The force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. It has the same form as gravitation and is far stronger.
The previous session gave the direction of the force between charges: like charges repel, unlike charges attract. In 1785 Charles-Augustin de Coulomb measured its size. He used a torsion balance: a light rod hung on a thin wire, with a charged ball at one end. The angle through which the wire twisted gave the force on the ball.
A point charge is a charged body whose size is very small compared with the distances involved. For two point charges q₁ and q₂ a distance r apart, Coulomb found that the force is proportional to the product q₁q₂ and inversely proportional to r².
Coulomb’s law. F is the size of the force, q₁ and q₂ are the charges, r is the distance between them, and k = 8.99 × 10⁹ N m² C⁻² in a vacuum.
In a vacuum, the constant is k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻², where ε₀ is the permittivity of free space. Two charges of 1 C each, 1 m apart, would exert a force of about 9 × 10⁹ N on each other. Charges met in practice are usually microcoulombs (10⁻⁶ C) or smaller.
Because r appears squared in the denominator, the force falls quickly with distance. Doubling r divides the force by 2² = 4. Tripling r divides it by 9. Ten times the distance gives one-hundredth of the force.
A force that depends on distance as 1/r² is called an inverse-square law. Newton’s law of gravitation and the intensity of light from a point source follow the same dependence.
Two point charges exert a force of 36 N on each other. They are moved three times further apart. What is the new force?
4 N.
F is proportional to 1/r². Multiplying r by 3 multiplies r² by 9, so F is divided by 9.
36 N ÷ 9 = 4 N.
The force acts along the straight line joining the two charges. If q₁ and q₂ have the same sign, the product q₁q₂ is positive and the force is repulsive. If they have opposite signs, the product is negative and the force is attractive.
By Newton’s third law, the force on q₁ due to q₂ is equal in size and opposite in direction to the force on q₂ due to q₁. This is true even when one charge is much larger than the other.
Newton’s law of gravitation, F = Gm₁m₂/r², has the same form: a product of the two sources divided by r². There are two differences.
First, gravitation is only attractive, because mass has only one sign. The electric force can attract or repel, because charge has two signs.
Second, the electric force is much stronger. For two protons, the electric repulsion is about 1.2 × 10³⁶ times the gravitational attraction. The ratio does not depend on distance, because both forces vary as 1/r².
The electric force between two protons is about 10³⁶ times their gravitational force. So why does gravity, not electricity, hold the Earth in orbit around the Sun?
The Sun and the Earth each contain almost exactly equal amounts of positive and negative charge. Their total charge is very close to zero, so the electric forces on their protons and electrons nearly cancel.
Mass has only one sign, so the gravitational effects of all the particles add up and do not cancel.
At the scale of atoms, electric forces are the larger effect: they bind electrons to nuclei and atoms to each other. At the scale of planets, where bodies are neutral overall, only gravity is left.
1According to Coulomb’s law, the force between two point charges is…
2The distance between two charges is doubled. The force becomes…
3Two charges feel a force of 36 N. Moved three times further apart, the force becomes…
4If the product q₁q₂ is negative, the force is…
5How is Coulomb’s law similar to Newton’s law of gravitation?
6Why does gravity, not the electric force, determine the motion of planets?
Next: the force on a charge due to several other charges, using superposition.
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