Question

Consider a parallel plate capacitor having an electric field $E$ inside it as shown in the figure. A particle of mass $m$ and charge $q$ is hanging inside the capacitor throgh a light string of length $l$. The time period, $T$ of this pendulum for small oscillations is:

• A. $T=2\pi \sqrt{\frac{l}{g+\left(\frac{qE}{m}\right)}}$
• B. $T=2\pi \sqrt{\frac{l}{{(g^2+\frac{q^2{E}^2}{m^2}+\frac{2qgE}{m}\cos \theta )}^{\frac{1}{2}}}}$
• C. $T=2\pi \sqrt{\frac{l}{{(g^2+\frac{q^2{E}^2}{m^2}-\frac{2qgE}{m})}^{\frac{1}{2}}}}$
• D. $T=2\pi \sqrt{\frac{l}{{(g^2+\frac{q^2{E}^2}{m^2}+\frac{2qgE}{m})}^{\frac{1}{2}}}}$

Answer 1

The correct option is B $T=2\pi \sqrt{\frac{l}{{(g^2+\frac{q^2{E}^2}{m^2}+\frac{2qgE}{m}\cos \theta )}^{\frac{1}{2}}}}$

Let us draw the $\text{F.B.D.}$ of the bob of the pendulum:


Here, the electrostatic force, $F_e=qE$,

Weight of the bob, $F_g=mg$

Now, the net force acting on the bob,

$F_{net}=\sqrt{F_e^2+F_g^2+2F_e{F}_g\cos \theta }$

$F_{net}=\sqrt{q^2{E}^2+m^2{g}^2+2qEmg\cos \theta }$

Time period of the oscillations,

$T=2\pi \sqrt{\frac{l}{g_{eff}}}$

Where, the effective acceleration on the bob,

$g_{eff}=\frac{F_{net}}{m}=\sqrt{\frac{q^2{E}^2+m^2{g}^2+2qEmg\cos \theta }{m^2}}$

$\Rightarrow g_{eff}=\sqrt{\frac{q^2{E}^2}{m^2}+g^2+\frac{2qgE}{m}\cos \theta }$

So,

$T=2\pi \sqrt{\frac{l}{{({\left(\frac{qE}{m}\right)}^2+g^2+\frac{2gqE\cos \theta }{m})}^{\frac{1}{2}}}}$

Hence, option (d) is the correct answer.