Compare Gravitational and Electric Forces - Grade 11 Physics

SECTION 1

Historical Context & Motivation

For thousands of years, humans noticed two mysterious phenomena: objects fall toward the ground, and rubbed amber attracts bits of straw. These observations seemed entirely unrelated until scientists began uncovering the mathematical laws that govern each force. The quest to understand gravitational force and electric force eventually revealed a striking parallel: both obey an inverse-square law, yet they differ in strength by an almost unimaginable factor. Understanding how these two fundamental forces compare is essential to explaining everything from why planets orbit stars to why atoms hold together.

1687 • Newton's Law of Universal Gravitation

Isaac Newton published the Principia, showing that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

1785 • Coulomb's Law

Charles-Augustin de Coulomb used a torsion balance to measure the force between charged objects, establishing that the electric force also follows an inverse-square relationship with distance, but depends on the magnitudes of the charges.

1798 • Cavendish Measures G

Henry Cavendish experimentally determined the gravitational constant G using a torsion balance, allowing scientists to calculate the gravitational force between any two known masses for the first time.

1909 • Millikan Oil Drop Experiment

Robert Millikan measured the charge of a single electron, providing the precise value of the elementary charge e, which anchors Coulomb's law to measurable quantities.

With both laws firmly established, a natural question emerged: why do gravitational and electric forces share the same mathematical form, and how do they differ in their effects on the matter around us? This lesson explores exactly that question.

SECTION 2

Core Principles & Definitions

Both gravitational and electric forces are non-contact forces—they act across empty space without the objects touching. They are also both field forces, meaning each source creates an invisible field that exerts influence on other objects placed within it. Despite these similarities, the two forces arise from fundamentally different properties of matter and behave differently in several important ways.

Source Property

Gravitational force arises from mass, a property all matter possesses. Electric force arises from electric charge, which comes in positive and negative varieties.

Inverse-Square Law

Both forces weaken with distance following the same pattern: when the distance doubles, the force drops to one-quarter of its original value. This is the inverse-square law.

Attractive vs. Repulsive

Gravity is always attractive—masses always pull toward each other. Electric force can be either attractive or repulsive depending on the signs of the charges.

Relative Strength

The electric force between two particles is enormously stronger than the gravitational force between them. For two protons, the electric force is about 10³⁶ times larger than their gravitational attraction.

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KEY TAKEAWAY

Think of gravitational and electric forces like two versions of a streaming service. Gravity is the basic plan—everyone has it because everything has mass, but it's relatively weak. The electric force is the premium plan—it's far more powerful, but only available to objects that carry charge. Both deliver content (force) the same way (inversely with distance squared), yet the premium version is overwhelmingly stronger and can also push things away, not just pull them in.

SECTION 3

Visual Comparison of Force Fields

A side-by-side diagram makes the structural similarities and key differences between these forces much clearer. In the following diagram, notice how both force arrows point along the line connecting the two objects, and both weaken with distance. The crucial difference is that gravitational field lines always point inward (attraction only), while electric field lines can point inward or outward depending on the charge combination.

Side-by-side comparison of gravitational and electric forces. Left: two masses always attract each other. Right: opposite charges attract (green arrows) while like charges repel (red arrows). Both forces act along the line connecting the objects and follow an inverse-square relationship with distance.

In the diagram, notice that both types of force arrows lie along the straight line connecting the two objects. This is because both are central forces—they act along the line joining the centers of the interacting objects. The color coding in the summary boxes highlights the key differences: gravity has one mode (attraction), while the electric force has two modes (attraction and repulsion).

SECTION 4

Mathematical Framework

The mathematical expressions for gravitational and electric forces are strikingly similar in structure. Both involve a constant multiplied by the product of two source quantities, divided by the square of the distance between them. This shared structure is one of the most elegant patterns in physics.

NEWTON'S LAW OF UNIVERSAL GRAVITATION

F_g = G × (m₁ × m₂) / r²

Where F_g = gravitational force (N), G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant), m₁ and m₂ = the two masses (kg), and r = distance between their centers (m).

