First slide

Electrostatic force

Question

Three identical spheres, each having a charge q and radius R, are kept in such a way that each touches the other two. The magnitude of the electric force on any sphere due to the other two is

Moderate

A

$\frac{1}{4 π ε_{0}} {(\frac{q}{R})}^{2}$
B

$\frac{\sqrt{3}}{4 π ε_{0}} {(\frac{q}{R})}^{2}$
C

$\frac{\sqrt{3}}{16 π ε_{0}} {(\frac{q}{R})}^{2}$
D

$\frac{\sqrt{5}}{16 π ε_{0}} {(\frac{q}{R})}^{2}$

Solution

For external points, a charged sphere behaves as if the whole of its charge is concentrated at its center.
Force on A due to B,
$\begin{array}{c} F_{A B} = \frac{1}{4 π ε_{0}} \frac{q^{2}}{(2 R)^{2}} \\ = \frac{1}{4 π ε_{0}} \frac{q^{2}}{4 R^{2}} along \vec{B A} \end{array}$
And force on A due to C,
$F_{A C} = \frac{1}{4 π ε_{0}} \frac{q^{2}}{(2 R)^{2}} = \frac{1}{4 π ε_{0}} \frac{q^{2}}{4 R^{2}} along \vec{C A}$
Now as angle between BA and CA is 60^o and
$\begin{array}{l} |F_{A B}| = |F_{A C}| = F \\ ∴ F_{A} = \sqrt{F^{2} + F^{2} + 2 F F \cos 60} = \sqrt{3} F \\ = \frac{1}{4 π ε_{0}} \frac{\sqrt{3}}{4} {(\frac{q}{R})}^{2} \end{array}$

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