Electric field:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi}{\mathbf{\epsilon}}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{r}}^{\mathbf{2}}}}$

**(a)**

The point x = 0 is the midpoint between the two positive charges. The electric field is zero at this point.

E = E_{a} + E_{-a}

$\begin{array}{rcl}\mathbf{E}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi}{\mathbf{\epsilon}}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{i}\mathbf{+}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi}{\mathbf{\epsilon}}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{(}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{2}}}\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi}{\mathbf{\epsilon}}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{i}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi}{\mathbf{\epsilon}}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{i}\end{array}$

Two positive point charges q are placed on the x-axis, one at x = a and one at x = -a.

(a) Find the magnitude and direction of the electric field at x = 0.

(b) Derive an expression for the electric field at points on the x -axis. Use your result to graph the x-component of the electric field as a function of x, for values of x between -4a and +4a.

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