COULOMB'S LAW

F_e = k × (|q₁| × |q₂|) / r²

Where F_e = electric force (N), k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant), q₁ and q₂ = the two charges (C), and r = distance between their centers (m). The absolute value bars indicate we use the magnitudes of the charges to find the force's size; the sign of the charges determines direction.

GENERAL INVERSE-SQUARE FORM

F = (constant) × (property₁ × property₂) / r²

Both laws fit this template. The constant sets the overall scale of the force, the properties (mass or charge) determine how much of the source each object carries, and r² ensures the force weakens as objects move apart.

The enormous difference between the constants G and k is the main reason gravity is so much weaker than the electric force. Notice that G is on the order of 10⁻¹¹ while k is on the order of 10⁹. That's a difference of about 20 orders of magnitude in the constants alone, before you even consider the tiny masses and charges of subatomic particles. This is why electric forces dominate at the atomic and molecular scale, while gravity only becomes significant when you have massive objects like planets and stars.

SECTION 5

Detailed Feature-by-Feature Comparison

Now that you've seen both equations, let's lay out every important similarity and difference in a structured format. The following diagram and table provide a thorough breakdown so you can see the full picture at a glance.

Both the electric force (solid pink curve) and gravitational force (dashed blue curve) follow the same 1/r² decay pattern. At distance r the force is F; at 2r it drops to F/4; at 3r it drops to F/9. The electric curve sits much higher on the graph because Coulomb's constant k is enormously larger than the gravitational constant G.

Feature	Gravitational Force	Electric Force
Source property	Mass (m)	Electric charge (q)
Formula	F = Gm₁m₂/r²	F = k\|q₁\|\|q₂\|/r²
Constant	G = 6.674 × 10⁻¹¹ N·m²/kg²	k = 8.99 × 10⁹ N·m²/C²
Direction	Always attractive	Attractive or repulsive
Distance dependence	Inverse-square (1/r²)	Inverse-square (1/r²)
Relative strength	Extremely weak at particle scale	~10³⁶ times stronger at particle scale
Can be shielded?	No — cannot be blocked or screened	Yes — conductors can shield electric fields
Dominates at…	Large (astronomical) scales	Small (atomic/molecular) scales

Comprehensive comparison of gravitational and electric forces

One of the most important entries in this table is the "dominates at" row. Even though the electric force is far stronger than gravity, matter on large scales is usually electrically neutral—the positive and negative charges cancel out. That's why gravity, despite its weakness, is the force that shapes galaxies, holds planets in orbit, and keeps your feet on the ground. At the atomic scale, however, the electric force is king: it holds electrons in orbit around nuclei, bonds atoms into molecules, and gives materials their structure.

SECTION 6

Worked Example: Comparing Forces Between Two Protons

Let's calculate both the gravitational and electric forces between two protons separated by 1.0 × 10⁻¹⁰ m (about the size of an atom) to see how dramatically different these forces are.

ℹ

Given Values

Mass of a proton: m = 1.67 × 10⁻²⁷ kg. Charge of a proton: q = 1.60 × 10⁻¹⁹ C. Separation: r = 1.0 × 10⁻¹⁰ m. G = 6.674 × 10⁻¹¹ N·m²/kg². k = 8.99 × 10⁹ N·m²/C².

Forces Between Two Protons

1
Step 1 — Calculate the Gravitational Force

Substitute into Newton's law: F_g = G × m₁ × m₂ / r² = (6.674 × 10⁻¹¹) × (1.67 × 10⁻²⁷) × (1.67 × 10⁻²⁷) / (1.0 × 10⁻¹⁰)².

F_g ≈ 1.86 × 10⁻⁴⁴ N
2
Step 2 — Calculate the Electric Force

Substitute into Coulomb's law: F_e = k × |q₁| × |q₂| / r² = (8.99 × 10⁹) × (1.60 × 10⁻¹⁹) × (1.60 × 10⁻¹⁹) / (1.0 × 10⁻¹⁰)².

F_e ≈ 2.30 × 10⁻⁸ N
3
Step 3 — Compare the Two Forces

Divide the electric force by the gravitational force to find the ratio: F_e / F_g = (2.30 × 10⁻⁸) / (1.86 × 10⁻⁴⁴).

