Question

When a proton is released from rest in a room, it starts with an initial acceleration $a_0$ towards west. When it is projected towards north with a speed $v_0$ it moves with an initial acceleration $3a_0$ towards west. The electric and magnetic field in the room are respectively ? (Assume gravity free space)

• A. $\frac{m{a}_0}{e}\text{west}$ , $\frac{2m{a}_0}{e{v}_0}\text{down}$
• B. $\frac{m{a}_0}{e}\text{east}$ , $\frac{3m{a}_0}{e{v}_0}\text{up}$
• C. $\frac{m{a}_0}{e}\text{east}$, $\frac{3m{a}_0}{e{v}_0}\text{down}$
• D. $\frac{m{a}_0}{e}\text{west}$ , $\frac{2m{a}_0}{e{v}_0}\text{up}$

Answer 1

The correct option is A $\frac{m{a}_0}{e}\text{west}$ , $\frac{2m{a}_0}{e{v}_0}\text{down}$

Using the equation, $F=q(\overrightarrow{E}+(\overrightarrow{v}\times \overrightarrow{B}\left)\right).........\left(1\right)$

Case $a:$ (a) proton released from rest which accelerates with acceleration $a_0$ towards west.


$\therefore \overrightarrow{F_B}=q(\overrightarrow{v}\times \overrightarrow{B})=0[\because \overrightarrow{v}=0]$

$\overrightarrow{F_E}=q\overrightarrow{E}=m{a}_0.........\left(2\right)$

Let, charge of the proton $q=e$ and $m=$ mass of proton.

From eqn (2) we get,

$\overrightarrow{E}=\frac{m{a}_0}{e}$ towards west

Case $b:$


From eq (1) we get,

$\overrightarrow{F_{net}}=\overrightarrow{F_E}+\overrightarrow{F_m}$

$\Rightarrow m\left(3a_0\right)=m{a}_0+q(\overrightarrow{v}\times \overrightarrow{B})$

$\Rightarrow 2m{a}_0=e(\overrightarrow{v}\times \overrightarrow{B})=e{v}_{0}B\sin {90}^{\circ }=e{v}_{0}B$

$\Rightarrow \overrightarrow{B}=\frac{2m{a}_0}{e{v}_0}\otimes$ into the plane/down.

Direction derived by right-hand thumb rule.

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, option (a) is the correct answer.

Why this question ? To familiarize oneself how to use Lorentz's force equation in different scenarios in order to get the asked parameters on which it depends. Also, to make oneself at ease in deciding directions of forces as it may affect the solution of whole problem.