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Electric Charges And Fields

Question

Three identical charges are placed at the vertices of an equilateral triangle. The force experienced by each charge, ( if k = 1/4 $π ε_{0}$ ) is

$2 k \frac{q^{2}}{r^{2}}$

$\frac{k q^{2}}{2 r^{2}}$

$\sqrt{3} k \frac{q^{2}}{r^{2}}$

$\frac{k q^{2}}{\sqrt{2} r^{2}}$

Solution

$\sqrt{3} k \frac{q^{2}}{r^{2}}$

The magnitude of the force is given by

F = $k \frac{|q_{1} q_{2}|}{r^{2}}$

Force on charge q at A:

( i ) Due to B

f = $k q \times \frac{q}{r^{2}}$

Along BA

f = $\frac{k q^{2}}{r^{2}}$

(ii) Due to C

f = $\frac{k q^{2}}{r^{2}}$

Along CA

f = $\frac{k q^{2}}{r^{2}}$

∴ Resultant force = F

F² = f² + f² + 2f × f × cos 60^o

F² = 2f² + $\frac{2 f^{2} \times 1}{2}$

F² = 3 f²

⇒ F = $\sqrt{3}$ f

⇒ Resultant force = $\sqrt{3} \frac{{kq}^{2}}{r^{2}}$