Question

Find the electric field a distance${}^s$from an infinitely long straight wire that carries a uniform line charge${}^{\lambda }$) ., Compare Eq. 2.9

Short answer

The electric field at a distance${}^s$from infinitely long wire is${}^{E=\frac{\lambda }{2{\mathrm{\pi s\epsilon }}_0}\hat{s}}$

Step-by-step solution

Describe the given information

It is given that an infinitely long straight wire carries a uniform volume charge density${}^{\lambda }$.The electric field at a distance s around the infinitely long wire has to be evaluated.

Define the Gauss law

If there is a surface area enclosing a volume, possessing a charge${}^q$inside the volume then the electric field due to the surface or volume charge is given as

Here${}^q$is the elemental surface area,${}^{{\epsilon }_0}$is the permittivity of free surface.

Obtain the electric field around the wire

Consider a Gaussian cylinder of finite length / located at a distance symmetrically around the infinitely long wire, as shown below:

It is known that the infinitely long wire consist the linear charge of density$\lambda$For a Gaussian cylinder of length${}^{/}$the total charge inside the Gaussian cylinder is${}^{\lambda /}$

Apply Gauss law, on the Gaussian surface, as,

Thus the electric field at a distance s from infinitely long wire is