Current Density

A hollow metallic sphere has an inner radius $a$, outer radius $b$ and resistivity $\rho$. If a potential difference $V$ is applied between the inner and outer surface. Find the (i) resistance between inner and outer surface (ii) total current
(iii) current density in the function of radial distance $J=f\left(r\right)$
(iv) electric field in the function of radial distance $E=f\left(r\right)$

The current density across a cylindrical conductor of radius $R$ varies in magnitude according to the equation, $J=J_0\left(1-\frac{r}{R}\right)$ where, $r$ is the distance from the central axis. Thus, the current density is maximum $\left(J_0\right)$ at that axis $(r=0)$ and decreases linearly to zero at the surface $(r=R)$. Calculate the current in terms of $J_0$ and the conductor's cross-sectional area $A=\pi R^2$.

A beam contains $2.0\times {10}^8$ doubly charged positive ions per cubic centimetre, all of which are moving towards the north with a speed of $1.0\times {10}^5{ \mathrm{m} \mathrm{s}}^{-1}$. (a) What are the magnitude and direction of the current density $\overrightarrow{J}$?

(b) Can you calculate the total current $i$ in this ion beam? If not, what additional information is needed?

An infinite line charge of uniform electric charge density $\lambda$ lies along the axis of an electrically conducting infinite cylindrical shell of radius $R$. At time $t=0$, the space inside the cylinder is filled with a material of permittivity $\epsilon$ and electrical conductivity $\sigma$. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density $j\left(t\right)$ at any point in the material?