A circular concentric disc whose outer radius(R)is 2a and inner radius(r) is 'a' has a surface charge density € (Sigma)find the equation of electric field on its axis 'h' distance away from centre.?

Question

Answer

Equation for Electric Field on the Axis of a Charged Disc

Given Parameters:

Outer radius of the disc, R = 2a

Inner radius of the disc, r = a

Surface charge density of the disc, Σ = ε

Distance of the point on the axis from the center of the disc, h

Permittivity of free space, ε₀

Electric Field due to a Charged Ring

Before we find the electric field on the axis of the charged disc, we need to find the electric field due to a charged ring at the same distance from its center. The formula for the electric field due to a charged ring at a distance h from its center is:

E = (k_eQ/h²)(1/(1 + x²)), where

k_e = 1/(4πε₀) is the Coulomb constant

Q is the charge on the ring

x = r/h, where r is the radius of the ring

Electric Field due to a Charged Disc

The electric field due to a charged disc can be obtained by integrating the electric field due to rings of charge along the axis of the disc. The electric field at a distance h from the center of the disc is:

E = k_eΣh(2z/((h² + z²)^1/2)), where

k_e = 1/(4πε₀) is the Coulomb constant

Σ is the surface charge density of the disc

h is the distance of the point on the axis from the center of the disc

z is the distance of the ring of charge from the point on the axis

Final Equation for Electric Field on the Axis of a Charged Disc

Substituting the values of Σ and z, we get:

E = (2k_eε)(h/(h² + a²)^3/2)

Therefore, the equation for electric field on the axis of a charged disc is given by:

E = (2k_eε)(h/(h² + a²)^3/2)

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