Question

A uniform magnetic field is directed into the page. A charged particle moving in the plane of the page follows a clockwise spiral of decreasing radius as shown in the given figure. Which of the following statements is correct?

A. The charge is positive and slowing down
B. The charge is negative and slowing down
C. The charge is positive and speeding up
D. The charge is negative and slowing up

Answer 1

Hint: Use left hand rule to determine the direction of magnetic force on the particle. We know that the direction of magnetic force is inward of its motion that determines whether it is negative charge or positive charge. The necessary centripetal force for the spiral path of the charge is provided by the magnetic force on the charge. Use this concept to determine the relation between velocity of particle and radius of the spiral path.

Complete step by step answer:
According to right hand rule to determine the direction of magnetic force acting on a positive charge, place the thumb in the direction of velocity of the particle and all the fingers in the direction of magnetic field; the direction perpendicular to the palm of the right hand is the direction of the magnetic force on the positive charge.
At the initial point, we point the thumb in the upward direction as the direction of velocity and we point the fingers into the page as the direction of magnetic field is into the page. Therefore, we see the palm is pointing outward of the spiral path. But we see the force is deflecting the particle inside the spiral path, therefore, the charge must be negative.
We use the formula for magnetic force acting on the charged particle, $$F=qvB$$.
The centripetal force for the spiral motion is provided by the magnetic force. Therefore, we can write,
$${\displaystyle \frac{m{v}^2}{r}}=qvB$$
$$\Rightarrow v={\displaystyle \frac{qBr}{m}}$$
Here, q is the charge on the particle, m is the mass and r is the radius of the spiral path.
We see that the velocity of the particle is proportional to the radius of the spiral path. Since the radius is decreasing, the velocity of the particle should also decrease.
Therefore, the charge is negative and slowing down.

So, the correct answer is “Option B”.

Note:
 The direction of magnetic force for negatively charged particles is opposite to the direction of magnetic force on the positive charge. Therefore, in this question, the magnetic force on the particle is deflecting the particle inward of the spiral path because it is negatively charged.