A thin ring of radius `R` meter has charge `q` coulomb uniformly spread on it. The ring rotates about its axis with a constant frequency of `f` revolution/s. The value of magnetic induction in `Wb m^(-2)` at the centre of the ring is

Options

A. `(mu_(0) q f)/(2pi R)`
B. `(mu_(0) q )/(2pi f R)`
C. `(mu_(0) q )/(2f R)`
D. `(mu_(0) q f)/(2R)`

Explanation

To find the magnetic induction (magnetic field) at the center of a rotating charged ring, we can follow these steps:

### Step 1: Understand the problem
We have a thin ring of radius \( R \) meters with a total charge \( q \) coulombs uniformly distributed on it. The ring rotates about its axis with a frequency \( f \) revolutions per second. We need to find the magnetic induction at the center of the ring.

### Step 2: Determine the current
The charge \( q \) is uniformly distributed over the ring. When the ring rotates, it creates a current. The current \( I \) can be calculated using the formula:
\[ I = \frac{\text{Charge}}{\text{Time period}} \]
The time period \( T \) for one complete revolution is given by:
\[ T = \frac{1}{f} \]
Thus, the current \( I \) can be expressed as:
\[ I = \frac{q}{T} = q \cdot f \]

### Step 3: Apply the Biot-Savart Law
For a circular current-carrying conductor, the magnetic field \( B \) at the center of the ring can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 4: Substitute the current into the magnetic field formula
Now, substituting the expression for current \( I \) into the magnetic field formula:
\[ B = \frac{\mu_0 (q \cdot f)}{2R} \]

### Step 5: Write the final expression
Thus, the magnetic induction at the center of the ring is:
\[ B = \frac{\mu_0 q f}{2R} \]

### Conclusion
The magnetic induction at the center of the ring is given by:
\[ B = \frac{\mu_0 q f}{2R} \text{ Wb/m}^2 \]

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