One end of an insulating rigid rod of negligible mass and length $l$ is pivoted to a fixed point O: A small ball of mass m having a negative charge of magnitude q is attached to the lower end of the rod. Another small ball carrying a positive charge Q is fixed at a height h above the point O as shown in the figure. What should the range of values of mass $m$ of the lower ball be so that the rod remains in a state of stable equilibrium? Acceleration due to gravity is g.

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$m > \frac{Q q h}{4 π ε_{0} g {(h + l)}^{3}}$

$m > \frac{Q q}{4 π ε_{0} g {(h + l)}^{2}}$

$m < \frac{Q q h}{8 π ε_{0} g {(h + l)}^{3}}$

$m < \frac{Q q h}{4 π ε_{0} g {(h + l)}^{3}}$

answer is C.

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Detailed Solution

For stable equilibrium
$\frac{\sin β}{l} = \frac{\sin (α - β)}{h} h = (\frac{l \sin (α - β)}{\sin β}) \frac{β}{l} = \frac{(α - β)}{h} h β = α l - β l β = (\frac{α l}{h + l}) M g l \sin α > \frac{K Q q}{{(h + l)}^{2}} \frac{(h) α l}{(h + l)} m > \frac{Q q h}{4 π ε_{0} g {(h + l)}^{3}}$