Question

Consider an infinitely long wire with charge per unit length ????centered at (x, y) = (0, d) parallel to the z-axis.A) Find the potential due to this line charge referenced to the origin so that ϕ=0.

Answer 1

Given that,

Charge per unit length = λ

Point (x, y)=(0. d) parallel to the z axis

We know that,

The electric field due to the infinitely long wire is

$E=\dfrac{\lambda}{2\pi\epsilon_{0}y}\hat{y}$

The electric potential is

$V=-\int_{d}^{r}{\dfrac{\lambda}{2\pi\epsilon_{0}y}dy}$....(I)

Here, $r=\sqrt{x^2+y^2}$

We need to calculate the potential due to this line charge

Using equation (I)

$V=-\int_{d}^{r}{\dfrac{\lambda}{2\pi\epsilon_{0}y}dy}$

On integratinting

$V=-\dfrac{\lambda}{2\pi\epsilon_{0}}ln(\dfrac{r}{d})$

$V=\dfrac{\lambda}{2\pi\epsilon_{0}}ln(\dfrac{d}{r})$

Put the value of r

$V=\dfrac{\lambda}{2\pi\epsilon_{0}}ln(\dfrac{d}{\sqrt{x^2+y^2}})$

$V=\dfrac{\lambda}{4\pi\epsilon_{0}}ln(\dfrac{d^2}{x^2+y^2})$

Hence, The potential due to this line charge is $\dfrac{\lambda}{4\pi\epsilon_{0}}ln(\dfrac{d^2}{x^2+y^2})$