Question

Analysis Model: Particle in a Field (Electric)
Four charged particles are at the corners of a square of side a as shown in Figure. Determine (a) the electric field at the location of charge $q$ and (b) the total electric force exerted on $q$.

Answer 1

We sum the electric fields from each of the other charges using Equation 23.7 for the definition of the electric field.
The field at charge $q$ is given by
$\overrightarrow{E}=\frac{k_e{q}_1}{r_1^2}+\frac{k_e{q}_2}{r_2^2}{\hat{r}}_2+\frac{k_e{q}_3}{r_3^2}\widehat{r_3}$
(a) Substituting for each of the charges gives
$\overrightarrow{E}=\frac{k_e\left(2q\right)}{a^2}\hat{i}+\frac{k_e\left(3q\right)}{2a^2}(\hat{i}\cos {45.0}^o+\hat{j}\sin {45.0}^o)+\frac{k_e\left(4q\right)}{a^2}\hat{j}$
$=\frac{k_{e}q}{a^2}[(2+\frac{3}{2}\cos {45.0}^o)\hat{i}+(\frac{3}{2}\sin {45.0}^o+4)\hat{j}]$
$=\frac{k_{e}q}{a^2}(3.06\hat{i}+5.06\hat{j})$
(b) The electric force on charge $q$ is given by
$\overrightarrow{F}=q\overrightarrow{F}=\frac{k_{e}q}{a^2}(3.06\hat{i}+5.06\hat{j})$