Question

An electron is accelerated through a potential difference V passes through a uniform transverse magnetic field and experiences force F. If the accelerating potential is increased to 2V, force experienced will be:
A)$F$
B)$\dfrac{F}{2}$
C)$\sqrt{2}F$
D)$2F$

Answer 1

Force experienced by a particle in a magnetic field is given by $qvB$. The kinetic energy of the particle in the field will be equal to $eV$. If we derive the velocity in terms of electric potential, we can relate the force acting on the particle and the electric potential of the particle.

Formula used:
$\begin{aligned} & F=qvB \\\ & \dfrac{1}{2}m{{v}^{2}}=eV \\\ \end{aligned}$

Complete answer:
The force acting on the particle in a magnetic field is given by
$F=qvB$
The kinetic energy of the particle is given by$\begin{aligned} & \dfrac{1}{2}m{{v}^{2}}=eV \\\ & v=\sqrt{\dfrac{2eV}{m}} \\\ \end{aligned}$
Substitute this velocity formula in the above force equation,
We get,
$\begin{aligned} & F=qB\sqrt{\dfrac{2eV}{m}} \\\ & F\alpha \sqrt{2V} \\\ \end{aligned}$
Now, given the potential difference is doubled, that means,
$\begin{aligned} & \dfrac{{{F}_{1}}}{{{F}_{2}}}=\sqrt{\dfrac{{{V}_{1}}}{{{V}_{2}}}} \\\ & {{F}_{2}}=\sqrt{\dfrac{2V}{V}}{{F}_{1}} \\\ & {{F}_{2}}=\sqrt{2}{{F}_{1}} \\\ \end{aligned}$

So, the correct answer is “Option C”.

Additional Information:
Moving electric charges produce some number of magnetic fields. The force exerted by the magnetic field on the charged particle moving with velocity v is called the Lorentz force. In this formula, the force acting must be perpendicular to the direction of the magnetic field B.
The direction of Lorentz force can be calculated by using the right-hand thumb rule. Let the fingers of our right hand point the velocity, then the Lorentz force is the direction of the thumb when the fingers curl in the direction of the magnetic field. The Lorentz force is vector quantity. The direction of the force can be calculated by using the right-hand thumb rule.

Note:
In the right-hand thumb rule, the thumb represents the direction of Lorentz force, fingers represent the velocity and direction of curl of fingers represent the magnetic field. Similarly, the index finger is for the current, middle finger is for the magnetic field and the thumb is for the force on the particle in the magnetic field.