Question

A particle of mass $m$ and charge $q$ is placed in an electric field $E$ which varies with time $t$ as $E=E_0\sin \omega t$. It will undergo simple harmonic motion of amplitude

• A. $\frac{q{E}_0^2}{m{\omega }^2}$
• B. $\frac{q{E}_0}{m{\omega }^2}$
• C. $\sqrt{\frac{q{E}_0}{m{\omega }^2}}$
• D. $\frac{q{E}_0}{m\omega }$

Answer 1

The correct option is B $\frac{q{E}_0}{m{\omega }^2}$

$E=E_o\sin \omega t$

Force on the charged particle $(q,m)$

$F=ma=qE=q{E}_o\sin \omega t$

$\Rightarrow m\frac{dv}{dt}=q{E}_o\sin \omega t$

After Integrating,

$m\int dt=q{E}_o\int \sin \omega t\cdot dt$

$mv=-\frac{q{E}_o}{\omega }\cos \left(\omega t\right)+C_1$

So,

$v=-\frac{q{E}_o}{m\omega }\cos \left(\omega t\right)+C_1=\frac{dx}{dt}$

After Integrating again,

$\int dx=-\frac{q{E}_o}{m\omega }\int \cos \omega t\cdot dt+\int C_{1}dt$

$x=-\frac{q{E}_o}{m{\omega }^2}\sin \left(\omega t\right)+C_{1}t+C_2$

The first term on the right provide the SHM part.

So, amplitude of the SHM part is $\frac{q{E}_o}{m{\omega }^2}$