Question

It is required to hold four equal point charges in equilibrium at the corner of a square. Find the point charge at the centre of the square.

Answer 1

Answer on Question #76057-Physics-Electromagnetism

It is required to hold four equal point charges in equilibrium at the corner of a square. Find the point charge at the center of the square.

Solution

At the corners: $q$, in the center: $-Q$.

For the equilibrium the net force is zero.

$$F_{D} + (F_{A} - F_{C}) \cos 45 = F \sin 45$$ $$F_{B} + (F_{A} - F_{C}) \sin 45 = F \cos 45$$ $$F = 0$$ $$\frac{k q^{2}}{a^{2}} + \left(\frac{k q^{2}}{\left(\sqrt{2} a\right)^{2}} - \frac{k Q q}{\left(\frac{\sqrt{2}}{2} a\right)^{2}}\right) \frac{\sqrt{2}}{2} = 0$$ $$\frac{k q^{2}}{a^{2}} + \left(\frac{k q^{2}}{2 a^{2}} - \frac{k Q q}{\frac{a^{2}}{2}}\right) \frac{\sqrt{2}}{2} = 0$$ $$q + \left(\frac{q}{2} - 2 Q\right) \frac{\sqrt{2}}{2} = 0$$ $$Q = \frac{q}{4} \left(1 + 2 \sqrt{2}\right)$$

The charge will be

$$- Q = - \frac{q}{4} \left(1 + 2 \sqrt{2}\right).$$

Answer provided by https://www.AssignmentExpert.com