A particle of mass m and charge q is kept at the top of a fixed frictionless sphere. A uniform&

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2. Electric Potential and Capacitance

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A particle of mass $m$ and charge $q$ is kept at the top of a fixed frictionless sphere. $A$ uniform horizontal electric field $E$ is switched on. The particle looses contact with the sphere, when the line joining the center of the sphere and the particle makes an angle $45^o$ with the vertical. The ratio $\frac{qE}{mg}$ is :-

A

$\frac{3}{{3 + 2\sqrt 2 }}$

B

$\frac{{3 + 2\sqrt 2 }}{3}$

C

$\frac{3}{{3 - 2\sqrt 2 }}$

D

$\frac{{3 - 2\sqrt 2 }}{3}$

Solution

$m g \cos \theta-q E \sin \theta=\frac{m v^{2}}{R}$ ………$(1)$

Applying work energy theorem

$\frac{1}{2} m v^{2}=m g R(1-\cos \theta)+q E R \sin \theta$ ……..$(2)$

Solving $(1)$ and $(2)$

$\frac{q E}{m g}=\frac{3-2 \sqrt{2}}{3}$

Standard 12

Physics

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(AIIMS-2009)