B M Sharma Solutions for Chapter: Electric Current and Circuits, Exercise 1: Concept Application Exercise

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.

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

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 $A=\pi R^2$.

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.

The space between the plates of a parallel plate capacitor is completely filled with a material of resistivity $2\times {10}^{11} \mathrm{\Omega } \mathrm{m}$ and dielectric constant $6$. The capacity of the capacitor with the given dielectric medium between the plates is $20 \mathrm{nF}$. Find the leakage current if a potential difference $2500 \mathrm{V}$ is applied across the capacitor.

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 $1.7\times {10}^{-8} \mathrm{\Omega } \mathrm{m}$.

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 $E=f\left(r\right)$

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.