Why Is the Current Density Inside a Sphere Constant?

[Nirmal Padwal]

Homework Statement

Question: A sphere of radius a centered at origin is made of linear isotropic homogeneous conducting material. The potential on surface is maintained at values given in spherical coordinates by $$V=V_0cosθ$$ $V_0$= constant. Find the free current density $J_f$ everywhere inside.

Relevant Equations

1. $$J_f = \sigma E$$
2. $$E = - \nabla V$$

Since sphere is made of l.i.h material, $$\vec{J_f}= \sigma \vec{E}$$

We compute electric field E using
$$\vec{E} = -\nabla V$$
$$= -\nabla \left(V_0cos\theta\right)$$
$$= -\frac{\hat\theta}{r}\frac{{\partial}}{{\partial\theta}}\left(V_0cos\theta\right)$$
$$\vec{E}= \frac{V_0sin\theta}{r}\hat\theta$$
This yields, $$J_f = \displaystyle\frac{\sigma V_0 sin\theta}{r}$$
I understand that this isn't the required answer as final answer should be a constant as there is no motion of free charge $$\left(\displaystyle\frac{\partial\rho_f}{\partial t} =0\right)$$ and continuity equation yields
$$\nabla . J_f + \frac{\partial\rho_f}{\partial t}=0$$ $$\nabla . J_f = 0$$ $$J_f = \mathrm{constant}$$

But I am stuck here. Furthermore, I don't understand the required final answer is pointing in direction of z axis.

 

[Orodruin]

$\nabla \cdot \vec J = 0$ does not imply that $\vec J$ is constant.