Question

A charged particle moves in a gravity free space without change in velocity. Which of the following is not possible in that space?

• A. $E=0,B=0$
• B. $E\ne 0,B=0$
• C. $E=0,B\ne 0$
• D. $E\ne 0,B\ne 0$

Answer 1

The correct option is B $E\ne 0,B=0$

Force on a charge $q$ moving with velocity $\overrightarrow{V}$ in the combination of electric field $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ is given by,$$\text{Lorentz force}=q[\overrightarrow{E}+(\overrightarrow{V}\times \overrightarrow{B}\left)\right]$$
Since there is no change in velocity,the net force on the charge $q$ is zero.i.e,

$\Rightarrow q[\overrightarrow{E}+\overrightarrow{V}\times \overrightarrow{B}]=0$

The possible cases for the above expression:

$\left(1\right)$ If $E=0,B=0$, $\Rightarrow {\overrightarrow{F}}_{net}=0$

$\left(2\right)$ If $E=0,B\ne 0$ and $\theta$ between $\overrightarrow{B}$ & $\overrightarrow{V}$ is $0^{\circ }\text{or}{180}^{\circ }$.
$\Rightarrow F_{net}=q\times 0+qVB\sin \left(0^{\circ }\text{or}{180}^{\circ }\right)=0$

$\left(3\right)$ If $E\ne 0,B\ne 0$ but $q\overrightarrow{E}=-q(\overrightarrow{V}\times \overrightarrow{B})$

$\because$ both forces are equal & opposite, so
${\overrightarrow{F}}_{net}=q\overrightarrow{E}+q(\overrightarrow{V}\times \overrightarrow{B})=0$

But for the option $\left(b\right)$ $B=0$ and $E\ne 0$
$\therefore {\overrightarrow{F}}_{net}=q\overrightarrow{E}$
and this force will change the velocity $\overrightarrow{V}$ of charge. Hence it is not possible.

Hence, option (b) is the correct answer.