Q13P - Introduction to Electrodynamics

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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