Electric potential due to long wire

[vs5813]

Homework Statement

Using Gauss' law, or otherwise, find an expression for electric potential at a distance r from an infinitely long straight line of charge of length h with linear charge density lambda per unit length.

Homework Equations

q=lambda*h

The Attempt at a Solution

The first part is easy, using gauss' law to find electric field..the result i get is

E=lambda/(2*pi*r*epsilon_0)

..which should be correct. But then I'm not sure how to find potential in this case. I tried using the equation:

E= - grad*V

so

V = integral[E*dl]

V= -lambda/(2*pi*epsilon_0) integral_from_infinity_to_r[(1/r) dr]

..which leads me to:

V = -lambda/(2*pi*epsilon_0) [ln(r)]

where the ln(r) is evaluated from infinity to r...and I am not sure that looks right, or where to go from there.. I don't know what the solution is supposed to be and I couldn't find any explanations when i googled it..anyone have any idea? thankyou sooo much!

 

[Dox]

Hello.

In fact that's the solution. That's life!

 

[vs5813]

heh thankyou!

 

[Cintdrix]

how do you evaluate ln at infinity and r??

 

[andurill13]

pick an arbitrary point a, and integrate with respect to that point. you're correct that you can't integrate to an infinite potential at the wire.

 

[damnedcat]

the only way i can think of is that u have to assume the boundary condition that V=0 at infinity. but then that seems forced because when evaluating the similar case for a sphere u get a 1/r term and when u plug in infinity there it goes to zero (see griffiths 3rd ed example 2.6). can anyone reconcile these cases? i don't see any explanation above addressing this.

 

[Tvdmeer]

Hi,

You cannot use infinity, because the problem states that there is an infinite length of wire. This tells us that the charge is infinite. Using a variable like "a" is your best bet for solving the problem.