Question

A particle of charge $–q$ and mass $m$ moves in a circle of radius $r$ around an infinitely long line charge of linear charge density $\lambda$. Then time period will be

• A. $T=2\pi r\sqrt{\frac{m}{2k\lambda q}}$
• B. $T=\frac{1}{2\pi r}\sqrt{\frac{m}{2k\lambda q}}$
• C. $T=2\pi r\frac{m}{k\lambda q}$
• D. $T=\frac{1}{2\pi r}\sqrt{\frac{2k\lambda q}{m}}$

Answer 1

The correct option is A $T=2\pi r\sqrt{\frac{m}{2k\lambda q}}$

The electric force on charge $-q$ due to infinite line charge will be balanced by the centrifugal force i.e.

net force on the charge particle with respect to charge is zero.
$F_{net}=ma$
$⟹qE-\frac{m{v}^2}{r}=m\left(0\right)$
$\frac{m{v}^2}{r}=qE$
$\frac{m{v}^2}{r}=q\frac{\lambda }{2\pi {ϵ}_{0}r}$
$⟹v=\sqrt{\frac{q\lambda }{2\pi {ϵ}_{0}m}}=\sqrt{\frac{2kq\lambda }{m}}$
$T=2\pi r\sqrt{\frac{m}{2kq\lambda }}$