Comparing Fields: Gravitational, Electric and Magnetic

This final lesson synthesises the entire Fields topic (AQA 3.7) by systematically comparing gravitational, electric, and magnetic fields. This is essential preparation for Paper 2, where synoptic questions frequently require you to draw parallels and identify key differences between field types.

Spec mapping: AQA 7408 A-Level Physics, Section 3.7 in full — gravitational fields (3.7.2), electric fields (3.7.3), capacitance (3.7.4) and magnetic fields with induction (3.7.5). This synthesis lesson does not introduce new content; it consolidates the mathematical parallels (inverse-square force laws, 1/r potentials, the field-potential relationship g = −dV/dr / E = −dV/dr) and the physical differences (always-attractive vs both-signs, work done vs no work for B). The aim is to prepare students for the synoptic comparison questions that appear regularly on Paper 2. Refer to the official AQA 7408 specification document for the authoritative wording.

Synoptic links: (i) Newtonian gravitation and orbital mechanics (3.7.2, 3.6) — orbital speed v = √(GM/r) is the gravitational analogue of cyclotron-like motion under electrostatic attraction in the Bohr atom. (ii) Capacitors (3.7.4) — the energy ½CV² stored in a capacitor parallels the gravitational ½gh² mass-per-area on a uniform-field disk; the analogy between RC discharge and radioactive decay (3.8) introduces the same exponential mathematics. (iii) Particle physics (3.2) — Thomson's measurement of e/m used the comparison of electric and magnetic deflection of an electron beam, the same physics underlying mass spectrometry.

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The Concept of a Field

A field is a region of space in which an object experiences a force without direct contact. Fields are the mechanism by which forces act at a distance.

Field Type	Source	Object That Feels the Force
Gravitational	Mass	Mass
Electric	Charge	Charge
Magnetic	Moving charge / current	Moving charge / current

Note: Magnetic fields are fundamentally different — they arise from and act on moving charges (or currents), not stationary ones.

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Inverse Square Laws

Both gravitational and electric forces between point objects obey inverse square laws. The magnetic force does not follow a simple inverse square law for point sources (it depends on the geometry of currents), but the field of a magnetic dipole falls off as 1/r³.

Property	Gravitational	Electric
Force law	F = GMm/r²	F = kQ₁Q₂/r²
Field strength	g = GM/r²	E = kQ/r²
Potential	V = −GM/r	V = kQ/r
Potential energy	E_p = −GMm/r	E_p = kQ₁Q₂/r

Key Parallels

1. Both gravitational and Coulomb forces are proportional to the product of the "sources" (masses or charges) and inversely proportional to r².
2. Both field strengths follow an inverse square law: field ∝ 1/r².
3. Both potentials follow a 1/r relationship.
4. In both cases, E (or g) = −dV/dr (field strength is the negative potential gradient).

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Attractive vs Repulsive Forces

Feature	Gravitational	Electric	Magnetic
Nature of force	Always attractive	Attractive or repulsive	Attractive or repulsive
Sign of potential	Always negative	Positive or negative	Not defined in the same way
Can be shielded?	No	Yes (Faraday cage)	Yes (mu-metal, superconductors)

Gravitational forces are always attractive because mass is always positive. This means:

• Gravitational potential is always negative (V = −GM/r, and G, M, r are all positive).
• Gravitational potential energy is always negative for bound systems.
• There is no gravitational equivalent of repulsion or a positive potential.

Electric forces can be attractive or repulsive because charge can be positive or negative. This means:

• Electric potential near a positive charge is positive; near a negative charge, it is negative.
• Two like charges have positive potential energy (repulsive; energy must be supplied to bring them together).
• Two unlike charges have negative potential energy (attractive; energy is released when they come together).

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Conservative Fields

Both gravitational and electrostatic fields are conservative: the work done in moving an object between two points depends only on the start and end positions, not on the path taken.

Consequences of conservative fields:

• Potential can be defined unambiguously at every point.
• The work done around any closed loop is zero.
• Energy is conserved (KE + PE = constant for an isolated system).

Magnetic forces, however, are not conservative in the traditional sense. The magnetic force is always perpendicular to the velocity, so it does no work on the particle. The concept of magnetic potential energy exists for magnetic dipoles in external fields, but the detailed treatment is beyond A-Level.

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Field Lines vs Equipotentials

In all three field types, field lines and equipotential surfaces (where applicable) are perpendicular.

