Potential for an infinite line charge

[blerb795]

Homework Statement

For the single line charge, derive an expression for Electric Potential.

Homework Equations

V(r)=-$\int$E$\bullet$dr
E for infinite line = $\frac{\lambda}{2\pi r\epsilon}$

The Attempt at a Solution

The integration is straightforward enough—my question is as to what the limits of the integration should be for a conceptual problem such as this. 0 to r makes the most sense to me but I am entirely stumped.
Not paying attention to the limits, I just got $\frac{\lambda ln(r)}{2\pi \epsilon}$ but I am again not sure of the sign.

 

[Simon Bridge]

$\vec{E}$ in the integrand is a vector. So is $d\vec{r}$.
Don't forget to include the coordinate system. Which direction is $d\vec{r}$?

Curious:
Why not start with the known solution for a point charge and use the superposition principle?

 

[frogjg2003]

You could do the indefinite integral. This will give you the potential plus a constant. Physically, the constant doesn't matter, so you can set it to any value you want, usually 0.

 

[blerb795]

I'm not sure if that was a rhetorical question to get me thinking, but dr⃗ is intended to be radially outward from the line charge.

I'm not sure how I would use the superposition principal here for a point charge...

frogjg, so in that case with the indefinite integral, my answer should still be negative?

 

[frogjg2003]

Think about it graphically. E=dV/dr is always positive, so V should be sloping up. Is +ln(r) or -ln(r) sloping up?

 

[Simon Bridge]

I'm not sure how I would use the superposition principal here for a point charge...

If you put the line on the z axis, a short length of the line, dz, at position z, has charge dq ... you know how to find the potential due to dq at some particular point a distance r away from the line. Adding up all the contributions (superposition principle) amounts to integrating along the line.

There are any number of worked examples online for this.

But you may be happier with the integral you've got. frogjg2003 can help you with that ;)

 

[TSny]

For an infinite line of charge there's a difficulty in integrating over the line if you use kdq/r as the potential of a charge element dq = λdz. The integral will not converge. That's because kdq/r assumes you're taking V = 0 at infinity. But, for an infinite line of charge, taking V = 0 at infinity will make V infinite at any finite distance from the line.

 

[Simon Bridge]

Yeah - one needs be more careful than that... though it is also why OP is having trouble picking limits to the integration. Excuse: I don't want to clutter the thread further with this side-chat.

I want to watch frogjg do this :)

 

[frogjg2003]

Now I feel stupid, I completely missed the minus sign when I was talking about the slope of the curve.
The correct way of thinking:
E=-dV/dr is positive, so V must have a negative slope. Since all the other constants are positive, that means there's only one source of negatives, the only place for there to be a negative is -ln(r).

As for what the constant at the end of the indefinite integral should be, it doesn't matter. The physics is in the E field, which doesn't see the constant. This usually means the constant is just assumed zero. Other times, the problem gives natural points to consider where to put the zero, and therefore what value to use for that constant.