Electrostatics

JEE Main level, single correct, medium

38 of 311

Question 38 of 311

A solid sphere of radius RR carries a charge Q+qQ+q distributed uniformly over its volume. A very small point like piece of it of mass mm gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge qq. If it acquires a speed ν\nu when it has fallen through a vertical height yy (see figure), then (assume the remaining portion to be spherical)

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Answer: (D) v2=2y[qQ4πϵ0R(R+y)m+g]{v}^{2}=2y\left[\dfrac{qQ}{4\pi {\epsilon }_{0}R(R+y)m}+g\right]

By using total energy conservation
ΔKE+(ΔPE)electo+(ΔPE)gravitation=0\Delta KE+(\Delta PE{)}_{\text{electo}} +(\Delta PE{)}_{\text{gravitation}} =0
12mV2+(kQqR+y−kQqR)+(−mgy)=0\dfrac{1}{2}m{V}^{2}+\left(k\dfrac{Qq}{R+y}-k\dfrac{Qq}{R}\right)+(-mgy)=0
12mV2=mgy+kQq(1R−1R+y);V2=2gy+2kQqmyR(R+y)\dfrac{1}{2}m{V}^{2}=mgy+kQq\left(\dfrac{1}{R}-\dfrac{1}{R+y}\right);{V}^{2}=2gy+\dfrac{2kQq}{m}\dfrac{y}{R(R+y)}
V2=2y[qQ4πϵ0R(R+y)m+g]{V}^{2}=2y\left[\dfrac{qQ}{4\pi {\epsilon }_{0}R(R+y)m}+g\right]

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