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Two long concentric conducting cylinders with cross section as shown in figure have radii R and 2R.The inner one carries a charge per unit length. A particle of mass m and negative charge of magnitude q is projected at a velocity $v_{0}$ perpendicular to common axis. The projection is done from a point at a distance $\frac{3 R}{2}$ from the axis of cylinders. Choose correct statement(s).

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v_{0} = \sqrt{\frac{q λ}{4 π ε_{0} m}}

then particle moves in a circular path

v_{0} = \sqrt{\frac{q λ}{2 π ε_{0} m}}

then particle moves in a circular path

Maximum

v_{0}

such that it does not touch the outer cylinder is

\sqrt{\frac{16 q λ}{7 π ε_{0} m} \ln (\frac{4}{3})}

Maximum

v_{0}

such that it does not touch the outer cylinder is

\sqrt{\frac{q λ}{2 π ε_{0} m} [1 + 2 \ln (\frac{4}{3})]}

answer is B.

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Detailed Solution

\frac{2 k λ q}{r} = \frac{m v^{2}}{r}

\sqrt{\frac{λ q}{2 π \in_{0} m}} = v

Question Image

$\int_{\frac{3 R}{2}}^{2 R} \frac{λ q}{2 π \in_{0} r} d r = \frac{λ q}{2 π \in_{0}} \ln \frac{4}{3}$

$\frac{1}{2} m v_{0}^{2} = \frac{1}{2} m \frac{λ q}{2 π \in_{0} m} + \frac{λ}{2 π \in_{0}} \ln \frac{4}{3}$

$v_{0} = \sqrt{\frac{λ q}{2 π \in_{0} m} [1 + 2 \ln \frac{4}{3}]}$

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