Figure shows two large cylindrical shells having uniform linear charge densities $+ λ$ and $- λ$ . Radius of inner cylinder is ‘a’ and that of outer cylinder is ‘b’. A charged particle of mass m, charge q revolves in a circle of radius r. Then, its speed ‘v’ is: (Neglect gravity and assume the radii of both the cylinders to be very small in comparison to their length.)

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$\sqrt{\frac{λ q}{2 {πε}_{0} m}}$

$\sqrt{\frac{2 λ q}{{πε}_{0} m}}$

$\sqrt{\frac{λ q}{{πε}_{0} m}}$

$\sqrt{\frac{λq}{4 {πε}_{0} m}}$

answer is A.

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Detailed Solution

Using Gauss's law for Gaussian surface shown in figure.
$\oint \vec{E} . \vec{dA} = \frac{q_{in}}{ε_{0}}$

$E . 2 πrl = \frac{λl}{ε_{0}}$ ;

$∴ E = \frac{λ}{2 {πε}_{0} r}$

For circular motion,
$qE = \frac{{mV}^{2}}{r} = \frac{qλ}{2 {πε}_{0} r}$

$V = \sqrt{\frac{qλ}{2 {πε}_{0} m}}$

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