A charged particle of mass m is moving with a speed u in a circle of radius r. If the magnetic field induction at the centre is B, the charge on the particle is

Options

A. `(4pi r^(2)B)/(mu_(0)u)`
B. `(2pir^(2)B)/(mu_(0)u)`
C. `(m u)/(Br)`
D. `(2m u)/(Br)`

Answer

To solve the problem, we need to find the charge \( q \) on a particle of mass \( m \) that is moving in a circular path of radius \( r \) with a speed \( u \), while experiencing a magnetic field \( B \) at the center of the circle.

### Step-by-Step Solution:

1. **Identify the Current**:
The current \( I \) due to the moving charge can be expressed as:
\[ I = \frac{q}{T} \]
where \( T \) is the time period of the motion.

2. **Calculate the Time Period**:
The time period \( T \) for a particle moving in a circular path is given by the circumference of the circle divided by the speed:
\[ T = \frac{2\pi r}{u} \]

3. **Substitute Time Period into Current**:
Now substituting \( T \) into the expression for current:
\[ I = \frac{q}{T} = \frac{q}{\frac{2\pi r}{u}} = \frac{q u}{2\pi r} \]

4. **Use the Magnetic Field Formula**:
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

5. **Substitute Current into Magnetic Field Equation**:
Substitute the expression for current \( I \) into the magnetic field equation:
\[ B = \frac{\mu_0}{2r} \cdot \frac{q u}{2\pi r} \]
Simplifying this gives:
\[ B = \frac{\mu_0 q u}{4\pi r^2} \]

6. **Rearranging to Find Charge \( q \)**:
Now, we can rearrange this equation to solve for \( q \):
\[ q = \frac{4\pi r^2 B}{\mu_0 u} \]

### Final Answer:
The charge \( q \) on the particle is given by:
\[ q = \frac{4\pi r^2 B}{\mu_0 u} \]