Feature	Gravitational	Electric
Field lines point...	Towards mass (inward)	Away from +Q, towards −Q
Equipotentials are...	Concentric spheres (point mass)	Concentric spheres (point charge)
In uniform fields...	Parallel lines (near surface)	Parallel lines (between plates)
Equipotential spacing	Closer = stronger field	Closer = stronger field

For magnetic fields, the concept of equipotentials does not apply in the same simple way. Instead, magnetic field lines form closed loops (from N to S outside the magnet, from S to N inside).

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The Field-Potential Relationship

For gravitational and electric fields, the field strength is related to the potential by:

g = −dV_grav/dr and E = −dV_elec/dr

On a potential-distance graph, the gradient at any point gives the field strength (with a negative sign).

For a uniform field (e.g., near Earth's surface or between parallel plates):

• g is constant, so V varies linearly with height: V = gh (approximately, for small h)
• E is constant, so V varies linearly with distance: V = Ed

For a radial field (point mass or point charge):

• The field and potential both follow power laws: field ∝ 1/r², potential ∝ 1/r.
• The V-r graph is a curve; its gradient becomes less steep at greater distances.

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Relative Strengths

The gravitational and electric forces differ enormously in strength:

Example: Consider the gravitational and electric forces between a proton and an electron in a hydrogen atom (separation r ≈ 5.3 × 10⁻¹¹ m).

Gravitational force: F_grav = Gm_pm_e/r² = (6.67 × 10⁻¹¹ × 1.67 × 10⁻²⁷ × 9.11 × 10⁻³¹) / (5.3 × 10⁻¹¹)² F_grav = (6.67 × 10⁻¹¹ × 1.522 × 10⁻⁵⁷) / (2.81 × 10⁻²¹) F_grav = 1.015 × 10⁻⁶⁷ / 2.81 × 10⁻²¹ F_grav = 3.61 × 10⁻⁴⁷ N

Electric force: F_elec = ke²/r² = (8.99 × 10⁹ × (1.60 × 10⁻¹⁹)²) / (5.3 × 10⁻¹¹)² F_elec = (8.99 × 10⁹ × 2.56 × 10⁻³⁸) / (2.81 × 10⁻²¹) F_elec = 2.301 × 10⁻²⁸ / 2.81 × 10⁻²¹ F_elec = 8.19 × 10⁻⁸ N

Ratio: F_elec/F_grav = 8.19 × 10⁻⁸ / 3.61 × 10⁻⁴⁷ ≈ 2.3 × 10³⁹

The electric force is approximately 10³⁹ times stronger than the gravitational force at the atomic scale! Gravity is only significant for very large masses (planets, stars) because it is always attractive and does not cancel out, while electric forces between the vast numbers of positive and negative charges in matter largely cancel.

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Comprehensive Comparison Table

Property	Gravitational	Electric	Magnetic
Source	Mass	Charge	Moving charge / current
Acted on by	Mass	Charge	Moving charge / current
Force law	F = GMm/r²	F = kQ₁Q₂/r²	F = BQv sin θ, F = BIL sin θ
Field strength	g = GM/r² (N kg⁻¹)	E = kQ/r² (N C⁻¹)	B (T) — depends on geometry
Potential	V = −GM/r (J kg⁻¹)	V = kQ/r (V)	No simple scalar potential
Field-potential link	g = −dV/dr	E = −dV/dr	Not applicable (same way)
Always attractive?	Yes	No	No
Inverse square?	Yes (force, field)	Yes (force, field)	Not for force; dipole field ∝ 1/r³
Can do work?	Yes	Yes	No (force ⊥ velocity)
Field lines	Radial inward	Radial out (+Q) or in (−Q)	Closed loops (N→S externally)
Uniform field example	Near planet surface	Between parallel plates	Between poles of a horseshoe magnet

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Synthesis Exam Questions — Approach

When answering synoptic comparison questions:

1. State the mathematical parallels clearly, writing both equations side by side.
2. Identify the physical differences — especially the sign conventions and the nature of the force (attractive only vs both).
3. Use specific values where possible to illustrate the vast difference in strength.
4. Link to practical contexts — gravity governs planetary orbits; electric fields govern atomic structure; magnetic fields govern electromagnetic devices.

Worked Example 1 — Comparing Field Strengths

Question: Calculate the gravitational and electric field strengths at the surface of a proton (radius ≈ 0.88 × 10⁻¹⁵ m, mass = 1.67 × 10⁻²⁷ kg, charge = 1.60 × 10⁻¹⁹ C).

