Compare gravitational and electric forces quantitatively - Physics

SECTION 1

Historical Context & Motivation

For most of human history, gravity seemed like the only force that mattered. Objects fell to Earth, planets orbited the Sun, and the heavens moved in predictable cycles. Yet by the 1700s, scientists began to realize that a second fundamental force — the electric force — governed interactions between charged particles. This force turned out to be staggeringly stronger than gravity at small scales. Understanding how these two forces compare quantitatively is essential for explaining why atoms hold together, why chemical bonds form, and why the universe is structured the way it is.

1687 • Newton's Principia

Isaac Newton publishes the law of universal gravitation, 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 Torsion Balance

Charles-Augustin de Coulomb uses a torsion balance to measure the force between charged spheres, establishing that electric force also follows an inverse-square law — mirroring gravity in mathematical form but differing dramatically in strength.

1897 • Discovery of the Electron

J.J. Thomson identifies the electron, revealing the particle responsible for electric forces in matter. Comparing the gravitational and electric forces on an electron inside an atom showed that electricity wins by an enormous factor — roughly 10³⁹.

1911 • Rutherford's Nuclear Model

Ernest Rutherford's gold-foil experiment confirms that atoms have a dense, positively charged nucleus. The electric force between the nucleus and orbiting electrons is what holds atoms together; gravity is completely negligible at this scale.

These discoveries raised a central question: both gravity and electricity follow the same mathematical inverse-square form, yet they produce wildly different effects at different scales. Why does gravity rule the cosmos while electric forces dominate the atom? The answer lies in the relative magnitudes of the fundamental constants and in the fact that most large objects are electrically neutral, allowing the weaker gravitational force to take center stage on astronomical scales.

SECTION 2

Core Principles & Definitions

Before comparing the two forces quantitatively, you need a clear picture of what each force law says and what the key variables mean. Both forces are non-contact forces that act over a distance, both obey an inverse-square law, and both are proportional to a product of properties of the two interacting objects. These structural similarities make a side-by-side comparison especially powerful.

Newton's Law of Gravitation

Every two masses attract each other with a force F_g = Gm₁m₂/r². Gravity is always attractive and depends on mass. The gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg² is extremely small.

Coulomb's Law

Two charges exert a force F_e = k|q₁q₂|/r² on each other. This force can be attractive or repulsive depending on the signs of the charges. Coulomb's constant k = 8.99 × 10⁹ N·m²/C² is enormously large.

Inverse-Square Dependence

Both forces weaken as 1/r². If you double the separation distance, both forces drop to one-quarter of their original value. This shared structure means the distance cancels when you take the ratio F_e/F_g.

Sign & Direction

Gravity is always attractive — masses pull toward each other. Electric force can attract (opposite charges) or repel (like charges). This means large bodies with balanced positive and negative charges experience negligible net electric force but still feel gravity.

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KEY TAKEAWAY

Think of gravity and electric force as two speakers in a concert hall. Gravity is a whisper — you can only hear it when the hall is otherwise silent (when charges cancel out in large, neutral objects). The electric force is a roaring amplifier that overpowers everything at close range. Both follow the same inverse-square pattern, but the 'volume knob' — determined by the constants G and k — differs by about 20 orders of magnitude.

SECTION 3

Visual Comparison of the Two Forces

The following diagram places Newton's law of gravitation and Coulomb's law side by side, highlighting their parallel structure. Notice that the mathematical form is nearly identical — the only differences are the constant out front and whether you use mass or charge as the source property. This structural parallel makes it straightforward to compute the force ratio for any pair of particles.

Both force laws share the same inverse-square structure. The ratio F_e/F_g depends only on the constants k and G and on the charges and masses of the particles involved — the separation distance r cancels out completely.

In the diagram above, the violet box represents gravity and the cyan box represents the electric force. The dashed line between them emphasizes the structural mirror. Below both boxes, the ratio formula shows why distance is irrelevant when comparing the two forces acting between the same pair of particles. The enormous value of k/G (about 1.35 × 10²⁰) means that even tiny charges can generate forces far exceeding those produced by large masses.

