Question

An alpha particle is an electric field of intensity E experiences the same force as when it moves perpendicular to a magnetic field of induction B. The magnitude of its velocity in the magnetic field is given by
A. ${\displaystyle \frac{B}{E}}$
B. ${\displaystyle \frac{E}{B}}$
C. ${\displaystyle \frac{B^2}{E}}$
D. ${\displaystyle \frac{E^2}{B}}$

Answer 1

Hint:-If a charged particle is moving or changing its direction, then it will have both electric as well as magnetic fields. It produces an electric field because it is charged. Also, it produces a magnetic field, because the particle is moving.

Complete step-by-step solution
Step I:
Since alpha particles are positively charged, so it will be deflected by the electric field. Since they are charged particles, alpha particles can also be deflected by the magnetic fields. But both electric and magnetic fields are also not dependent on each other.
Step II:
When a particle travels through both electric and magnetic fields, the net force on the charge is known as Lorentz force.
Step III:
Given that both electric and magnetic fields are experiencing same force, therefore
$f_E=f_m$---(i)
Step IV:
The force due to the electric field will act parallel to the charge and is independent of the velocity of the charge. Force due to electric field is written as
$f_E=q.\overrightarrow{E}$
Where q is the charge and
E is the electric field
The magnetic force will always act perpendicular to the velocity. Therefore it is written as
$f_m=q\overrightarrow{v}\times \overrightarrow{B}$
Where v is the velocity of the particle
B is the magnetic field
Step V:
Substituting the values in equation (i),
$q.\overrightarrow{E}=q\overrightarrow{v}\times \overrightarrow{B}$
$v={\displaystyle \frac{E}{B}}$
Step VI:
Hence, Option B is the right answer.

Note:-
It is to be noted that the strength of the electric field and magnetic fields can be increased by increasing the amount of charge and the strength. When the amount of charge is increased, then their velocities will also increase. The magnetic field helps in the charge moving and the electric field accelerates and imparts energy to the charged particles.