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.

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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