SECTION 4

Mathematical Framework

The quantitative comparison of gravitational and electric forces begins with writing each law in its standard algebraic form. By dividing one equation by the other, you derive a dimensionless ratio that reveals which force dominates for a given pair of particles. This ratio is independent of distance because both forces share the same r² dependence.

GRAVITATIONAL FORCE

F_g = G × (m₁ × m₂) / r²

G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant), m₁ and m₂ are masses in kilograms, and r is the center-to-center distance in meters.

ELECTRIC (COULOMB) FORCE

F_e = k × |q₁ × q₂| / r²

k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant), q₁ and q₂ are charges in coulombs, and r is the same separation distance. The absolute value ensures we compute the magnitude.

FORCE RATIO (DISTANCE CANCELS)

F_e / F_g = (k × |q₁ × q₂|) / (G × m₁ × m₂)

Because both forces divide by r², the distance term cancels when you form the ratio. The result depends only on the constants and the charge-to-mass properties of the two objects.

Applying the Ratio to a Proton-Electron Pair

Consider the classic case of a proton and an electron inside a hydrogen atom. The proton has mass m_p = 1.673 × 10⁻²⁷ kg and charge +e = 1.602 × 10⁻¹⁹ C. The electron has mass m_e = 9.109 × 10⁻³¹ kg and charge −e = −1.602 × 10⁻¹⁹ C. Plugging into the ratio:

PROTON–ELECTRON RATIO

F_e / F_g = (8.99 × 10⁹ × (1.602 × 10⁻¹⁹)²) / (6.674 × 10⁻¹¹ × 1.673 × 10⁻²⁷ × 9.109 × 10⁻³¹) ≈ 2.27 × 10³⁹

The electric force between a proton and electron is roughly 2.3 × 10³⁹ times stronger than their gravitational attraction. This explains why gravity plays essentially no role in atomic structure. The NGSS Crosscutting Concept of Scale, Proportion, and Quantity is central here: the scale of the system (subatomic particles) determines which force dominates.

SECTION 5

Scale Dependence: From Atoms to Galaxies

The reason gravity dominates at astronomical scales even though it is intrinsically far weaker than the electric force comes down to one key fact: large objects are almost perfectly electrically neutral. A planet like Earth contains roughly 10⁵⁰ protons and an almost identical number of electrons. The positive and negative charges nearly cancel, leaving negligible net charge. Mass, by contrast, never cancels — every kilogram adds to the gravitational pull. This asymmetry is why gravity, despite its tiny constant G, governs the motion of planets, stars, and galaxies.

At atomic scales, the electric force exceeds gravity by roughly 10³⁶ to 10³⁹. As objects become macroscopic and electrically neutral, the net electric force approaches zero and gravity takes over. This illustrates the NGSS Crosscutting Concept of Scale, Proportion, and Quantity: the dominant force depends dramatically on the scale of the system.

The bar chart above shows representative systems arranged from smallest to largest. At the top, subatomic particles have the most extreme electric-to-gravitational force ratio. As you move down to macroscopic neutral objects, the net electric force effectively vanishes. The crossover is not gradual — even a small net charge on a macroscopic object can produce a measurable electric force, as demonstrated by a rubbed balloon sticking to a wall. But for astronomical bodies with essentially zero net charge per kilogram of mass, gravity is the only game in town.

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Why Doesn't Charge Cancel Gravity?

Charge comes in two signs — positive and negative — so opposite charges in a large object cancel each other's fields. Mass, however, is always positive. There is no 'negative mass' to cancel gravitational attraction, so mass accumulates without limit. This fundamental asymmetry is why gravity rules at cosmic scales.

SECTION 6

Worked Example: Two Protons in a Nucleus

Let's compute both forces between two protons separated by 1.0 × 10⁻¹⁵ m (approximately the size of a nucleus) and then find their ratio. Each proton has mass m_p = 1.673 × 10⁻²⁷ kg and charge q = +1.602 × 10⁻¹⁹ C.

