Coulomb's Law (Vector Form)

Force on charge q₂ due to q₁. r̂₁₂ is the unit vector from q₁ to q₂. Positive F₁₂ means repulsion.

Class 11Class 12

Derivation

Setting up the geometry

Let charge $q_1$ be at position $\vec{r}_1$ and charge $q_2$ at $\vec{r}_2$ . Define:

\vec{r}_{12} = \vec{r}_2 - \vec{r}_1, \qquad r = |\vec{r}_{12}|, \qquad \hat{r}_{12} = \frac{\vec{r}_{12}}{r}

$\hat{r}_{12}$ is the unit vector pointing from $q_1$ toward $q_2$ .

\vec{F}_{12} = k\frac{q_1 q_2}{r^2}\hat{r}_{12}

When $q_1 q_2 > 0$ : $\vec{F}_{12}$ points along $\hat{r}_{12}$ , i.e., away from $q_1$ — repulsion.

When $q_1 q_2 < 0$ : $\vec{F}_{12}$ points opposite to $\hat{r}_{12}$ , i.e., toward $q_1$ — attraction.

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

\vec{F}_{21} = k\frac{q_1 q_2}{r^2}\hat{r}_{21} = -\vec{F}_{12}

since $\hat{r}_{21} = -\hat{r}_{12}$ . The two forces are equal in magnitude and opposite in direction.

\boxed{\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{r}_{12}}

Note

The scalar and vector forms are equivalent. The vector form is essential when computing net forces from multiple charges, where direction matters.