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=(\lambda)/(2\pi\epsilon_(0)y)\hat{y}`

The electric potential is

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

Here,
`r=√(x^2+y^2)`

We need to calculate the potential due to this line charge

Using equation (I)

`V=-\int_(d)^(r){(\lambda)/(2\pi\epsilon_(0)y)dy}`

On integratinting

`V=-(\lambda)/(2\pi\epsilon_(0))ln((r)/(d))`

`V=(\lambda)/(2\pi\epsilon_(0))ln((d)/(r))`

Put the value of r

`V=(\lambda)/(2\pi\epsilon_(0))ln((d)/(√(x^2+y^2)))`

`V=(\lambda)/(4\pi\epsilon_(0))ln((d^2)/(x^2+y^2))`

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