Question

When a positively charged particle enters a uniform magnetic field with uniform velocity, then the trajectory can be
(i) A circle
(ii) A straight line
(iii) A helix

• A. Only (i)
• B. (i) or (ii)
• C. (i) or (iii)
• D. Anyone of (i), (ii), or (iii)

Answer 1

The correct option is D

Anyone of (i), (ii), or (iii)

The magnetic force

The magnetic force can be expressed as follows:

$F=q\left(v\times B\right)$

Here, $F$ is the magnetic force, $v$ is the velocity of the particle, $q$ is the charge of the particle, and $B$ is the magnetic field.

Consider the magnetic force to be zero $\left(F=0\right)$.

1. The magnetic force becomes zero when the particle moves along the direction of the magnetic field.
2. Since the direction of the magnetic field and the velocity is parallel, the cross product of these two becomes zero.

$$\begin{gathered} v\times B=0 \\ F=0 \end{gathered}$$

3. Since the magnetic force is zero, the particle moves in a straight line in the direction of the incident.

Consider the magnetic force to be perpendicular to the direction of the magnetic field

1. The magnetic force becomes perpendicular to the direction of the magnitude when the particle moves to the perpendicular of the magnetic field.
2. When the particle moves perpendicular to the magnetic field, the cross-product of velocity and the magnetic field becomes negative. This results in a perpendicular magnetic force acting on the system.

$$\begin{gathered} v\times B=-v\times B \\ F=-F \end{gathered}$$

3. In this case, the trajectory of the particle is circular.

Consider the velocity of the particle having both parallel and perpendicular components

1. Due to the parallel component, the particle will tend to move along a straight line.
2. Due to the perpendicular component, the particle will tend to move along a circular path.
3. Hence, the trajectory of the particle will become a helix.

Therefore, the trajectory can be a straight line, a circle, or a helix. Hence option (D) is correct.