Question

A uniform vertical electric field $E$ is established in the space between two large parallel plates. A small conducting sphere of mass $m$ is suspended in the field from a string of length $L$. If the sphere is given a $+q$ charge and the lower plate is charged positively, the period of oscillation of this pendulum is :

• A. $2\pi \sqrt{\frac{L}{g}}$
• B. $2\pi \sqrt{\frac{L}{g+\frac{qE}{m}}}$
• C. $2\pi \sqrt{\frac{L}{g-\frac{qE}{m}}}$
• D. $2\pi \sqrt{\frac{L}{{(g^2+{\left(\frac{qE}{m}\right)}^2)}^{1/2}}}$

Answer 1

Answer: (C)

$2\pi \sqrt{\frac{L}{g-\frac{qE}{m}}}$

The equation of motion of sphere, $ma=mg-qE\Rightarrow a=g-\frac{qE}{m}$
Thus the effective acceleration of sphere oscillation will become: $g^{\prime }=a=g-\frac{qE}{m}$
Time period, $T=2\pi \sqrt{\frac{L}{g^{\prime }}}=2\pi \sqrt{\frac{L}{g-\frac{qE}{m}}}$