Coulomb's law:

$\overline{){\mathbf{F}}{\mathbf{=}}\frac{\mathbf{k}{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}}$

**(a) **

A symmetry is created if another charge of magnitude Q is placed directly above the center of the square.

This makes the horizontal components of the forces to cancel leaving the electric force to be equal to the sum of the vertical component of all the forces.

r = sqrt [(half the diagogan)^{2} + (distance between Q and the center of the square)^{2})

$\begin{array}{rcl}\mathbf{r}& \mathbf{=}& \sqrt{{\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{d}\sqrt{\mathbf{2}}\mathbf{\right)}}^{\mathbf{2}}\mathbf{+}{\mathbf{d}}^{2}}\\ & \mathbf{=}& \sqrt{\frac{\mathbf{3}}{\mathbf{2}}}\mathbf{d}\end{array}$

Therefore,

$\begin{array}{rcl}\mathbf{F}& \mathbf{=}& \frac{\mathbf{k}\mathbf{q}\mathbf{Q}}{{\mathbf{r}}^{\mathbf{2}}}\\ & \mathbf{=}& \frac{\mathbf{2}}{\mathbf{3}}\frac{\mathbf{q}\mathbf{Q}}{{\mathbf{d}}^{\mathbf{2}}}\\ {\mathbf{F}}_{\mathbf{y}}& \mathbf{=}& \mathbf{F}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta}\\ \mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta}& \mathbf{=}& \frac{\mathbf{d}}{\sqrt{\frac{\mathbf{2}}{\mathbf{3}}}\mathbf{d}}\\ & \mathbf{=}& \sqrt{\frac{\mathbf{2}}{\mathbf{3}}}\end{array}$

The component of force in the y-direction is expressed as:

$\begin{array}{rcl}{\mathbf{F}}_{\mathbf{y}}& \mathbf{=}& \mathbf{4}\mathbf{F}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta}\\ & \mathbf{=}& \mathbf{4}\mathbf{\left(}\sqrt{\frac{\mathbf{2}}{\mathbf{2}}}\mathbf{\right)}\mathbf{\left(}\frac{\mathbf{2}}{\mathbf{3}}\mathbf{\right)}\mathbf{\left(}\frac{\mathbf{k}\mathbf{q}\mathbf{Q}}{{\mathbf{d}}^{\mathbf{2}}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{177}\frac{\mathbf{k}\mathbf{q}\mathbf{Q}}{{\mathbf{d}}^{\mathbf{2}}}\end{array}$

(a) In Figure 18.59, four equal charges q lie on the corners of a square. A fifth charge Q is on a mass m directly above the center of the square, at a height equal to the length d of one side of the square. Determine the magnitude of q in terms of Q , m , and d , if the Coulomb force is to equal the weight of m.

(b) Is this equilibrium stable or unstable? Discuss.

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