Validity and Failure of Ohm's Law

A spherical shell of inner radius $a$ and outer radius $b$ is filled with a poorly conducting material whose resistivity $\rho$ varies with distance $r$ from its center as $\rho ={\rho }_0{r}^3.$ Find the resistance between the sphere's inner surface and the outer surface.

A uniform copper wire of mass $2.23\times {10}^{-3} \mathrm{kg}$ carries a current of $1 \mathrm{A}$ when $1.7 \mathrm{V}$ is applied across it. Calculate its length and area of cross-section. If the wire is uniformly stretched to double its length, calculate the new resistance. The density of copper is $8.92\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ and resistivity is $1.7\times {10}^{-8} \mathrm{\Omega } \mathrm{m}$.

The figure shows a conductor of length $l$ having a circular cross-section.

The radius of cross-section varies linearly from $a$ to $b$. The resistivity of the material is $\rho$. Assuming that $b-a\ll l$, find the resistance of the conductor.

If a copper wire is stretched to make it $0.1\%$ longer, what is the percentage change in its resistance?

A wire has a resistance $R.$ What will be its resistance if

$\left(\mathrm{a}\right)$ it is double on itself and

$\left(\mathrm{b}\right)$ it is stretched so that
$\left(\mathrm{i}\right)$ its length is doubled and $\left(\mathrm{ii}\right)$ its radius is halved.

In an experiment, a graph (figure) was plotted of the potential difference $V$ between the terminals of a cell against circuit current $I$ by varying load rheostat. Find the internal conductance of the cell.
(a) 

(b) 

The current in a simple series circuit is $5 \mathrm{A}$. When an additional resistance of $2 \mathrm{\Omega }$ is introduced, the current is reduced to $4 \mathrm{A}$. Calculate the resistance of the original circuit. Assume that the applied potential difference is the same in both the cases.

A battery of emf $2 \mathrm{V}$ and internal resistance $0.1 \mathrm{\Omega }$ is charged with a current of $5 \mathrm{A}$. In what direction will the current flow inside the battery? What is the potential difference between the two terminals of the battery?