Principle of Superposition/Examples/Electric Field

Example of Use of Principle of Superposition

The Principle of Superposition applies to an electric field:

The total electric field caused by an assemblage of point charges is equal to the sum of the electric fields caused by each of the point charges individually.

Proof

This is apparent from Electric Field Strength from Assemblage of Point Charges:

$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {q_i} {\size {\mathbf r - \mathbf r_i}^3} \paren {\mathbf r - \mathbf r_i}$

where:

$q_1, q_2, \ldots, q_n$ are point charges

$\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ are the position vectors of $q_1, q_2, \ldots, q_n$ respectively

$\map {\mathbf E} {\mathbf r}$ is the electric field strength at a point $P$ whose position vector is $\mathbf r$.

It follows that the electric field strength caused by $q_i$ alone is:

$\ds \map {\mathbf E_i} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q_i \paren {\mathbf r - \mathbf r_i} } {\size {\mathbf r - \mathbf r_i}^3} $

as shown in Electric Field caused by Point Charge.

Thus we have:

$\ds \map {\mathbf E} {\mathbf r} = \sum_i \ds \map {\mathbf E_i} {\mathbf r}$

Hence the result.

$\blacksquare$

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.2$ The Electric Field