Electric Potential for an infinity line of charge

[henry3369]

Homework Statement

Find the potential at a distance from a very long line of charge with linear charge density (charge per unit length) λ.

I actually have this solved with the help of my book, but I need an explanation of the results.

V = (λ/2πε)ln(r_b/r_a)
Where the electric potential at r_b is zero.

Homework Equations

ΔV = -∫E⋅ds

The Attempt at a Solution

V = (λ/2πε)ln(r_b/r_a)
From this equation, it is clear that if you can't set the electric potential at the line of charge to be zero, otherwise you would have ln(0). Additionally, if it is zero at r_b you get ln(infinity) = infinity. So does that mean it is not possible to find the electric potential with respect to the line of charge or infinity? I'm not sure how I would solve a problem like this on an exam if electric potential is relative, and r_b is some arbitrary finite point. Wouldn't different points for r_b yield different results for the electric potential?

 

[Orodruin]

Yes, picking different rb gives you different potentials. However, note that the potential is only physical up to the addition of an arbitrary constant (compare with the gravitational potential in a homogeneous gravitational field, you can chose where you put the zero level). As you can/should verify, different rb give potentials that only differ by a constant and therefore give rise to the same field.

Usually in 3D, we chose the reference point of zero potential to be at infinity but, as you have noticed, this is not possible for the line charge so you must instead pick a different zero level in order to have a finite potential for finite values of r.