# (a) Use Gauss’ law to derive the expression for the electric field $\left ( \vec{E} \right )$due to a straight uniformly charged infinite line of charge density $\lambda \; C/m\cdot$ (b) Draw a graph to show the variation of E with perpendicular distance r from the line of charge. (c) Find the work done in bringing a charge q from perpendicular distance $r_{1}\, to\, r_{2}\left ( r_{2}> r_{1} \right )\cdot$

a) Electric field due to an infinitely long straight charged wire

Consider infinitely long straight charged wire of linear charge density $\lambda \, C/m$
For calculating electric field consider an imaginary cylindrical Gaussian Surface of radius r and length l.
Here the field is radial everywhere, so flux through the two ends of cylinder is Zero.
At the Gaussian cylindrical surface, the electric field E is normal to the surface at every point. The magnitude of E depends only on radius 'r', so it is constant.
Therefore flux through Gaussian surface
$= E\times 2\pi rl---(1)$
According to Gauss's law , flux

$\phi = \frac{charge\; enclosed}{\varepsilon _{0}}$
Here, total charge enclosed = linear charge density $\times$ length
$= \lambda l$
Therefore flux $\phi= \frac{\lambda l}{\varepsilon _{0}}---(2)$
Using equation (1) and (2)  $E\times 2\pi rl= \frac{\lambda l}{\varepsilon _{0}}$
That is ,

$E= \frac{\lambda }{2\pi \varepsilon _{0}r}$
The vector notation is

$\vec{E}= \frac{\lambda }{2\pi \varepsilon _{0}r}\hat{n}$
where $\hat{n}$ is the radial unit vector normal to the line charge
b)

$E\alpha \frac{1}{r}$
c) work done to move the charge 'q' through a small displacement 'dr'
$dW= F\cdot dr$
We know , $F= qE$
where E = electric field
Therefore $dW= qEdr\cos \theta = qEdr\left ( \theta = 0^{\circ} \right )$
Substitute the value of E
$dW= q\times \frac{\lambda }{2\pi \varepsilon _{0}r}dr$
work done in moving the charge from r1 to r2 $\left ( r_{2}> r_{1} \right )$ is
$W=\int_{r_{1}}^{r_{2}}dW= \int_{r_{1}}^{r_{2}}q\, \frac{\lambda }{2\pi \varepsilon _{0}r}dr$
$W=q\, \frac{\lambda }{2\pi \varepsilon _{0}}\left [ l_{n} r_{2}- l_{n} r_{1}\right ]$
$W=q\, \frac{\lambda }{2\pi \varepsilon _{0}}ln\frac{r_2}{r_1}$

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