Gravitational vs. Electric Force Between Two Protons

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Step 1 — Identify Given Values

m₁ = m₂ = 1.673 × 10⁻²⁷ kg, q₁ = q₂ = 1.602 × 10⁻¹⁹ C, r = 1.0 × 10⁻¹⁵ m. Constants: G = 6.674 × 10⁻¹¹ N·m²/kg², k = 8.99 × 10⁹ N·m²/C².
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Step 2 — Calculate Gravitational Force

F_g = G × m² / r² = (6.674 × 10⁻¹¹)(1.673 × 10⁻²⁷)² / (1.0 × 10⁻¹⁵)². First compute m² = (1.673)² × 10⁻⁵⁴ = 2.799 × 10⁻⁵⁴ kg². Then r² = 1.0 × 10⁻³⁰ m². So F_g = (6.674 × 10⁻¹¹ × 2.799 × 10⁻⁵⁴) / (1.0 × 10⁻³⁰) = 1.868 × 10⁻⁶⁴ / 10⁻³⁰ = 1.87 × 10⁻³⁴ N.

$F_g$ ≈ 1.87 × 10⁻³⁴ N
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Step 3 — Calculate Electric Force

F_e = k × q² / r² = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)² / (1.0 × 10⁻³⁰). First compute q² = (1.602)² × 10⁻³⁸ = 2.566 × 10⁻³⁸ C². Then F_e = (8.99 × 10⁹ × 2.566 × 10⁻³⁸) / (1.0 × 10⁻³⁰) = 2.307 × 10⁻²⁸ / 10⁻³⁰ = 230.7 N.

$F_e$ ≈ 230 N
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Step 4 — Compute the Ratio

F_e / F_g = 230 / (1.87 × 10⁻³⁴) ≈ 1.24 × 10³⁶. Alternatively, use the ratio formula directly: (k/G) × (q²/m²) = (8.99 × 10⁹ / 6.674 × 10⁻¹¹) × (2.566 × 10⁻³⁸ / 2.799 × 10⁻⁵⁴) = 1.347 × 10²⁰ × 9.168 × 10¹⁵ ≈ 1.24 × 10³⁶. Both methods agree.

$F_e$ / $F_g$ ≈ 1.24 × 10³⁶
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Step 5 — Interpret the Result

The electric repulsion between two protons is about 10³⁶ times stronger than their gravitational attraction. This enormous ratio explains why the strong nuclear force — a completely different interaction — is required to hold protons together in a nucleus. Gravity alone is far too weak to overcome the electric repulsion.

SECTION 7

Strengths and Limitations of Each Force

Property	Gravitational Force	Electric Force
Source property	Mass (always positive)	Electric charge (positive or negative)
Direction	Always attractive	Attractive (opposite charges) or repulsive (like charges)
Constant magnitude	G = 6.674 × 10⁻¹¹ (very small)	k = 8.99 × 10⁹ (very large)
Shieldable?	No — no known way to shield or cancel gravity	Yes — opposite charges cancel, Faraday cages shield
Dominant scale	Astronomical (planets, stars, galaxies)	Atomic and molecular (chemistry, biology)
Distance dependence	1/r² (inverse-square)	1/r² (inverse-square)

Comparison of gravitational and electric force properties

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KEY TAKEAWAY

Gravity is like a universal background hum — always present, never shielded, never repulsive, but incredibly quiet. The electric force is like a high-powered signal that can be either positive or negative. In a crowded room of neutral atoms (a large object), all the positive and negative electrical signals cancel to near-silence, leaving only gravity's steady hum to guide planets and stars.

SECTION 8

Connection to the Four Fundamental Forces

Gravity and the electromagnetic force (of which the electric force is one aspect) are two of the four fundamental forces of nature. The other two — the strong nuclear force and the weak nuclear force — operate only at subatomic distances. In advanced physics (beyond this course), Coulomb's law is extended into a full electromagnetic theory by James Clerk Maxwell, and gravity is reinterpreted through Einstein's general relativity as the curvature of spacetime.

