart.
2. A planet has mass 6.4 × 10²³ kg and radius 3.4 × 10⁶ m. Calculate g at its surface.
3. V at Earth’s surface is −6.25 × 10⁷ J kg⁻¹. State V at 2R from the centre.
4. A satellite’s orbital radius is made four times larger. By what factor does its period change?
5. Plates 1.5 cm apart have a p.d. of 300 V. Calculate E and the force on an electron.
6. A 220 μF capacitor is charged to 9.0 V. Calculate Q and the energy stored.
7. R = 10 kΩ, C = 220 μF. Calculate the time to halve.
8. What fraction of the charge remains after a discharge time of 3RC?
9. An electron moves at 4.0 × 10⁶ m s⁻¹ perpendicular to a 2.0 mT field. Calculate the radius of its path.
10. A 50-turn coil of area 0.020 m² is in a 0.40 T field with its normal at 60° to the field. Calculate the flux linkage.
11. A flux linkage of 0.20 Wb-turns falls to zero in 0.050 s. Calculate the average induced emf.
12. An oscilloscope shows a sine wave 6.0 divisions peak-to-peak at 2.0 V/div; one cycle spans 4.0 divisions at 5.0 ms/div. Find V₀, V_rms and f.

Answers

1. F = 6.67 × 10⁻¹¹ × 50 × 50 / 2.0² = 4.2 × 10⁻⁸ N
2. g = GM/R² = 3.7 N kg⁻¹
3. V ∝ 1/r, so −3.13 × 10⁷ J kg⁻¹
4. T ∝ r^(3/2) = 4^1.5 = ×8
5. E = 300/0.015 = 2.0 × 10⁴ V m⁻¹; F = eE = 3.2 × 10⁻¹⁵ N
6. Q = CV = 1.98 × 10⁻³ C; E = ½CV² = 8.9 × 10⁻³ J
7. RC = 2.2 s; T½ = 0.69 × 2.2 = 1.5 s
8. e⁻³ = 0.050 (5.0%)
9. r = mv/BQ = 1.1 × 10⁻² m
10. NΦ = 50 × 0.40 × 0.020 × cos 60° = 0.20 Wb-turns
11. ε = 0.20/0.050 = 4.0 V
12. V₀ = 3.0 × 2.0 = 6.0 V; V_rms = 4.2 V; T = 20 ms, so f = 50 Hz

Where marks are usually lost

• Using height above the surface instead of distance from the centre of mass in GM/r² and −GM/r.
• Omitting the negative sign on gravitational potential, or failing to explain it (attractive force, zero at infinity).
• Quoting T² ∝ r³ when the question asks you to derive it.
• Using an hour or a day rather than seconds for T in orbit calculations.
• Saying a dielectric “stores more charge” without explaining the opposing field of the aligned polar molecules.
• Reading the half-time from a graph and calling it the time constant.
• Forgetting the current direction reverses for negative particles in Fleming’s left-hand rule.
• Using θ to the coil plane instead of to the normal in BAN cos θ.
• Writing “Lenz’s law: the emf opposes the flux” instead of “opposes the change in flux linkage”.
• Calculating transmission loss as V²/R using the supply voltage rather than I²R with the line current.

Official syllabus

AQA AS and A-level Physics specification (7407/7408), version 1.4, July 2026, AS and A-level exams June 2016 onwards, published by AQA. Section 3.7 Fields and their consequences.