Question

A beam of ions with velocity\[2 \times {10^5}m/s\] enters normally into a uniform magnetic field of\[4 \times {10^{ - 2}}\] tesla. If the specific charge of the ion is\[5 \times {10^7}C/kg\] then radius of the circular path described will be
A. 0.10 m
B. 0.16 m
C. 0.20 m
D. 0.25 m

Answer 1

Hint: It is given that the charge is entering into a uniform magnetic field, so it will start moving in a circular path. It is basically a cyclotron, in which the charged particle gains acceleration if electric and magnetic fields are present.




Formula used:
Radius of the circular path
\[r = \dfrac{{mv}}{{Be}}\]




Complete answer:
The specific charge of the ion is defined as the ratio of the charge of an electron to the mass of the electron. The value of specific charge on an ion is given, so according to formula it can be written that
\[\dfrac{e}{m} = 5 \times {10^7}C/kg\]
Also, in case of a cyclotron the radius of the circular path in a magnetic field is given by the following formula:
\[r = \dfrac{{mv}}{{Be}}\]
It is given that\[B = 4 \times {10^{ - 2}}Tesla\]
Also the velocity is given as, \[v = 2 \times {10^5}m/s\]
Substituting all the given values in the formula and solving for radius we get,
\[r = \dfrac{{2 \times {{10}^5}}}{{5 \times {{10}^7} \times 4 \times {{10}^{ - 2}}}}\]
The radius of the charged ion will hence be 0.1m.
Therefore, the radius of the circular path described by the ion will be 0.10m.

Hence, Option A is the correct answer.




Note: If an electric charge or subatomic particle has both electric and magnetic fields, then the net force acting on the charge will be calculated by applying the formula of Lorentz force as done in this case. Lorentz force is the sum of electric force and the magnetic force. The force experienced by the charge due to the magnetic field varies according to the strength of the charge and the magnetic field. Also, if the velocity with which the charge is moving is high, then the charge will experience more force.