Question

A uniform magnetic field acts at right angles to the direction of motion of electrons. As a result, the electron moves in a circular path of radius \[2{\rm{ cm}}\], if the speed of electrons is doubled, then find the radius of the circular path.
A. \[2{\rm{ cm}}\]
B. \[0.5{\rm{ cm}}\]
C. \[4{\rm{ cm}}\]
D. \[1{\rm{ cm}}\]

Answer 1

Hint: In the given question, we need to find the radius of the circular path if the seed of the electron is doubled. For this, we need to use the formula for force experienced by a charged particle in an external magnetic field to get the desired result.




Formula used:
The following formula is used for solving the given question.
The radius of circular path is given by
 \[r = \dfrac{{mv}}{{qB}}\]
Here, \[m\]is the mass, \[v\] is the velocity, \[B\] is the magnetic field strength, and \[q\] is charge.


Complete answer:
We know that the radius of a circular path is given by
\[r = \dfrac{{mv}}{{qB}}\]
Here, \[m\]is the mass, \[v\] is the velocity, \[B\] is the magnetic field strength, and \[q\] is charge.
This indicates that the radius of a circular path is directly proportional to velocity. That means as velocity increases, radius also increases and vice versa.
Mathematically, it is given by
\[r\alpha v\]
If \[{v_2} = 2{v_1}\] then \[{r_2} = 2{r_1}\].
That means, we get
\[{r_2} = 2 \times 2\]
\[{r_2} = 4{\rm{ cm}}\]
Hence, the radius of a circular path is \[4{\rm{ cm}}\].

Therefore, the correct option is (C).

Note: The field lines must be both parallel and equally spaced in order for the region under examination to have a uniform magnetic field, which must possess the same strength and direction throughout. Many students make mistakes in calculation as well as writing the formula of radius of a circular path. This is the only way through which we can solve the example in the simplest way. Also, it is essential to do calculations carefully to get the correct value of the force.