Section 1.3: Coulomb’s Law (Vector Form)

This section introduces Coulomb’s law in vector form, describing the force between two point charges and how to compute forces using vectors.

Coulomb’s Law (Vector Form):

The force on charge \( q_1 \) due to \( q_2 \) is:

\( \vec{F}_{12} = k_e \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12} \)

Where \( \hat{r}_{12} \) is the unit vector pointing from \( q_1 \) to \( q_2 \).

Example 1

Two charges \( q_1 = 2 \, \mu C \) and \( q_2 = -3 \, \mu C \) are 0.5 m apart. Find the force vector on \( q_1 \).

Magnitude: \( F = k_e \frac{|q_1 q_2|}{r^2} = 8.99 \times 10^9 \frac{(2 \times 10^{-6})(3 \times 10^{-6})}{0.5^2} \approx 0.215 \, \mathrm{N} \)

Direction: Attractive, along the line connecting the charges.

Charges \( q_1 = 1 \, \mu C \) and \( q_2 = 2 \, \mu C \) are 0.3 m apart. Find the force vector on \( q_1 \).
Three charges at vertices of an equilateral triangle: calculate net force on one charge.
Charge \( -5 \, \mu C \) at origin, \( +5 \, \mu C \) at (0,1 m). Determine the force vector.
Charges \( q_1 = q_2 = 1 \, \mu C \) are 0.1 m apart. Express force in vector components assuming they lie along x-axis.
Explain why Coulomb’s law is vectorial and how to apply superposition.