F_e / F_g ≈ 1.24 × 10³⁶ — the electric force is about a trillion trillion trillion times stronger!
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Step 4 — Interpret the Result

The gravitational force between two protons is so tiny it is completely negligible at the atomic scale. The electric repulsion between two protons, on the other hand, is large enough to influence atomic structure. This is why the strong nuclear force is needed inside the nucleus to overcome this electric repulsion and hold protons together.

SECTION 7

Strengths and Limitations of Each Force

Both gravitational and electric forces are essential in different domains of the physical world. Understanding when each one dominates—and why—is crucial for applying these concepts correctly in physics problems and real-life situations.

Aspect	Gravitational Force	Electric Force
Strength	Always present between any two masses; accumulates over large scales because it cannot cancel out	Overwhelmingly strong at short range; drives chemistry, electricity, and material properties
Limitation	Negligibly weak at atomic or molecular scales; cannot explain atomic structure or chemical bonding	Positive and negative charges tend to neutralize on large scales, so electric force has limited range in bulk matter
Real-world role	Governs planetary orbits, tides, and the large-scale structure of the universe	Holds atoms and molecules together, powers electronics, generates lightning
Shielding	Cannot be blocked by any known material	Can be shielded by conductors (Faraday cage effect)

Practical strengths and limitations of gravitational vs. electric forces

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KEY TAKEAWAY

Imagine gravity as the bass drum in a concert—it's a deep, steady beat that fills the entire stadium (the universe). The electric force is like the lead guitar—it's much louder and more versatile, but mostly only heard near the stage (at atomic scales). In a huge arena, the bass drum is what everyone feels, even in the back row. Up close near the stage, though, the guitar completely overpowers it. That's why gravity shapes galaxies while electric forces shape atoms.

SECTION 8

Connection to Advanced Theory

The inverse-square laws you've learned here are part of a much larger story in physics. Both Newton's gravitational law and Coulomb's law describe static situations—objects that are stationary or moving slowly. When objects move near the speed of light or when you need to account for fields that change over time, more advanced frameworks take over. Here's a preview of where these ideas lead.

What You've Learned	Where It Leads
Newton's law of universal gravitation (F = Gm₁m₂/r²)	Einstein's General Relativity: gravity is a curvature of spacetime caused by mass-energy, not simply a force
Coulomb's law for static charges (F = kq₁q₂/r²)	Maxwell's Equations: unify electricity and magnetism into electromagnetism, predicting light as an electromagnetic wave
Comparing gravity and electric force as separate entities	Grand Unified Theories: physicists attempt to show that all fundamental forces (gravity, electromagnetic, strong, weak) are aspects of a single underlying force
Forces act instantaneously at a distance	Field theory: forces propagate at the speed of light through fields; gravitational waves were detected in 2015 by LIGO

From classical force laws to modern physics

Even though general relativity and electromagnetism are more complete theories, the classical force laws you're learning now remain extremely useful and accurate for the vast majority of everyday situations. Engineers use Newton's gravitational law to plan satellite orbits, and electrical engineers use Coulomb's law to design circuits and electronic devices. Mastering these foundations gives you the tools to tackle more advanced physics in college courses.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Gravity is much weaker than the electric force, yet gravity is the dominant force holding the solar system together. Explain why the electric force doesn't play a significant role at the scale of planets and stars.

Show answer

Large astronomical objects like planets and stars contain roughly equal numbers of protons and electrons, making them electrically neutral overall. Because positive and negative charges cancel, the net electric force between two planets is essentially zero. Gravity, however, has no "negative mass"—every bit of matter contributes to gravitational attraction, so the force accumulates as objects get more massive. This is why gravity dominates at astronomical scales despite being far weaker on a particle-by-particle basis.

Question 2: BASIC CALCULATION

Two small spheres, each carrying a charge of +3.0 × 10⁻⁶ C, are separated by 0.20 m. Calculate the electric force between them using Coulomb's law. Is this force attractive or repulsive? (k = 8.99 × 10⁹ N·m²/C²)

Show answer

Using Coulomb's law: F = k × |q₁| × |q₂| / r² = (8.99 × 10⁹) × (3.0 × 10⁻⁶) × (3.0 × 10⁻⁶) / (0.20)² = (8.99 × 10⁹)(9.0 × 10⁻¹²) / 0.04 = 0.0809 / 0.04 ≈ 2.0 N. Because both charges are positive (like charges), the force is repulsive. This is roughly the weight of a small apple, showing that even microcoulombs of charge can produce a noticeable force at close range.

