01 / 21

Gravitational vs.
Electric Forces

Learn to quantitatively compare the two fundamental forces that shape everything from atoms to galaxies.

High School Physics

What Are Fundamental Forces?

In physics, a fundamental force is a basic interaction between objects that cannot be broken down into simpler interactions. Two of the most important are gravity and the electric (Coulomb) force. Both act over a distance without physical contact.

Gravitational Force
Acts between any two objects that have mass. Always attractive.
Electric Force
Acts between any two objects that have electric charge. Can attract or repel.

The Inverse-Square Law

Both gravitational and electric forces obey an inverse-square law: the force between two objects is proportional to \(\frac{1}{r^2}\), where \(r\) is the distance between their centres. Double the distance and the force drops to one-quarter.

Distance \(r\) Force \(F\) r 2r F F/4 \(F \propto \frac{1}{r^2}\)

Newton's Law of Gravitation

\(m_1\) \(m_2\) \(r\) \(F_g\)

Gravity is always attractive: it pulls the two masses toward each other. It is extremely weak at the atomic scale, but dominates at planetary and galactic scales because mass is always positive and it accumulates.

Coulomb's Law

\(q_1\) \(q_2\) \(r\) \(F_e\) Like charges: repulsion shown

Unlike gravity, the electric force can be attractive (opposite charges) or repulsive (like charges). The magnitude depends on the absolute values of the charges.

Key Distinctions

Gravity
Acts on mass. Always attractive. Constant \(G\) is tiny: \(\sim 10^{-11}\).
Coulomb
Acts on charge. Attractive or repulsive. Constant \(k\) is huge: \(\sim 10^{9}\).

Despite this enormous difference, gravity dominates large-scale structure because large bodies are electrically neutral: positive and negative charges cancel out, but mass never cancels.

Structural Comparison

Gravitational Force \(F = G \frac{m_1 m_2}{r^2}\) Constant: \(G \approx 6.67 \times 10^{-11}\) Property: mass (kg) Electric Force \(F = k \frac{|q_1||q_2|}{r^2}\) Constant: \(k \approx 8.99 \times 10^{9}\) Property: charge (C) ≅

Why the Huge Difference in Strength?

Consider a proton and an electron separated by a distance \(r\). They interact through both gravity (they have mass) and the electric force (they have charge).

1
Gravitational force: \(F_g = G \frac{m_p \, m_e}{r^2}\). The masses are tiny: \(m_p \approx 1.67 \times 10^{-27}\) kg, \(m_e \approx 9.11 \times 10^{-31}\) kg.
2
Electric force: \(F_e = k \frac{e^2}{r^2}\). The charges are small but \(k\) is enormous: \(e = 1.60 \times 10^{-19}\) C.
3
The ratio \(\frac{F_e}{F_g} \approx 2.3 \times 10^{39}\). The electric force wins by 39 orders of magnitude.

How to Compare Quantitatively

1
Identify the objects: What are their masses \(m_1, m_2\) and charges \(q_1, q_2\)?
2
Write both force expressions using \(F_g = G\frac{m_1 m_2}{r^2}\) and \(F_e = k\frac{|q_1||q_2|}{r^2}\).
3
Form the ratio \(\frac{F_e}{F_g} = \frac{k \, |q_1||q_2|}{G \, m_1 m_2}\). Notice \(r^2\) cancels.
4
Substitute values and compute. Use powers-of-ten arithmetic to simplify.
5
Interpret: State which force dominates and by how many orders of magnitude.

Example 1: Proton-Electron Pair

Problem: Find the ratio \(\frac{F_e}{F_g}\) for a proton and electron at any separation \(r\).

The electric force is about \(2.3 \times 10^{39}\) times stronger: gravity between subatomic particles is utterly negligible.

Example 2: Two Electrons

Problem: Compare the electric repulsion and gravitational attraction between two electrons.

The electric repulsion between two electrons is \(4.2 \times 10^{42}\) times stronger than their gravitational attraction: even more extreme than the proton-electron case because the electron mass is so much smaller.

Example 3: Earth and Moon

Problem: Suppose the Earth and Moon each carried a net charge of \(q = 1\) C. How would the electric force compare to their gravitational attraction?

Here \(F_g \gg F_e\): gravity wins by 28 orders of magnitude. With such enormous masses and tiny charges, gravity dominates completely.

When Does Each Force Dominate?

Gravity Dominates
Large, electrically neutral bodies: planets, stars, galaxies. Charges cancel; mass accumulates.
Electricity Dominates
Charged particles: atoms, molecules, circuits. Tiny masses but non-zero net charge.
Both Matter
Astrophysical plasmas, neutron stars with charge, cosmological models balancing both.

The Master Ratio Formula

1
Constants ratio: \(\frac{k}{G} = \frac{8.99 \times 10^{9}}{6.67 \times 10^{-11}} \approx 1.35 \times 10^{20}\)
2
Properties ratio: \(\frac{|q_1||q_2|}{m_1 m_2}\) depends on the specific objects.
3
Multiply the two factors to get the final comparison.

This is the most efficient approach for any quantitative comparison problem: compute \(\frac{k}{G}\) once, then adjust by the charge-to-mass product ratio.

Common Mistakes

✕ Forgetting to Square \(r\)
Both laws use \(r^2\), not \(r\). Forgetting the square gives a force that falls off too slowly with distance.
✕ Mixing Up Constants
Using \(G\) in Coulomb's law or \(k\) in gravity. Remember: \(G \sim 10^{-11}\) for mass, \(k \sim 10^{9}\) for charge.
✕ Wrong Units for Charge
Coulomb's law requires charge in Coulombs (C). Using microcoulombs or elementary charges without converting leads to errors of many orders of magnitude.

Summary Table

Property Gravitational Electric (Coulomb)
Law \(F = G\frac{m_1 m_2}{r^2}\) \(F = k\frac{|q_1||q_2|}{r^2}\)
Acts on Mass Charge
Direction Always attractive Attractive or repulsive
Constant \(6.67 \times 10^{-11}\) \(8.99 \times 10^{9}\)
Relative strength Extremely weak \(\sim 10^{20}\) times stronger
Dominates at Planetary / galactic scale Atomic / molecular scale
Can cancel? No (mass always positive) Yes (charges can be + or −)

Question 1 of 5

Both Newton's law of gravitation and Coulomb's law depend on distance as \(\frac{1}{r^2}\). If the distance between two charges is tripled, the electric force between them:

Question 2 of 5

Which of the following is a key difference between gravitational and electric forces?

Question 3 of 5

For a proton and an electron, the ratio \(\frac{F_e}{F_g}\) is approximately \(2.3 \times 10^{39}\). If the separation is doubled, the new ratio is:

Question 4 of 5

Two identical metal spheres, each of mass \(m = 1\) kg and charge \(q = 1\) C, are placed 1 m apart. Which statement is correct?

Question 5 of 5

Why does gravity dominate at astronomical scales even though the electric force is intrinsically much stronger?