Question

A charge of $+10\mu C$ and mass of $1gm$ is moving with velocity of ${10}^{6}m/s$ at an angle of ${45}^o$ to $x$ & $y$- axis as shown. It experiences a magnetic force along $-$ve $z$-direction. When it is projected with ${10}^{6}m/s$ along +ve $z$-direction, it experiences a magnetic force of ${10}^{-2}N$ in +ve $x$-direction.
The magnetic field $\overrightarrow{B}$ is

• A. $-{10}^{-3}(\hat{i}+\hat{j})T$
• B. $-{10}^{-3}(\hat{j}+\hat{k})T$
• C. $-{10}^{-3}\hat{i}T$
• D. $-{10}^{-2}\hat{j}T$

Answer 1

The correct option is D $-{10}^{-2}\hat{j}T$

Since force is in $z-$direction or in the $x-$direction, $\overrightarrow{B}$ should be in $y-$direction only.

Using the second condition,
$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})$

$\Rightarrow {10}^{-2}\hat{i}={10}^{-6}({10}^6\hat{k}\times \overrightarrow{B})\frac{{10}^{-2}}{{10}^{-6}}\hat{i}=({10}^6\hat{k}\times B\hat{j})$
${10}^4\hat{i}=({10}^6\times B)\hat{i}$
$\Rightarrow B=-{10}^{-2}\text{T}$
$\Rightarrow \overrightarrow{B}=-{10}^{-2}\hat{j}\text{T}$

Hence, the magnetic field will be in $y-$direction with $-{10}^{-2}\text{T}$ magnitude.