On a semicircular ring of radius = 4 R, charge +3q is distributed in such a way that on one quarter +q is uniformly distributed and on another quarter +2q is uniformly distributed. Along its axis a sm

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On a semicircular ring of radius = 4 R, charge +3q is distributed in such a way that on one quarter +q is uniformly distributed and on another quarter +2q is uniformly distributed. Along its axis a smooth nonconducting and uncharged pipe of length 6R is fixed axially as shown. A small ball of mass m and charge +q is thrown from the other end of pipe. The ball can come out of the pipe if:

TestHub · Accepted Answer

On the pipe's axis every charge element is equidistant, so only the total charge $3 q$ matters (the $q$ / $2 q$ split is irrelevant here).

The ring threads the 6R pipe at its middle, so the ball starts $3 R$ from the ring plane; the potential is maximum at the ring plane.

$V_{e n d} = \frac{3 k q}{( 4 R ) ^{2} + ( 3 R ) ^{2}} = \frac{3 k q}{5 R}$ ; $V_{ma x} = \frac{3 k q}{4 R}$ .

$\frac{1}{2} m u^{2} > q (V_{ma x} - V_{e n d}) = \frac{3 k q ^{2}}{20 R}$ ⇒ $u > \frac{3 q ^{2}}{40 π ε _{0} R m}$ - (B)

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