Question

Two charged particles A and B having the same charge,
mass and speed enter into a magnetic field in such a
way that the initial path of A makes an angle of 30° and
that of B makes an angle of 90° with the field. Then the
trajectory of
(a) B will have smaller radius of curvature than that of
А
(b) both will have the same curvature
(c) A will have smaller radius of curvature than that of
B.
(d) both will move along the direction of their original
velocities
​

Answer 1

Answer:

The answer is option (c)

Answer 2

B will have smaller radius of curvature than that of A. option (A).

Given:

• 2 charged particles A and B having same mass, speed and charge.

• A makes 30° with the field.

• B makes 90° with the field.

To find:

Which particle has the longest trajectory.

Explanation:

• A charged particle when enters a magnetic field, it experiences a force that deflects it from its mean position.

• This force is called the Lorentz Force , mathematically given by

                         F = q ( v ×B ) ; or

                         F = q v B sinθ

• If we consider a cyclotron, a same concept follows there too. That is cyclotron is a circular device.

• Hence, when a charged particle enters the cyclotron, it experiences a centripetal force that makes it move on a circular path that is given by the Lorentz force itself.

• Hence, technically the Lorentz force is converted to centripetal force.

Solution:

Let radius of curvature made by A be R₁ and radius by B be R₂.

We know,

Lorentz force = q v B sinθ

Centripetal force  $= \frac{mv}{r} ^{2}$

Since we are asked the curvature, the Lorentz force shall be converted to centripetal force that will make the particle move in a parabolic path.

Therefore,

q v B sinθ = $\frac{m v^{2} }{R}$

We can see that,

sinθ ∝ $\frac{1}{R}$

Hence, greater the angle, smaller will be the trajectory of the path.

and, sin90° > sin30°

Therefore,

R₁ > R₂

Final answer :

Hence, curvature of trajectory will be greater for A and smaller for B . option (A).