Question

A charged particle with some initial velocity is projected in a region where non-zero electric and/or magnetic fields are present. In column I, information about the existence of electric and/or magnetic field and direction of initial velocity of charged particle are given; while in Column II, the probable path of the charged particle is mentioned. Match the entries of Column I with the entries of Column II.
$\begin{array}{cccc}& \text{Colum I}& & \text{Colum II}\\ \text{i.}& \overrightarrow{E}=0,\overrightarrow{B}\ne 0\text{and initial velocity may}& \text{a.}& \text{Straight line}\\ & \text{be at any angle with}\overrightarrow{B}.& & \\ \text{ii.}& \overrightarrow{E}=0,\overrightarrow{B}=0\text{and initial velocity may}& \text{b.}& \text{Parabola}\\ & \text{be at any angle with}\overrightarrow{E}.& & \\ \text{iii}& \overrightarrow{E}\ne 0,\overrightarrow{B}\ne 0,\overrightarrow{E}||\overrightarrow{B}\text{and initial velocity is}& \text{c.}& \text{Circular}\\ & \perp \text{to both.}& & \\ & & \text{d.}& \text{Helical path}\end{array}$

• A. $\text{(i)}\to \text{(a),(b),(c); ~(ii)}\to \text{(a),(b);~ (iii)}\to \text{(a),(d)}$
• B. $\text{(i)}\to \text{(a),(c),(d);~ (ii)}\to \text{(a),(b);~ (iii)}\to \text{(d)}$
• C. $\text{(i)}\to \text{(a),(c),(d); ~(ii)}\to \text{(a),(b); ~(iii)}\to \text{(c),(d)}$
• D. $\text{(i)}\to \text{(c),(d); ~(ii)}\to \text{(d),(b);~ (iii)}\to \text{(d)}$

Answer 1

The correct option is B $\text{(i)}\to \text{(a),(c),(d);~ (ii)}\to \text{(a),(b);~ (iii)}\to \text{(d)}$

If $\overrightarrow{E}=0,\overrightarrow{B}\ne 0$ then let us take 3 cases,
Case1 :
When $\overrightarrow{V}\parallel \overrightarrow{B}$ then $\overrightarrow{V}\times \overrightarrow{B}=0$
$⟹F_m=q(\overrightarrow{V}\times \overrightarrow{B})=0$
So, the particle will not experience any force and will continue to move in its path without getting deflected (i.e straight).
Case 2:
When $\overrightarrow{V}\perp \overrightarrow{B}$ , then
$F_m=q\overrightarrow{V}\overrightarrow{B}$ ($\because$ $\overrightarrow{V}\perp \overrightarrow{B}$)
So, in this case the path followed by the particle is going to be circular.
Case 3:
When $\overrightarrow{V}$ is at some angle with $\overrightarrow{B}$
$⟹F_m=q\overrightarrow{V}\overrightarrow{B}sin\theta$
So, in this case the particle will follow a hellical path.
According to this, (a), (c) and (d) matches with (i).

If $\overrightarrow{E}\ne 0$ and $\overrightarrow{B}=0$
In this if $\overrightarrow{E}$ is parallel to $\overrightarrow{V}$ then the particle will follow straight line path.
Secondly, if $\overrightarrow{E}$ is at some angle with $\overrightarrow{V}$ then it will follow parabolic path.
According to this, (a) and (b) matches with (ii).
If $\overrightarrow{E}\ne 0,\overrightarrow{B}\ne 0$, $\overrightarrow{E}\parallel \overrightarrow{B}$ and initial velocity is $\perp$ to both.
So, in this case the particle will follow hellical path as it also experiences an electric force which accelerates it in the forward direction.
According to this, option (d) matches with (iii).