Gravitational vs Electric Fields

Overview

Gravitational fields and electric fields are two important examples of force fields. Both describe interactions at a distance without direct contact.

They share many mathematical similarities:

• inverse-square force laws,
• field strength concepts,
• potential concepts,
• conservative forces,
• superposition.

They differ in physical source, sign behaviour, and typical strength.

This comparison connects:

• Gravitational Fields
• Electric Fields
• Work, Energy and Power

Why It Matters

Once one field model is understood, the other becomes easier.

General framework:

• source creates field,
• field exerts force,
• field has potential,
• systems have potential energy,
• field strength relates to potential gradient,
• multiple sources combine by superposition.

This comparison also helps prevent sign and direction errors.

Definition

A gravitational field is produced by mass and acts on mass. An electric field is produced by charge and acts on charge.

Key Representations

$$\vec{g}=-\frac{GM}{r^2}\hat{r}$$ $$\vec{E}=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}\hat{r}$$ $$V_g=-\frac{GM}{r}$$ $$V=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}$$

Core Comparison Table

Aspect	Gravitational Field	Electric Field
Source quantity	Mass	Electric charge
Acts on	Mass	Charge
Nature of force	Always attractive	Attractive or repulsive
Range	Infinite	Infinite
Point-source field	Radial central field	Radial central field
Relative strength	Very weak	Much stronger

Force Laws

Gravitational Force

Magnitude between two point masses:

$$F_G=\frac{G{m}_1{m}_2}{r^2}$$

Vector form on test mass:

$${\vec{F}}_G=-\frac{G{m}_1{m}_2}{r^2}\hat{r}$$

Always attractive.

Electric Force

Magnitude between two point charges:

$$F_E=\frac{1}{4\pi {ϵ}_0}\frac{|q_1{q}_2|}{r^2}$$

Direction depends on signs:

• like charges repel,
• unlike charges attract.

Vector force depends on chosen geometry and charge signs.

Shared Inverse-Square Behaviour

Both magnitudes satisfy:

$$F\propto \frac{1}{r^2}$$

So if distance doubles:

$$F\to \frac{1}{4}F$$

Field Strength

Gravitational Field Strength

Defined as force per unit mass:

$$\vec{g}=\frac{{\vec{F}}_G}{m}$$

For point mass $M$:

$$\vec{g}=-\frac{GM}{r^2}\hat{r}$$

Magnitude:

$$g=\frac{GM}{r^2}$$

Units:

$$\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$$

or

$$\mathrm{m}{\mathrm{s}}^{-2}$$

Direction: toward the mass.

Electric Field Strength

Defined as force per unit positive charge:

$$\vec{E}=\frac{{\vec{F}}_E}{q}$$

For point charge $Q$:

$$\vec{E}=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}\hat{r}$$

Interpretation:

• $Q>0$: field points outward,
• $Q<0$: field points inward.

Units:

$$\mathrm{N}{\mathrm{C}}^{-1}$$

or

$$\mathrm{V}{\mathrm{m}}^{-1}$$

Important Direction Difference

Gravity

Mass is positive in H2 Physics treatment, so gravity is always attractive.

Electricity

Charge can be positive or negative, so electric force may attract or repel.

For a negative charge:

$$\vec{F}=q\vec{E}$$

Force is opposite to $\vec{E}$.

Potential

Gravitational Potential

Work done per unit mass by external agent bringing a small test mass from infinity.

For point mass:

$$V_g=-\frac{GM}{r}$$

Properties:

• scalar,
• zero at infinity,
• negative near isolated mass.

Units:

$$\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$$

Electric Potential

Work done per unit positive charge by external agent bringing a small test charge from infinity.

For point charge:

$$V=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}$$

Properties:

• scalar,
• sign depends on $Q$.

Units:

$$\mathrm{J}{\mathrm{C}}^{-1}=\mathrm{V}$$

Potential Energy

Gravitational

For mass $m$:

$$U_g=m{V}_g$$

For two masses:

$$U_g=-\frac{GMm}{r}$$

Always negative for two attracting masses.

