Question

1. Assertion(A). The centripetal force on the test charge q. is qo VB, where v is the velocity of a particle and B is the magnetic field. Reason (R): When a charged particle is fired at right angles to the magnetic field, the radius of its circular path is directly proportional to the kinetic energy of the particle​

Answer 1

Assertion(A) is correct but Reason (R) is wrong.

Assertion(A). The centripetal force on the test charge $q_{0}$ is $q_{0}$vB, where v is the velocity of a particle and B is the magnetic field.

• This statement is correct.
• When a test charge $q_{0}$ is projected into a magnetic field perpendicularly, magnetic force puts the charge in a circular motion.
• This magnetic force is given by:

                                          F= $q_{0}$vB                    

• Since this force keeps the charge in a circular motion, thus we can say that it acts as the centripetal force for this charge.
• So the centripetal force on the test charge $q_{0}$ is $q_{0}$vB, where v is the velocity of a particle and B is the magnetic field.

Reason (R): When a charged particle is fired at right angles to the magnetic field, the radius of its circular path is directly proportional to the kinetic energy of the particle​.

• This statement is wrong.
• As established in the explanation for Assertion, centripetal force is given as:

                   F = $q_{0}$vB = $\frac{mv^{2} }{R}$                           (1)

where m is the mass of the particle

    and R is the radius of the particle.

• Further simplifying (1) we get:

                      ⇒ $q_{0}$B = $\frac{mv}{R}$

                      ⇒ R = $\frac{mv}{q_{0}B }$                                   (2)

• In (2) we can see that R is directly proportional to mv(momentum).
• Now,

                                   p=mv

      and Kinetic energy = $\frac{1}{2}$mv²

                          KE = $\frac{(m^{2}v^{2}) }{2m}$                           (multiplying and dividing by m)

                          KE = $\frac{p^{2} }{2m}$

                         p = $\sqrt{2m KE}$                        (3)

• Substituting value of p from (3) in (2)

                       R = $\frac{\sqrt{2m KE} }{q_{0}B }$

• Thus as we can see from the expression above, the radius of the circular path is directly proportional to the square root of Kinetic Energy.

Therefore, the Assertion(A) is correct but the Reason(R) is wrong.