Why Does Gravity Hold the Planets in Orbit if Electric Forces Are Stronger?

The claim that electric force is stronger than gravity is true for certain pairs of particles, but it does not mean electricity controls every large object. The key difference is that ordinary matter is nearly electrically neutral, while gravity acts on mass whether or not matter carries a net electric charge.

That is why the Sun and planets move mainly under gravity. Their electric charges largely cancel at large distances, but their masses do not.

How much stronger is electric force between particles?

For two point charges, Coulomb’s law gives the electric force. For two masses, Newton’s law gives the gravitational force. Comparing the two forces between a proton and another proton gives

$$\frac{F_{\mathrm{electric}}}{F_{\mathrm{gravity}}} =\frac{k e^2}{Gm_p^2} \approx 1.2\times 10^{36},$$

where $k$ is Coulomb’s constant, $e$ is the elementary charge, $G$ is the gravitational constant, and $m_p$ is the proton’s mass. The electric repulsion between two protons is therefore about $10^{36}$ times the gravitational attraction between them. Between two electrons, the ratio is even larger, about $4\times10^{42}$, because electrons have much less mass.

This comparison applies to charged particles. It does not directly describe the force between two electrically neutral planets.

Why electric forces mostly cancel in planets and stars

Atoms contain positively charged nuclei and negatively charged electrons. In ordinary matter, those charges nearly balance. A planet may contain an enormous number of charged particles, but if it has nearly equal total positive and negative charge, its electric field far away is small compared with the fields those charges would produce if they did not cancel.

Gravity behaves differently. There is no known negative gravitational mass that cancels positive mass in ordinary matter. The gravitational attraction from each part of a planet adds to the attraction from every other part. A planet can therefore have a large gravitational field even when its net electric charge is close to zero.

Electrical forces are still essential inside matter. They hold atoms and molecules together and govern chemical bonds, friction, and many contact forces. The point is that those forces do not usually produce a large net electric pull or push between neutral astronomical bodies.

A numerical comparison for the Earth and Sun

At the Earth’s average distance from the Sun, Newton’s law gives a gravitational force of about

$$F_g=\frac{GM_{\mathrm{Earth}}M_{\mathrm{Sun}}}{r^2} \approx 3.5\times10^{22}\ \mathrm{N}.$$

If both bodies had net charges, the magnitude of their electric force would be $F_e=k|Q_{\mathrm{Earth}}Q_{\mathrm{Sun}}|/r^2$. Dividing by the gravitational force cancels the distance:

$$\frac{F_e}{F_g} =\frac{k|Q_{\mathrm{Earth}}Q_{\mathrm{Sun}}|}{GM_{\mathrm{Earth}}M_{\mathrm{Sun}}}.$$

For the electric force to match gravity, the product of the two bodies’ charge-to-mass ratios would need to be about

$$\left|\frac{Q_{\mathrm{Earth}}}{M_{\mathrm{Earth}}} \frac{Q_{\mathrm{Sun}}}{M_{\mathrm{Sun}}}\right| =\frac{G}{k} \approx 7.4\times10^{-21}\ \mathrm{C^2/kg^2}.$$

If their charge-to-mass ratios had equal magnitudes, each would need about $8.6\times10^{-11}\ \mathrm{C/kg}$. That corresponds to roughly $5\times10^{14}$ coulombs for Earth and $2\times10^{20}$ coulombs for the Sun. These are hypothetical values, not estimates of their actual net charges. Such enormous charges would create intense electric fields and attract or repel surrounding charged particles, making them very unlike the nearly neutral bodies observed in the solar system.

The sign also matters: like net charges repel, while opposite net charges attract. Gravity between positive-mass bodies is attractive in either case.

What about plasma, magnetic fields, and solar wind?

The Sun and space around it are not perfectly electrically inactive. The Sun emits charged particles; planets can have magnetic fields; and plasma can carry electric currents. These effects shape auroras, radiation belts, solar storms, and the motion of small charged particles. They do not usually compete with gravity as the main force setting the large-scale orbits of planets.

A small net charge can matter for dust grains or spacecraft because these have little mass. Solar radiation pressure and the solar wind can also affect small particles. For the massive planets, however, such influences are small orbital perturbations, not the force that keeps them in orbit.

Common mistakes

• Comparing forces on particles and then applying the ratio to whole planets. The particle comparison assumes the particles carry charge. A neutral planet’s positive and negative charges mostly cancel at long range.
• Assuming every electric force is attractive. Electric forces can attract or repel depending on charge signs; gravity between ordinary masses is attractive.
• Saying electromagnetic forces do not matter in space. They matter greatly for plasma, radiation, and small charged objects. They simply do not dominate the planets’ overall orbital motion.
• Thinking a planet must be exactly neutral. It can have a small net charge or electrical structure. The relevant point is that its large-scale electric effect is far smaller than it would be if all its constituent charges failed to cancel.