Question

A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of $\lambda$, and the cylinder has a net charge per unit length of $2\lambda$. From this information, use Gauss's law to find (i) the charge per unit length on the inner and outer surface of the cylinder and (ii) the electric field outside the cylinder at a distance $r$ from the axis.

Answer 1

$a).$ $q_1=-\lambda L$ because, wire induces the inner surface of cylinder.

$\Rightarrow q_{cylinder}=q_{in}+q_{out}$

$\Rightarrow \left({\lambda }_{cylinder}\right)L=-\lambda L+q_{out}$

$\Rightarrow q_{out}=\lambda L+\left(2\lambda \right)L$

$\Rightarrow q_{out}=3\lambda L$

$b).$ $E_{outsidecylinder}$ is given by :

$\Rightarrow E\left(2\pi rL\right)=\frac{\lambda L+2\lambda L}{{\epsilon }_0}$ $\{q_{out}=3\lambda L\}$

$\Rightarrow E=\frac{3\lambda L}{{\epsilon }_0\times 2\pi rL}$

$\Rightarrow E=6k\frac{\lambda }{r}$ $\therefore [k=\frac{1}{4\pi {\epsilon }_0}]$

Hence, solved.