Coulomb's Law

What is Coulomb’s Law?

Coulomb’s law is a quantitative statement that gives the electrostatic force between two point charges. A point charge is a charged body whose size is negligible compared to the distance separating it from other charges. The law was discovered experimentally by Charles-Augustin de Coulomb using a torsion balance.

The force between two point charges:

• Varies directly as the product of the magnitudes of the charges.
• Varies inversely as the square of the distance between them.
• Acts along the line joining the two charges.

How Coulomb arrived at the law

Coulomb used a torsion balance to measure the force between two charged metallic spheres. He did not know the exact amount of charge on each sphere initially. To vary the charge in a known way, he used a clever method:

• If a charged sphere (charge $q$) is touched to an identical uncharged sphere, the charge distributes equally, giving each sphere $q/2$.
• Repeating this process yields charges $q/2$, $q/4$, etc.

By measuring the force for different separations (with fixed charges) and for different charge pairs (with fixed separation), he deduced the inverse-square law.

Mathematical form of Coulomb’s law

In vacuum, the magnitude of the electrostatic force $F$ between two point charges $q_1$ and $q_2$ separated by a distance $r$ is:

where $k$ is a proportionality constant. In SI units, the value of $k$ is chosen as:

Here, $\varepsilon_0$ is the permittivity of free space, with value:

Thus, Coulomb’s law in SI units becomes:

This gives the magnitude of the force. The direction is along the line joining the charges — repulsive for like charges, attractive for unlike charges.

Vector form of Coulomb’s law

Let the position vectors of charges $q_1$ and $q_2$ be $\mathbf{r}_1$ and $\mathbf{r}_2$. Define:

• $\mathbf{r}_{21} = \mathbf{r}_2 - \mathbf{r}_1$ (vector from $q_1$ to $q_2$)
• $\mathbf{r}_{12} = \mathbf{r}_1 - \mathbf{r}_2 = -\mathbf{r}_{21}$
• $r_{21} = |\mathbf{r}_{21}|$ (magnitude)
• Unit vector from $q_1$ to $q_2$: $\hat{\mathbf{r}}_{21} = \frac{\mathbf{r}_{21}}{r_{21}}$

The force on $q_2$ due to $q_1$ is:

Similarly, the force on $q_1$ due to $q_2$ is:

This shows that Coulomb’s law obeys Newton’s third law: action and reaction are equal and opposite.

Important remarks

• The vector form works for any sign of $q_1$ and $q_2$. If the product $q_1 q_2 > 0$ (like charges), $\mathbf{F}_{21}$ is along $\hat{\mathbf{r}}_{21}$ (repulsion). If $q_1 q_2 < 0$ (unlike charges), $\mathbf{F}_{21}$ is along $-\hat{\mathbf{r}}_{21}$ (attraction).
• The law is valid for point charges in vacuum. In matter, the situation is more complex due to the presence of other charged particles.
• Coulomb’s law has been verified down to subatomic distances ($r \sim 10^{-10}$ m).

Definition of the coulomb

The constant $k$ is chosen so that:

• If $q_1 = q_2 = 1$ C and $r = 1$ m, then $F = 9 \times 10^9$ N.

Thus, 1 coulomb is the charge that, when placed 1 m away from an equal charge in vacuum, experiences a repulsive force of $9 \times 10^9$ N. This is a very large unit; in practice, smaller units like $\mu$C ($10^{-6}$ C) or mC ($10^{-3}$ C) are used.

Comparison with gravitational force

Both Coulomb’s law and Newton’s law of gravitation have an inverse-square dependence on distance. However, the electrostatic force is enormously stronger than the gravitational force.

For an electron and a proton:

For two protons:

Inside a nucleus (distance $\sim 10^{-15}$ m), the electrostatic force between two protons is about 230 N, while the gravitational force is only about $1.9 \times 10^{-34}$ N.

Example: Acceleration due to Coulomb force

For an electron and a proton separated by $1$ Å ($10^{-10}$ m):

• Magnitude of force: $|F| = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2} = 2.3 \times 10^{-8}$ N
• Acceleration of electron: $a_e = \frac{F}{m_e} = 2.5 \times 10^{22}$ m/s² …