Question

A particle is moving in a region having a uniform magnetic field and a parallel uniform electric field. At some instant, the velocity of the particle is perpendicular to the field direction. The path of the particle will be a

• A. circle
• B. straight line
• C. helix with uniform pitch
• D. helix with non-uniform pitch

Answer 1

The correct option is D helix with non-uniform pitch

We know that

$\overrightarrow{F}=q[\overrightarrow{E}+(\overrightarrow{v}\times \overrightarrow{B}\left)\right]$

Assuming $\overrightarrow{E}$ and $\overrightarrow{B}$ are along $+vez-$axis and $\overrightarrow{v}$ is along $+vex-$axis.

$\overrightarrow{F}=q[E_0\hat{k}+(v_0\hat{i}\times B_0\hat{k}\left)\right]$

$\overrightarrow{F}=q[E_0\hat{k}+B_0{v}_0(-\hat{j}\left)\right]$

$\Rightarrow \overrightarrow{F}=q[E_0\hat{k}-B_0{V}_0\hat{j}]$

Hence, the magnetic force is along $-vey-$axis, it causes change in the direction of velocity.

Electric force is acting along $+vez-$axis, so it accelerates the charged particle. So the velocity along $+vez-$axis increases continuously.

So the path of the charged particle is a helix with non-uniform pitch.

Hence, option $\left(d\right)$ is the correct answer.