Feature	This Course (Classical)	Advanced Physics
Electric force model	Coulomb's law (static point charges)	Maxwell's equations (includes magnetism, radiation, and time-varying fields)
Gravity model	Newton's law of gravitation (instantaneous action at a distance)	General relativity (gravity as spacetime curvature, gravitational waves)
Force ratio use	Compare magnitudes between two particles to determine which force dominates	Dimensional analysis and coupling constants in quantum field theory; running coupling strengths at high energies

Classical vs. advanced treatment of gravitational and electric forces

The quantitative comparison you learned in this lesson — taking the ratio F_e/F_g — remains a valid and useful tool in advanced physics. It becomes the starting point for understanding why gravity is called the 'weakest' force and why unifying gravity with the other forces remains one of the greatest unsolved problems in physics.

SECTION 9

Practice Problems

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NGSS Practices in This Section

These problems integrate multiple NGSS dimensions. Each problem is labeled with the Science and Engineering Practice (SEP) and Crosscutting Concept (CCC) it emphasizes. Use these labels to reflect on which scientific skill you are exercising.

Question 1: CONCEPTUAL

[SEP: Constructing Explanations | CCC: Scale, Proportion, and Quantity] Two protons experience both a gravitational attraction and an electric repulsion. Why does the electric force overwhelmingly dominate over gravity at this scale? (A) The electric force does not follow the inverse-square law, so it falls off more slowly. (B) The proton's charge-to-mass ratio is very large, and the constant k is vastly larger than G, making the electric force roughly 10³⁶ times stronger. (C) Gravity only acts on neutral objects, so it cannot affect charged protons. (D) The two protons are too close together for gravity to apply.

Show answer

Correct answer: (B). Both forces follow the same inverse-square law, so the distance dependence is identical and cancels in the ratio. The electric force dominates because the ratio k/G ≈ 1.35 × 10²⁰ is enormous, and the proton's charge squared divided by its mass squared further amplifies this. The net result is

F_e

F_g

≈ 1.24 × 10³⁶ for two protons. Why other choices are wrong: • (A) is incorrect because both forces obey the 1/r² law — neither falls off faster or slower. • (C) is incorrect because gravity acts on all objects with mass, whether charged or not. Charge does not shield gravity. • (D) is incorrect because there is no minimum distance requirement for gravity; Newton's gravitational law applies at all separations (within classical physics). This problem illustrates CCC: Scale, Proportion, and Quantity — the scale of the system (subatomic) determines that the electric force dominates by an astronomically large factor.

Question 2: BASIC CALCULATION

[SEP: Using Mathematics and Computational Thinking | CCC: Patterns] Two electrons are separated by 1.0 × 10⁻¹⁰ m (approximately one atomic diameter). Calculate the magnitude of the electric force between them. Use k = 8.99 × 10⁹ N·m²/C², e = 1.602 × 10⁻¹⁹ C. (A) 2.31 × 10⁻⁸ N (B) 2.31 × 10⁻¹⁸ N (C) 1.44 × 10⁻⁹ N (D) 8.99 × 10⁻⁸ N

Show answer

Correct answer: (A). Step-by-step:

F_e

= k × q² / r² = (8.99 × 10⁹)(1.602 × 10⁻¹⁹)² / (1.0 × 10⁻¹⁰)². First, q² = (1.602)² × 10⁻³⁸ = 2.566 × 10⁻³⁸ C². Then r² = 1.0 × 10⁻²⁰ m².

F_e

= (8.99 × 10⁹ × 2.566 × 10⁻³⁸) / (1.0 × 10⁻²⁰) = (2.307 × 10⁻²⁸) / (10⁻²⁰) = 2.31 × 10⁻⁸ N. Why other choices are wrong: • (B) 2.31 × 10⁻¹⁸ N results from forgetting to square r, so the student divides by 10⁻¹⁰ instead of 10⁻²⁰, losing 10 orders of magnitude. • (C) 1.44 × 10⁻⁹ N comes from computing k × e / r² (using q instead of q²), omitting the squaring of the charge. • (D) 8.99 × 10⁻⁸ N comes from incorrectly using q² = (1.6)² × 10⁻¹⁹ (failing to square the exponent on 10), yielding a numerator that is too large. This problem reinforces the CCC of Patterns: inverse-square force laws appear across multiple domains.

