If a charged particle having charge `q` is moving in a circular path of radius R with a uniform speed u, what would be the magnetic field at the centre?

Options

A. `(mu_(0) qv)/(2 pi R^(2))`
B. `(mu_(0) qv)/(3 pi R^(2))`
C. `(mu_(0)qv)/(pi R^(2))`
D. `(mu_(0) qv)/(4 pi R^(2))`

Explanation

To find the magnetic field at the center of a circular path where a charged particle with charge \( q \) is moving with uniform speed \( u \) and radius \( R \), we can follow these steps:

### Step 1: Understand the relationship between charge motion and current
When a charged particle moves in a circular path, it creates a current. The current \( I \) can be defined as the charge passing through a point per unit time.

### Step 2: Calculate the time period of the circular motion
The time period \( T \) for one complete revolution of the charged particle can be calculated using the formula:
\[ T = \frac{2\pi R}{u} \]
where \( R \) is the radius of the circular path and \( u \) is the speed of the particle.

### Step 3: Calculate the current produced by the moving charge
The current \( I \) can be expressed as:
\[ I = \frac{q}{T} \]
Substituting the expression for \( T \):
\[ I = \frac{q}{\frac{2\pi R}{u}} = \frac{qu}{2\pi R} \]

### Step 4: Use the formula for the magnetic field at the center of a circular loop
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.

### Step 5: Substitute the expression for current into the magnetic field formula
Now substituting \( I = \frac{qu}{2\pi R} \) into the magnetic field formula:
\[ B = \frac{\mu_0 \left(\frac{qu}{2\pi R}\right)}{2R} = \frac{\mu_0 qu}{4\pi R^2} \]

### Step 6: Conclusion
Thus, the magnetic field at the center of the circular path is:
\[ B = \frac{\mu_0 qu}{4\pi R^2} \]