What Is the Potential at an Infinite Uniformly Charged Line?

[ExtravagantDreams]

I know it seems a bit trivial, but what is the potential right at an infinite uniformly charged line?

Irregardless of reference point, the Potential will have a ln|s|, where s is the perpendicular distance to the line. Obviously this would result in infinity.

At the same time when I visualize a test charge on the line charge, the charge on either side of the test charge cancels and the result might be zero electric field on the line. This would result in a potential with respect to a reference point. But I suppose even this gives a potential of infinity, since there is the infinite potential difference between the reference point and essentially to the line.

 

[topsquark]

I know it seems a bit trivial, but what is the potential right at an infinite uniformly charged line?

Irregardless of reference point, the Potential will have a ln|s|, where s is the perpendicular distance to the line. Obviously this would result in infinity.

At the same time when I visualize a test charge on the line charge, the charge on either side of the test charge cancels and the result might be zero electric field on the line. This would result in a potential with respect to a reference point. But I suppose even this gives a potential of infinity, since there is the infinite potential difference between the reference point and essentially to the line.

The electric field ON a line charge will be zero, since a charge placed there will not move. You could say that the potential is zero there for that reason. However, we can't take a derivative of E there since the potential, as you say, blows up as it approaches that point from the exterior. So my conclusion would be that the potential is cannot be defined on the line charge.

-Dan

 

[ExtravagantDreams]

Ok, so then if I wanted to find the capacitance between a line charge and a cylindrical conducting shell around it, since I would have to define a potential difference, could I simply say the potential on the line is zero and the potential at the shell is the negative integral of the electric field due to the line charge at the shell, using a reference point where the potential is zero. And then use that for the capacitance. Or would the potential difference be infinite, and then the capacitance zero?

 

[topsquark]

Ok, so then if I wanted to find the capacitance between a line charge and a cylindrical conducting shell around it, since I would have to define a potential difference, could I simply say the potential on the line is zero and the potential at the shell is the negative integral of the electric field due to the line charge at the shell, using a reference point where the potential is zero. And then use that for the capacitance. Or would the potential difference be infinite, and then the capacitance zero?

Is this a homework problem? The capacitance does go to zero in the limit of a line charge at the center of a hollow cylindrical conductor, but normally we do this problem for a cylindrical charge with a nonzero radius at the center. Physically that's what we would have in that situation anyway. I would recommend assuming a small radius r for the line charge and do the problem that way. (And if your professor doesn't like it, just take the limit as r goes to zero of your answer!)

-Dan