Electric

For charge $q$:

$$U_e=qV$$

For two charges:

$$U_e=\frac{1}{4\pi {ϵ}_0}\frac{Qq}{r}$$

Can be:

• positive,
• negative,
• zero (reference choice).

Conservative Nature

Both gravitational and electric forces are conservative.

Therefore:

• work done is path independent,
• depends only on initial and final positions,
• potential energy can be defined.

For both:

$$W=-\Delta U$$

Relationship Between Field and Potential

Gravity

$$\vec{g}=-\nabla V_g$$

For radial motion:

$$g=-\frac{d{V}_g}{dr}$$

Electricity

$$\vec{E}=-\nabla V$$

For one-dimensional motion:

$$E=-\frac{dV}{dx}$$

Field points toward decreasing potential.

Superposition

Both obey superposition.

Fields Add Vectorially

$${\vec{g}}_{\text{net}}={\vec{g}}_1+{\vec{g}}_2+\cdots$$ $${\vec{E}}_{\text{net}}={\vec{E}}_1+{\vec{E}}_2+\cdots$$

Potentials Add Algebraically

$$V_{g,\text{net}}=V_{g1}+V_{g2}+\cdots$$ $$V_{\text{net}}=V_1+V_2+\cdots$$

Important distinction:

• fields are vectors,
• potentials are scalars.

Uniform Fields

Gravitational Field Near Earth

Approximately uniform:

$$g\approx 9.81\mathrm{m}{\mathrm{s}}^{-2}$$

Field lines nearly parallel.

Electric Field Between Parallel Plates

Approximately uniform:

$$E=\frac{V}{d}$$

Field lines parallel and equally spaced.

This analogy is useful in problem solving.

Relative Strength

Electric force between elementary particles is vastly stronger than gravitational force.

Yet gravity dominates astronomy because:

• masses are always positive,
• charges often cancel overall,
• celestial bodies contain enormous mass.

Memory Map

Gravity	Electricity
Mass creates field	Charge creates field
$\vec{F}=m\vec{g}$	$\vec{F}=q\vec{E}$
$\vec{g}=-\frac{GM}{r^2}\hat{r}$	$\vec{E}=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}\hat{r}$
$V_g=-\frac{GM}{r}$	$V=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}$
$U_g=m{V}_g$	$U_e=qV$

Worked Example 1

Two identical masses are separated by distance $r$. If distance doubles, gravitational force becomes:

$$F^{\prime }=\frac{1}{4}F$$

Same inverse-square logic applies to electric force magnitude.

Worked Example 2

A negative charge is placed in electric field $\vec{E}$ to the right.

Since:

$$\vec{F}=q\vec{E}$$

and $q<0$, force acts to the left.

Common Exam Pitfalls

1. Mixing Units

• $g$: $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$
• $E$: $\mathrm{N}{\mathrm{C}}^{-1}$

2. Assuming Potential Is Always Negative

• Gravitational potential near isolated mass is negative.
• Electric potential may be positive or negative.

3. Confusing Field and Force

Field is property of space.

Force acts on inserted object:

$$\vec{F}=m\vec{g}$$ $$\vec{F}=q\vec{E}$$

4. Forgetting Charge Sign

Negative charges experience force opposite to $\vec{E}$.

5. Adding Potentials as Vectors

Potentials are scalars.

6. Thinking Gravity Can Repel

Not in classical H2 treatment.

Summary

Gravitational and electric fields share similar mathematical structure but differ in physical origin.

Gravity

• sourced by mass,
• always attractive,
• dominant in astronomy.

Electricity

• sourced by charge,
• attractive or repulsive,
• stronger microscopically.

Understanding the analogy improves transfer across force, field, potential, and energy problems.

Links

• Gravitational Fields
• Electric Fields
• Work, Energy and Power
• Vectors