Question

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in xy-plane centred at origin. A point charge $q$ is moving towards the ring along the z-axis and has speed $u$ at $z = 4a$. The minimum value of $u$ such that it crosses the origin is :

• A. $\sqrt{\frac{2}{m}} \left(\frac{1}{15} \frac{q^{2}}{4\pi\varepsilon_{0}a}\right) ^{1/2} $
• B. $\sqrt{\frac{2}{m}} \left(\frac{2}{15} \frac{q^{2}}{4\pi\varepsilon_{0}a}\right) ^{1/2} $
• C. $\sqrt{\frac{2}{m}} \left(\frac{2}{15} \frac{q^{2}}{3\pi\varepsilon_{0}a}\right) ^{1/2} $
• D. $\sqrt{\frac{2}{m}} \left(\frac{1}{5} \frac{q^{2}}{4\pi\varepsilon_{0}a}\right) ^{1/2} $

Answer 1

Answer: (B)

$U_i + K_i = U_f + K_f$
$\frac{kq^{2}}{\sqrt{16a^{2}+9a^{2}}} + \frac{1}{2}mv^{2} = \frac{kq^{2}}{3a} $
$ \frac{1}{2} mv^{2} = \frac{kq^{2} }{a} \left(\frac{1}{3} - \frac{1}{5}\right) = \frac{2kq^{2}}{15a} $
$ v= \sqrt{\frac{4kq^{2}}{15ma}} $