Question 3: INTERMEDIATE

[SEP: Using Mathematics and Computational Thinking | CCC: Scale, Proportion, and Quantity] Two identical small spheres each have mass m = 0.020 kg and charge q = 3.0 × 10⁻⁶ C. Find the ratio

F_e

F_g

of the electric force to the gravitational force between them. Use k = 8.99 × 10⁹ N·m²/C² and G = 6.674 × 10⁻¹¹ N·m²/kg². (A) 3.03 × 10¹² (B) 8.09 × 10⁻² (C) 3.03 × 10¹⁶ (D) 2.67 × 10⁻¹⁴

Show answer

Correct answer: (A) 3.03 × 10¹². Step-by-step: Since both spheres have the same mass and charge,

F_e

F_g

= (k × q²) / (G × m²). The r² terms cancel. Numerator: k × q² = (8.99 × 10⁹) × (3.0 × 10⁻⁶)² = (8.99 × 10⁹) × (9.0 × 10⁻¹²) = 8.091 × 10⁻² N·m². Denominator: G × m² = (6.674 × 10⁻¹¹) × (0.020)² = (6.674 × 10⁻¹¹) × (4.0 × 10⁻⁴) = 2.670 × 10⁻¹⁴ N·m². Ratio: (8.091 × 10⁻²) / (2.670 × 10⁻¹⁴) = 3.03 × 10¹². Why other choices are wrong: • (B) 8.09 × 10⁻² is the value of k × q² alone (the numerator) — the student computed the numerator but forgot to divide by G × m². • (C) 3.03 × 10¹⁶ results from failing to square q. Using q instead of q² gives k × q = (8.99 × 10⁹)(3.0 × 10⁻⁶) = 2.697 × 10⁴, and then dividing by 2.670 × 10⁻¹⁴ yields approximately 1.01 × 10¹⁸ — but a student who also neglects squaring m (using m instead of m²) gets a denominator of G × m = 1.335 × 10⁻¹² and a ratio of about 2.0 × 10¹⁶, in the neighborhood of this distractor. The key error is neglecting to square both q and m consistently. • (D) 2.67 × 10⁻¹⁴ is the value of G × m² alone (the denominator) — the student computed only the denominator and mistakenly reported it as the answer. This problem demonstrates CCC: Scale, Proportion, and Quantity. Even for macroscopic charged objects (20-gram spheres with microcoulomb charges), the electric force exceeds gravity by a factor of ~10¹².

Question 4: APPLIED

[SEP: Constructing Explanations and Designing Solutions | CCC: Cause and Effect] A rubber balloon of mass 0.003 kg is rubbed on hair and acquires a net charge of about 10 nC (1.0 × 10⁻⁸ C). When placed near a neutral wall, it sticks due to an electric force from induced charges on the wall's surface. Suppose the net attractive electric force on the balloon is measured to be 0.05 N. The balloon's weight is W = mg ≈ 0.003 × 10 = 0.03 N (using g ≈ 10 m/s²). Which statement best explains the balloon's behavior? Note: The electric force of 0.05 N is a given datum based on the specific geometry and induced charge distribution on the wall. (A) The electric force (0.05 N) exceeds the balloon's weight due to gravity (0.03 N), so the balloon sticks without sliding down. (B) The balloon sticks because gravity pulls it into the wall. (C) The balloon sticks because its weight is zero near the wall. (D) The balloon sticks because rubbing removed all the electrons from the balloon.

