A non conducting sphere of radius r charged uniformly with surface charge density sigma rotates with angular velocity omega about the axis passing through its centre. Find the magnetic induction at the centre of sphere.

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https://www.physicsforums.com/threads/magnetic-field-of-rotating-sphere.482886/

To calculate the magnetic induction at the center of a uniformly charged, non-conducting sphere rotating about its axis, we can use Ampere's Law and the concept of current density.

First, let's understand the setup: We have a non-conducting sphere of radius r, and it is spinning with an angular velocity ω. The surface charge density (σ) is uniformly distributed on the surface of the sphere.

Ampere's Law states that the line integral of magnetic field (B) around a closed path is equal to the product of the permeability of free space (μ₀) and the current passing through the surface enclosed by that path.

In this case, we can consider a circular loop path of radius R, concentric with the sphere and passing through its center. The current flowing through this loop is given by the current density (J) multiplied by the area of the loop.

At any point on the sphere's surface, the velocity vector of the charge is tangential to the surface and its magnitude is given by the product of the angular velocity (ω) and the distance (r) from the rotation axis to the point. Therefore, the current density (J) is given by J = σωr.

Now, let's calculate the magnetic field using Ampere's Law:

∮B·dl = μ₀Ienclosed,

where Ienclosed is the current enclosed by the loop.

We can rewrite the line integral as B * 2πR (since the path is a circle) and rearrange the equation to solve for B:

B * 2πR = μ₀(J * πR²),

since the area of the loop is πR².

From the equation J = σωr, we can substitute J in the above equation:

B * 2πR = μ₀(σωr * πR²).

Since we are interested in finding the magnetic induction (B) at the center of the sphere (R = 0), the equation simplifies to:

B * 2π(0) = μ₀(σωr * π(0)²).

This simplifies further to B * 0 = 0.

Therefore, at the center of the sphere, the magnetic induction (B) is zero.

To summarize, the magnetic induction at the center of a uniformly charged, non-conducting sphere rotating about its axis is zero.