To determine the magnetic field at point P due to a uniformly charged rotating disc, we can apply the principles of electromagnetism, particularly focusing on how a rotating charged body generates a magnetic field. Let's break this down step by step.

Understanding the System

We have a disc that is rotating with an angular velocity \( \omega \) and has a uniform charge density \( \sigma \). The point P is located at a distance \( d \) from the center of the disc, where \( d \) is much greater than the radius of the disc. This means that we can treat the disc as a point magnetic dipole for our calculations.

Magnetic Moment of the Disc

The first step is to calculate the magnetic moment \( \mu \) of the disc. The magnetic moment for a rotating charged object can be expressed as:

• Magnetic Moment \( \mu = \frac{1}{2} \sigma A r \omega \)

Where:

• \( A \) is the area of the disc, given by \( A = \pi R^2 \) (with \( R \) being the radius of the disc).
• \( r \) is the distance from the center of the disc to the point where we are calculating the magnetic field.

Substituting the area into the equation gives us:

• \( \mu = \frac{1}{2} \sigma (\pi R^2) R \omega = \frac{1}{2} \sigma \pi R^3 \omega \)

Magnetic Field at Point P

Next, we need to find the magnetic field \( B \) at point P due to this magnetic moment. For a magnetic dipole, the magnetic field at a distance \( d \) along the axis of the dipole is given by:

• \( B = \frac{\mu_0}{4\pi} \cdot \frac{2\mu}{d^3} \)

Here, \( \mu_0 \) is the permeability of free space. Substituting our expression for \( \mu \) into this formula, we get:

• \( B = \frac{\mu_0}{4\pi} \cdot \frac{2 \cdot \frac{1}{2} \sigma \pi R^3 \omega}{d^3} \)

Simplifying this expression leads to:

• \( B = \frac{\mu_0 \sigma R^3 \omega}{4 d^3} \)

Final Expression

Thus, the magnetic field at point P, which is located at a distance \( d \) from the center of the disc, is given by:

• \( B = \frac{\mu_0 \sigma R^3 \omega}{4 d^3} \)

This result shows how the magnetic field decreases with the cube of the distance from the disc, which is a common characteristic of magnetic dipole fields. The presence of the charge density \( \sigma \), the radius \( R \), and the angular velocity \( \omega \) all contribute to the strength of the magnetic field at point P.

In summary, we derived the magnetic field at a point far from a rotating charged disc by first calculating the magnetic moment and then applying the formula for the magnetic field of a dipole. This approach is effective for understanding the behavior of magnetic fields generated by rotating charged bodies.