JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Elasticity & SHM

Two identical positive charges Q each are fixed at a distance of '2a' apart from each other. Another point charge

q_{0}

with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge

q_{0}

executes SHM. The time period of oscillation of charge

q_{0}

will be :

Answer: (A)

Solution

For small displacement

x

from the midpoint, the net force is

F = \frac{k Q q _{0}}{( a - x ) ^{2}} - \frac{k Q q _{0}}{( a + x ) ^{2}} \approx \frac{4 k Q q _{0} x}{a ^{3}}

for

x ≪ a

. This gives acceleration

a = \frac{4 k Q q _{0}}{m a ^{3}} x

, characteristic of SHM with

ω^{2} = \frac{4 k Q q _{0}}{m a ^{3}} = \frac{Q q _{0}}{π ε _{0} m a ^{3}}

. The time period is

T = 2 π \frac{m a ^{3}}{k Q q _{0}} = 2 π \frac{4 π ε _{0} m a ^{3}}{Q q _{0}} = \frac{4 π ^{3} ε _{0} m a ^{3}}{Q q _{0}}

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option