Question

A charged particle begins to move from the origin in a region which has a uniform magnetic field in the $x-$direction and a uniform electric field in the $y-$ direction. Its speed is $v$ when it reaches the point $(x,y,z)$. Then, $v$ will depend

• A. only on $x$
• B. only on $y$
• C. on both $x$ and $y,$ but not $z$
• D. on $x,y$ and $z$

Answer 1

Answer: (B)

only on $y$

As given,
$\overrightarrow{B}=B_0\hat{i}$
$\overrightarrow{E}=E_0\hat{j}$
Velocity in the $y-$direction will increase due to the electric field.

The particle will move in yz plane and $v_y$ will increase continuously due to acceleration $a_y=\frac{qE}{m}$

$v_y=\frac{qE}{m}t$ ..................................................$\left(1\right)$

$y=\frac{1}{2}\frac{qE}{m}{t}^2$ ....................................$\left(2\right)$

Substituting $t$ from $\left(2\right)$ in $\left(1\right)$

$v_y=\frac{qE}{m}\sqrt{\frac{2ym}{qE}}=\sqrt{\frac{2yqE}{m}}$

Thus, $v_y$ depends on $y$.