A circular ring of radius a carries a uniform charge ρLC/m and placed on XY- plane with axis the same as the Z-axis then examine the electric field intensity E at (0,0,h), (0,0,-h), (0,0,h), and (0,0,0)

The electric field E due to a ring of charge can be found using the principle of superposition; that is, the total electric field at a point is simply the vector sum of the electric fields due to the individual charges.

1. At (0,0,h):

The electric field at a point on the axis of a ring of charge is directed along the line joining the center of the ring and the point. The magnitude of the electric field E at a distance z from the center of the ring along the axis is given by:

E = (1/4πε₀) * (Qz) / (z² + a²)^(3/2)

where Q is the total charge on the ring, z is the distance from the center of the ring to the point, and a is the radius of the ring. In this case, Q = ρL * 2πa, where ρL is the linear charge density. Substituting this into the equation gives:

E = (1/4πε₀) * (ρL * 2πa * h) / (h² + a²)^(3/2)

2. At (0,0,-h):

The electric field at this point is the same as at (0,0,h), but directed in the opposite direction, because the electric field is a vector quantity.

3. At (0,0,h):

This is the same point as in step 1, so the electric field is the same.

4. At (0,0,0):

At the center of the ring, the electric field is zero, because the electric fields due to the individual charges on the ring cancel each other out.