A sphere of radius a centered at the origin is made of an ohmic material of conductivity g. The potential at the surface is maintained at the values ​​expressed in spherical coordinates by:Φ=ΦoCosθ.Find the current density at any point inside the sphere.

To find the current density at any point inside the sphere, we can use the following steps:

  1. Calculate the electric field inside the sphere. The electric field inside the sphere can be calculated using the formula:
E = -∇V

where V is the electrostatic potential and ∇ is the gradient operator. In spherical coordinates, the gradient of a scalar function is given by:

∇V = (1/r)∂V/∂r + (1/r sinθ)∂V/∂θ + (1/rsinθ∂ϕ)∂V/∂ϕ

Substituting the given expression for V, we get:

E = -∇V = -((1/r)∂(ΦoCosθ)/∂r + (1/r sinθ)∂(ΦoCosθ)/∂θ)

Evaluating the derivatives, we get:

E = (Φo/r²)(Cosθ + 2sinθ)
  1. Calculate the current density inside the sphere. The current density inside the sphere is given by:
J = σE

where σ is the conductivity of the sphere. Substituting the given expression for E, we get:

J = (σΦo/r²)(Cosθ + 2sinθ)

Therefore, the current density at any point inside the sphere is given by:

J = (σΦo/r²)(Cosθ + 2sinθ)

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