Current density inside a sphere

In summary, the conversation discusses the computation of electric field E using the equation $$\vec{E} = -\nabla V$$ and the resulting equation for current density $$J_f = \displaystyle\frac{\sigma V_0 sin\theta}{r}$$. The speaker is unsure of the required final answer, which should be a constant, and is stuck on the fact that the continuity equation $$\nabla . J_f + \frac{\partial\rho_f}{\partial t}=0$$ does not imply that ##\vec J## is constant.
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.

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

1. What is current density inside a sphere?

Current density inside a sphere refers to the amount of electric current flowing through a unit area inside a spherical object. It is typically measured in amperes per square meter (A/m^2).

2. How is current density calculated inside a sphere?

Current density inside a sphere can be calculated by dividing the total current flowing through the sphere by the surface area of the sphere. This can also be expressed as the product of the electric field strength and the electrical conductivity of the material.

3. What factors affect the current density inside a sphere?

The current density inside a sphere is affected by the electric field strength, the electrical conductivity of the material, and the size and shape of the sphere. It can also be influenced by external factors such as temperature and the presence of other conductive materials nearby.

4. How does the current density change as you move closer to the center of the sphere?

As you move closer to the center of the sphere, the current density typically increases due to the decreasing surface area. This is because the same amount of current is flowing through a smaller area, resulting in a higher current density.

5. What are some practical applications of understanding current density inside a sphere?

Understanding current density inside a sphere is important in various fields, such as electrical engineering, physics, and materials science. It can help in designing and optimizing electrical systems, studying the behavior of materials under electric fields, and predicting the effects of current on different objects and structures.

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