Solution:

Gravitational field strength: g = GM/r² = (6.67 × 10⁻¹¹ × 1.67 × 10⁻²⁷) / (0.88 × 10⁻¹⁵)² g = 1.114 × 10⁻³⁷ / 7.74 × 10⁻³¹ g = 1.44 × 10⁻⁷ N kg⁻¹ (unimaginably weak)

Electric field strength: E = kQ/r² = (8.99 × 10⁹ × 1.60 × 10⁻¹⁹) / (0.88 × 10⁻¹⁵)² E = 1.438 × 10⁻⁹ / 7.74 × 10⁻³¹ E = 1.86 × 10²¹ N C⁻¹ (enormously strong)

The electric field at the proton's surface is roughly 10²⁸ times stronger than the gravitational field.

Worked Example 2 — Satellite vs Electron in Orbit

Question: Compare the orbital speed of a satellite at the Earth's surface with the speed of an electron orbiting a proton in the Bohr model of hydrogen (r = 5.3 × 10⁻¹¹ m).

Solution:

Satellite at Earth's surface (hypothetical, ignoring atmosphere): v = √(GM/r) = √(6.67 × 10⁻¹¹ × 5.97 × 10²⁴ / 6.37 × 10⁶) v = √(3.98 × 10¹⁴ / 6.37 × 10⁶) = √(6.25 × 10⁷) = 7.91 × 10³ m s⁻¹ ≈ 7.9 km s⁻¹

Electron orbiting a proton: The centripetal force is provided by the Coulomb force: ke²/r² = mv²/r → v² = ke²/(mr) where m is the electron mass. v = √(8.99 × 10⁹ × (1.60 × 10⁻¹⁹)² / (9.11 × 10⁻³¹ × 5.3 × 10⁻¹¹)) v = √(8.99 × 10⁹ × 2.56 × 10⁻³⁸ / 4.83 × 10⁻⁴¹) v = √(2.301 × 10⁻²⁸ / 4.83 × 10⁻⁴¹) v = √(4.76 × 10¹²) = 2.18 × 10⁶ m s⁻¹

The electron moves at about 2200 km s⁻¹ — roughly 280 times faster than the satellite. This reflects the enormous strength of the electric force compared to gravity at atomic scales.

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Common Exam Mistakes to Avoid

1. Confusing field strength with potential. Field strength is a vector (force per unit mass/charge); potential is a scalar (energy per unit mass/charge).

2. Forgetting the negative sign in gravitational potential. V = −GM/r, always negative. Students who write V = GM/r lose marks immediately.

3. Using the wrong sign convention for electric potential. V = +kQ/r for a positive charge (positive potential); V = −k|Q|/r for a negative charge (negative potential).

4. Confusing the formulas for capacitors and resistors in series/parallel. Capacitors in parallel add directly; resistors in parallel combine reciprocally. The rules are swapped compared to each other.

5. Stating that g = 0 in orbit. Astronauts experience apparent weightlessness because they are in free fall, not because g = 0.

6. Forgetting that magnetic force does no work. The magnetic force is always perpendicular to velocity, so it changes direction but not speed. This is fundamentally different from gravitational and electric forces.

7. Using peak values instead of rms in power calculations. P = V_rms × I_rms, not P = V₀ × I₀.

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Final Summary — Key Equations

Gravitational Fields

g = F/m	g = GM/r²	V = −GM/r	F = GMm/r²

Electric Fields

E = F/Q	E = kQ/r²	V = kQ/r	F = kQ₁Q₂/r²	E = V/d (uniform)

Capacitors

C = Q/V	C = ε₀εᵣA/d	W = ½CV²	τ = RC	Q = Q₀e^(−t/RC)

Magnetic Fields

F = BIL sin θ	F = BQv sin θ	r = mv/(BQ)	ε = BLv	ε = NBAω sin(ωt)

Electromagnetic Induction & AC

Φ = BA cos θ	ε = −NdΦ/dt	V_rms = V₀/√2	P_loss = I²R	Ns/Np = Vs/Vp

Fundamental Constants

Constant	Symbol	Value	Units
Gravitational constant	G	6.67 × 10⁻¹¹	N m² kg⁻²
Coulomb's constant	k	8.99 × 10⁹	N m² C⁻²
Permittivity of free space	ε₀	8.85 × 10⁻¹²	F m⁻¹
Elementary charge	e	1.60 × 10⁻¹⁹	C
Electron mass	m_e	9.11 × 10⁻³¹	kg
Proton mass	m_p	1.67 × 10⁻²⁷	kg

Exam Tip: In the final weeks before exams, practise converting between the different representations of each field type — force law, field strength, potential, and energy. Being fluent in moving between these forms is the hallmark of a top-grade physicist. Draw up your own comparison tables and test yourself until you can reproduce them from memory.

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Specimen Exam Question

Specimen question modelled on the AQA A-Level Physics 7408 Paper 2 format. Not from any published paper.