Electric potential outside an insulator in a uniform field

[frimidis]

Homework Statement

An Ohmic material with some conductivity has a uniform current density J initially. Let's say the current is flowing in the direction of the z-axis. A small insulating sphere with radius R is brought inside the material. Find the potential outside the sphere.

Homework Equations

Continuity equation $$\nabla\cdot J=-\dfrac{\partial\rho}{\partial t}$$
Ohmic material electric field $$J=\sigma E$$
Maxwell's equations boundary conditions


$$E_1^{\parallel}-E_2^{\parallel}=0$$

and

$$\epsilon_1E_2^{\perp}-\epsilon_2E_2^{\perp}=\sigma_f$$

Presumably there's no free charge so the charge density of it is 0.

The Attempt at a Solution

I assume the insulator will make the current flow such that there is no change in the charge density. Hence $$\nabla\cdot J=0$$
and so $$\nabla\cdot E=0$$
and so the electric potential obeys the Laplace equation outside the sphere
$$\nabla^2V=0$$

I know the general solution for the Laplace equation in spherical coordinates when there's azimuthal symmetry. I'm stuck with the boundary conditions.

1) The first one is easy. Far away from the insulator the potential is just due to the electric field that causes the current to flow in the Ohmic material so far away in spherical coordinates
$$V_{far}=-\dfrac{J}{\sigma}z=-\dfrac{J}{\sigma}r\cos\theta$$
2) I'm unsure about the other boundary conditions. And they seem to be only about the discontinuity of the electric field components at the boundary of the sphere. Apparently we can/need only consider one of the electric field components because we don't have enough information what is going on in the insulator. Is the electric field supposed to be 0 inside it or does it just mean that there doesn't flow any current? I'm stuck at the boundary conditions and I'm not sure how exactly is the current supposed to be affected by the presence of the insulator. Does the current just flow around it tangentially? Presumably no current can go inside as it is an insulator.

Thanks

 

[TSny]

2) Does the current just flow around it tangentially? Presumably no current can go inside as it is an insulator.

Yes. In the steady state, any free surface charge on the sphere must be constant in time. What does this tell you about the radial component of $\vec J$ at the outside surface of the sphere? In turn, what boundary condition on $V$ does this imply at the outside surface of the sphere?