A charged particle of mass m and charge q describes circular motion of radius r in a uniform magnetic field of strength B the frequency of revolution is

Options

A. `(Bq)/(2pim)`
B. `(Bq)/(2pim)`
C. `(2pim)/(Bq)`
D. `(Bq)/(2piq)`

Answer

To find the frequency of revolution of a charged particle moving in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the relationship between frequency, angular velocity, and time period.
The frequency \( f \) of revolution is defined as the reciprocal of the time period \( T \):
\[ f = \frac{1}{T} \]
The time period \( T \) can also be expressed in terms of angular velocity \( \omega \):
\[ T = \frac{2\pi}{\omega} \]
Thus, we can express the frequency in terms of angular velocity:
\[ f = \frac{\omega}{2\pi} \]

### Step 2: Determine the expression for angular velocity \( \omega \).
In circular motion, the centripetal force required to keep the particle moving in a circle is provided by the magnetic force acting on the charged particle. The magnetic force \( F_B \) acting on the particle is given by:
\[ F_B = qvB \]
where \( v \) is the velocity of the particle and \( B \) is the magnetic field strength.

This magnetic force acts as the centripetal force, which can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle and \( r \) is the radius of the circular path.

### Step 3: Set the magnetic force equal to the centripetal force.
Equating the two forces gives:
\[ qvB = \frac{mv^2}{r} \]

### Step 4: Solve for the velocity \( v \).
Rearranging the equation to solve for \( v \):
\[ qvB = \frac{mv^2}{r} \implies qB = \frac{mv}{r} \implies v = \frac{qBr}{m} \]

### Step 5: Substitute \( v \) into the expression for angular velocity \( \omega \).
The angular velocity \( \omega \) is related to the linear velocity \( v \) and the radius \( r \) by the relation:
\[ \omega = \frac{v}{r} \]
Substituting the expression for \( v \):
\[ \omega = \frac{qBr}{m} \cdot \frac{1}{r} = \frac{qB}{m} \]

### Step 6: Substitute \( \omega \) into the frequency formula.
Now substituting \( \omega \) into the frequency formula:
\[ f = \frac{\omega}{2\pi} = \frac{qB}{m \cdot 2\pi} \]

### Final Result:
Thus, the frequency of revolution of the charged particle in the magnetic field is:
\[ f = \frac{qB}{2\pi m} \]