In an uniformly charged ( charge density $ρ)$ dielectric sphere a very thin tunnel has been made along the diameter as shown in fig. A charged particle -q having mass m is released from rest at one end of tunnel. For this situation mark the correct statements (neglect gravity)

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Time period of the particle is $2 π \sqrt{\frac{ρq}{3 ε_{0} m}}$

Speed of the particle while crossing mean position is $R \sqrt{\frac{ρ q}{3 ε_{0} m}}$

Particle will perform oscillation but not SHM

Charge particle will perform SHM about center of the sphere as mean position

answer is A, D.

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Detailed Solution

Electric field inside the sphere E = $\frac{ρ r}{3 ε_{0}}$ (r $\leq R)$

At a radial distance r , m a = -q ( $\frac{ρ r}{3 ε_{0}})$ $\Rightarrow a = - (\frac{ρ q}{3 m ε_{0}}) r . T h i s e q u a t i o n r e p r e s e n t s S H M w h o s e t i m e p e r i o d i s T = 2 π \sqrt{\frac{3 m ε_{0}}{ρ q}}$

Angular frequency is $ω = \sqrt{\frac{ρ q}{3 ε_{0} m}}$

Therefore maximum velocity = $ω R = R \sqrt{\frac{q ρ}{3 m ε_{0}}}$

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