Question

A particle of mass m and charge -q moves diametrically through a uniformly charged sphere of radius R with total charge Q. The angular frequency of the particle's simple harmonic motion, if its amplitude < R, is given by :

• A. $\sqrt{\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{mR}}$
• B. $\sqrt{\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{m{R}^2}}$
• C. $\sqrt{\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{m{R}^3}}$
• D. $\sqrt{\frac{1}{4\pi {\epsilon }_0}\frac{m}{mQ}}$

Answer 1

Answer: (C)

$\sqrt{\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{m{R}^3}}$

$F=\frac{kQrq}{R^3}=-ma$
$a=-\left(\frac{Qq}{4\pi {\epsilon }_{0}m{R}^3}\right)r$
For simple harmonic motion $a=-{\omega }^{2}r$ where $\omega$ is angular frequency
so$-{\omega }^{2}r=-\left(\frac{Qq}{4\pi {\epsilon }_{0}m{R}^3}\right)r$
or $\omega =\sqrt{\frac{Qq}{4\pi {\epsilon }_{0}m{R}^3}}$