Question

Determine the potential of a non-conducting long straight wire that carries uniform linear charge density lambda and length k at a point that falls on a line that splits the wire into two equivalent parts

Answer 1

Given that,

Charge density = λ

Length  = l

According to figure,

Let us consider a small element of the charge distribution between  y and dy.

The charge in this cell is $dq=\lambda dy$

The distance from the cell to the field point P is $\sqrt{x^2+y^2}$.

We need to calculate the potential of non-conducting long straight wire

Using formula of potential

$V_{P}=k\int{\dfrac{dq}{r}}$

$V_{P}=k\int_{-\frac{l}{2}}^{\frac{l}{2}}{\dfrac{\lambda dy}{\sqrt{x^2+y^2}}}$

$V_{P}=k\lambda(ln(y+\sqrt{y^2+x^2}))_{-\frac{l}{2}}^{\frac{l}{2}}$

$V_{P}=k\lambda(ln((\dfrac{l}{2})+\sqrt{(\dfrac{l}{2})^2+x^2}))-ln((-\dfrac{l}{2})+\sqrt{(\dfrac{l}{2})^2+x^2})$

$V_{P}=k\lambda\ ln(\dfrac{l+\sqrt{l^2+4x^2}}{-l+\sqrt{l^2+4x^2}})$

Hence, The potential of non-conducting long straight wire is $k\lambda\ ln(\dfrac{l+\sqrt{l^2+4x^2}}{-l+\sqrt{l^2+4x^2}})$