A conducting sphere of radius R and net charge Q is immersed in a large dielectric medium whose permittivity can be expressed in the form

Qn:

A conducting sphere of radius R and net charge Q is immersed in a large dielectric medium whose permittivity can be expressed in the form:

E(r)=Eo(1+a/r)

r: distance from the center of the sphere to any point in space.

a) Determine the electrostatic potential at any point in space

b) Find the polarization charges.

ANS:

(a) Electrostatic Potential

To determine the electrostatic potential at any point in space, we can use Gauss’s law. Gauss’s law states that the electric flux through a closed surface is proportional to the charge enclosed by the surface. In this case, we can use a spherical surface with radius r centered at the center of the sphere.

The electric flux through the spherical surface is given by:

Φ = ∫E dA

where E is the electric field and dA is the element of area. The electric field is given by:

E = -∇V

where V is the electrostatic potential. The element of area for a sphere is given by:

dA = r^2sinθdθdϕ

where θ is the angle from the z-axis and ϕ is the azimuthal angle.

Substituting these equations into Gauss’s law, we get:

∫E dA = 4πQ/ε0

Integrating both sides of the equation, we get:

V = 4πQ/(ε0r) - (1/3)ar^2/ε0

Therefore, the electrostatic potential at any point in space is:

V = 4πQ/(ε0r) - (1/3)ar^2/ε0

(b) Polarization Charges

The polarization charges are the induced charges that arise from the interaction of the electric field with the dielectric medium. The polarization charges are given by:

P = ε0(εr - 1)E

where εr is the relative permittivity of the dielectric medium. Substituting the given expression for E, we get:

P = ε0(εr - 1)Eo(1 + a/r)

The polarization charges are zero at infinity and increase linearly with distance from the center of the sphere. The polarization charges are responsible for the induced electric field, which is opposite to the external electric field.

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