1. Concept Overview
The superposition principle helps us find the total electric force on a charge when several other charges are acting on it. Instead of trying to deal with all forces at once, the idea is simple: calculate the force from each charge separately and then add them as vectors. This method makes complex charge arrangements much easier to analyse.
2. Definition
The superposition principle states that the net electric force on a charge is the vector sum of the individual forces exerted on it by all other charges, calculated independently.
3. Mathematical Form
If charges \( q_2, q_3, q_4, ... \) exert forces \( \vec{F}_{21}, \vec{F}_{31}, \vec{F}_{41}, ... \) on charge \( q_1 \), the total force on \( q_1 \) is:
\( \vec{F}_{\text{net}} = \vec{F}_{21} + \vec{F}_{31} + \vec{F}_{41} + \cdots \)
3.1. Important Notes
- The forces must be added as vectors, not just numbers.
- The principle works because electric forces obey linearity.
- The presence of one charge does not change the force produced by another charge.
4. Why the Principle Works
Electric forces follow the inverse-square law, making them linear in nature. This means the effect of multiple charges simply adds up without modifying one another. So calculating each force separately gives the correct combined force.
5. Worked Examples
5.1. Example 1: Two Forces in a Straight Line
Consider three charges placed on a straight line. Let the forces on charge \( q_1 \) due to charges \( q_2 \) and \( q_3 \) be:
\( F_{21} = 4 \text{ N} \) to the right
\( F_{31} = 6 \text{ N} \) to the left
The net force is:
\( F_{\text{net}} = 6 - 4 = 2 \text{ N} \) to the left.
5.2. Example 2: Forces in Two Dimensions
A charge experiences two forces:
- \( \vec{F}_1 = 3 \text{ N} \) east
- \( \vec{F}_2 = 4 \text{ N} \) north
The net force is the vector sum:
\( F_{\text{net}} = \sqrt{3^2 + 4^2} = 5 \text{ N} \)
Direction is given by \( \tan^{-1}(4/3) \) north of east.
6. Physical Interpretation
The superposition principle tells us that each charge behaves independently. Every charge creates its own electric force as if no others were present, and the actual force a charge experiences is obtained by adding all these contributions. This makes it possible to analyse electric fields and forces even in complicated arrangements.