Question 3: INTERMEDIATE

Two identical objects are separated by a distance d and experience a gravitational force of 4.0 × 10⁻⁸ N. If the distance between them is tripled to 3d, what is the new gravitational force?

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According to Newton's law of gravitation, force is inversely proportional to the square of the distance: F ∝ 1/r². When the distance is tripled (r → 3r), the force is divided by 3² = 9. Therefore the new force is F_new = (4.0 × 10⁻⁸ N) / 9 ≈ 4.4 × 10⁻⁹ N. This demonstrates the inverse-square relationship—tripling the distance reduces the force to one-ninth, not one-third. This same principle applies equally to the electric force because it also obeys 1/r².

Question 4: APPLIED

An electron (mass = 9.11 × 10⁻³¹ kg, charge = −1.60 × 10⁻¹⁹ C) orbits a proton (mass = 1.67 × 10⁻²⁷ kg) at a distance of 5.3 × 10⁻¹¹ m (the Bohr radius of hydrogen). Calculate both the gravitational and electric forces, then find the ratio F_e/F_g. What does this ratio tell you about which force keeps the electron in orbit?

Show answer

Gravitational force:

F_g

= (6.674 × 10⁻¹¹)(9.11 × 10⁻³¹)(1.67 × 10⁻²⁷) / (5.3 × 10⁻¹¹)² ≈ (1.014 × 10⁻⁶⁷) / (2.809 × 10⁻²¹) ≈ 3.6 × 10⁻⁴⁷ N. Electric force:

F_e

= (8.99 × 10⁹)(1.60 × 10⁻¹⁹)(1.60 × 10⁻¹⁹) / (5.3 × 10⁻¹¹)² ≈ (2.30 × 10⁻²⁸) / (2.809 × 10⁻²¹) ≈ 8.2 × 10⁻⁸ N. The ratio

F_e

F_g

≈ 2.3 × 10³⁹, meaning the electric force is over 10³⁹ times stronger. Gravity is utterly irrelevant in holding the hydrogen atom together; the Coulomb attraction between the proton and electron is entirely responsible.

Question 5: CRITICAL THINKING

Both gravitational and electric forces obey the inverse-square law. Explain why physicists believe this shared mathematical structure is not a coincidence but instead reflects a deeper geometric property of three-dimensional space. Hint: think about how the surface area of a sphere changes as you increase its radius.

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In three-dimensional space, a force emanating from a point source spreads uniformly over the surface of an expanding sphere. The surface area of a sphere is 4πr², so as you double the distance, the same total force is spread over four times the area, making the force per unit area (and the force on a test object) one-quarter as strong. This geometric spreading explains why any central force in 3D space naturally follows a 1/r² law. Both mass and charge produce fields that radiate outward in all directions, so they both experience this geometric dilution. If space had four dimensions instead of three, the spreading surface would grow as r³ and both forces would follow 1/r³ instead. The inverse-square law is therefore a fundamental consequence of living in a universe with three spatial dimensions, not just a coincidence between gravity and electricity.

SUMMARY

Lesson Summary

Summary

Gravitational force (F = Gm₁m₂/r²) and electric force (F = k|q₁||q₂|/r²) are both non-contact, inverse-square forces that act along the line connecting two objects. Gravity arises from mass and is always attractive, while the electric force arises from electric charge and can be either attractive or repulsive. The electric force is roughly 10³⁶ times stronger than gravity at the particle scale.

Despite its enormous strength, the electric force's effect is largely neutralized at large scales because matter contains balanced amounts of positive and negative charge. Gravity has no such cancellation, so it dominates on astronomical scales, while electric force dominates on atomic and molecular scales. Both forces obey the inverse-square law because they spread through three-dimensional space, a deep geometric connection that hints at the unity of nature's fundamental interactions.