Question

Assuming mass m of the charged particle, find time period of oscillation

• A. $2\pi \sqrt{\frac{q_{1}q}{4\pi {\epsilon }_0{a}^{3}m}}$
• B. $2\pi \sqrt{\frac{4\pi {\epsilon }_0{a}^{3}m}{q_{1}q}}$
• C. $\frac{\pi }{2}\pi \sqrt{\frac{q_{1}q}{4\pi {\epsilon }_0{q}^{3}m}}$
• D. none of these

Answer 1

Answer: (B)

$E_0=2E_{1}cos\theta =\frac{q}{4\pi {\epsilon }_0(x^2+a^2)}\frac{x}{\sqrt{x^2+a^2}}$
$-q_{1}E=F=mf=\frac{-q_{1}qx}{4\pi {\epsilon }_0(x^2+a^2{)}^{3/2}}$ as x << a, neglecting
x${}^2$ as compared to acceleration, $f=\frac{-q_1{q}_{2}x}{4\pi {\epsilon }_0{a}^{3}m}$
Comparing it with $f={\omega }^{2}x$
$\omega \sqrt{\frac{q_{1}q}{4\pi {\epsilon }_0{a}^{3}m}}$ or $T=2\pi \sqrt{\frac{4\pi {\epsilon }_0{a}^{3}m}{q_{1}q}}$