Calculation of Magnetic Field at the Centre of the Ring

To calculate the magnetic field at the centre of the ring, we can use Biot-Savart's law. According to this law, the magnetic field (B) at a point due to a current-carrying element is given by the formula:

B = (μ0 * I * dl * sinθ) / (4π * r^2)

Where:
- B is the magnetic field
- μ0 is the permeability of free space (4π × 10^-7 Tm/A)
- I is the current
- dl is the length element
- θ is the angle between the current element and the line joining the element to the point of observation
- r is the distance between the current element and the point of observation

Calculating the Current

The charge q on the ring is uniformly spread, which means that the linear charge density λ (charge per unit length) is given by:

λ = q / (2πR)

Where R is the radius of the ring.

To calculate the current, we can consider a small element of length dl on the ring. The charge on this element is dq, given by:

dq = λ * dl

The current (I) in this small element is the rate of flow of charge, which can be expressed as:

I = dq / dt

As the ring is rotating with a constant frequency of frevolution/s, the time taken for one revolution (T) is given by:

T = 1 / frevolution

Therefore, the time taken for a small element to complete one revolution is dt = T * (dl / 2πR)

Substituting the values of dq and dt, we get:

I = λ * dl / (T * (dl / 2πR))
I = 2πRλ / T

Substituting the value of λ, we have:

I = 2πR * (q / (2πR)) / T
I = q / T

Calculating the Magnetic Field

Now that we have the current (I) in terms of q and T, we can calculate the magnetic field (B) at the centre of the ring.

Considering a small element of length dl on the ring, the angle θ between the current element and the line joining the element to the point of observation (centre of the ring) is 90 degrees. Therefore, sinθ = 1.

Using Biot-Savart's law, the magnetic field (B) at the centre of the ring is given by:

B = (μ0 * I * dl * sinθ) / (4π * r^2)
B = (μ0 * I * dl) / (4π * r^2)

As the magnetic field is due to the cumulative effect of all the current elements on the ring, we need to integrate this expression over the entire ring.

Integrating both sides of the equation, we have:

∫B = ∫(μ0 * I * dl) / (4π * r^2)

Since the magnetic field is the same at all points on the ring, we can take it out of the integral:

B * ∫1 = ∫(μ0 * I * dl) / (4π * r^2)

The integral on the left