Show answer

Correct answer: (A). The balloon's weight due to Earth's gravitational pull is about 0.03 N directed downward. The electric force of 0.05 N (provided as a given value) acts horizontally toward the wall due to attraction between the balloon's net charge and the induced opposite charges on the wall's surface. The normal force from the wall and the resulting static friction are large enough to prevent the balloon from sliding down. Thus, the electric force from charge induction overcomes the tendency to fall. Why other choices are wrong: • (B) is incorrect because gravity pulls the balloon downward, not into the wall. Gravity acts vertically, while the electric force acts horizontally toward the wall. • (C) is incorrect because the balloon's weight is not zero — it has mass and is in Earth's gravitational field, giving it a weight of about 0.03 N. • (D) is incorrect because rubbing transfers only a tiny fraction of the balloon's electrons (or transfers electrons to it). The balloon acquires a small net charge of ~10 nC, which represents a minuscule fraction of its total charge — roughly 10⁻⁸ C compared to the enormous total positive and negative charge in all its atoms. This connects to the lesson's key idea that macroscopic objects remain nearly neutral overall. This problem uses CCC: Cause and Effect — the cause (charge transfer by rubbing and subsequent charge induction on the wall) produces the effect (an electric force sufficient to hold the balloon against gravity).

Question 5: CRITICAL THINKING

[SEP: Engaging in Argument from Evidence | CCC: Scale, Proportion, and Quantity] A student claims: 'Since the electric force is 10³⁶ times stronger than gravity between protons, electric forces must also dominate at the scale of planets and stars.' (A) The student is correct — electric forces are always stronger regardless of scale. (B) The student is incorrect — at planetary scales, objects are nearly electrically neutral, so the net electric force approaches zero while gravitational force accumulates with total mass. (C) The student is incorrect — the electric force weakens faster than gravity at large distances. (D) The student is incorrect — gravity becomes a repulsive force at large distances, overpowering electricity. After selecting your answer, write a 2–3 sentence explanation constructing a counterargument to the student's claim. Cite specific evidence from this lesson (the ratio formula, the role of charge cancellation, or the bar chart of force dominance by scale) to support your reasoning.

Show answer

Correct answer: (B). The 10³⁶ ratio applies to individual protons, which are charged. Planets and stars contain roughly equal numbers of positive and negative charges, so their net charge is nearly zero. When net charge approaches zero, the numerator of the ratio formula

F_e

F_g

= (k|q₁q₂|)/(Gm₁m₂) shrinks toward zero as well. Meanwhile, mass is always positive and never cancels, so the gravitational force accumulates with every added kilogram. As shown in the scale diagram (Section 5), the electric force dominance drops from ~10³⁹ at atomic scales to effectively zero for neutral astronomical bodies. Why other choices are wrong: • (A) ignores the critical role of electrical neutrality in large objects. The force law's strength constant k is large, but multiplying k by near-zero net charge yields near-zero force. • (C) is incorrect because both forces obey the same 1/r² dependence — neither weakens faster than the other with distance. • (D) is incorrect because gravity is always attractive; it never becomes repulsive. Sample counterargument: 'The claim fails because the 10³⁶ ratio assumes both objects carry the full elementary charge of a proton, but planets are electrically neutral — their positive and negative charges cancel almost perfectly. Using the ratio formula

F_e

F_g

= (k|q₁q₂|)/(Gm₁m₂), if q₁q₂ ≈ 0 then the electric force vanishes regardless of how large k is, while m₁m₂ continues to grow with the planet's total mass, making gravity the controlling force.'

SUMMARY

Lesson Summary

Summary

Both Newton's law of gravitation and Coulomb's law describe inverse-square forces between pairs of objects. Their shared mathematical structure makes it possible to form the force ratio F_e/F_g = (k|q₁q₂|)/(Gm₁m₂), where the distance dependence cancels completely. For subatomic particles like the proton and electron, this ratio reaches approximately 10³⁹, showing that the electric force overwhelmingly dominates gravity at the atomic scale.

At macroscopic and astronomical scales, large objects are nearly electrically neutral — their positive and negative charges cancel — so the net electric force approaches zero. Because mass never cancels (it is always positive), gravitational force accumulates with every kilogram and becomes the dominant interaction for planets, stars, and galaxies. This lesson illustrates the NGSS Crosscutting Concept of Scale, Proportion, and Quantity: the scale of a system determines which fundamental force governs its behavior.