- #1
fab123
Homework Statement
I am trying to figur out whether there is or isn t an Electric Field along the surface of a cylindrical current carrying wire With radius r and length L, current I and resistance R. I was trying to see if someone already asked this and i found one discussion, however where a Clear answer wasn't given, so i thought i d asked the same question (and something concering the question that confuses me).
Homework Equations
The Attempt at a Solution
I first thought about using Gauss Law. Since the net charge of the current element would be zero, I d get that the flux through a surface of the same shape as the current element would be zero. Concluding that the E Field has to be zero along the surface would be false since zero flux doesn t imply e Field Equal to zero which got me thinking about how one would argue for that the e Field inside a conducting symmetric sphere With excess charge q, is zero by using gauss Law where one takes a concentric sphere With radius less than the actual sphere as a surface such that the enclosed charge would be zero. I remember that we did exactly this in class but thinking about using the same approach on a dipole would give a wrong result. For a dipole we have two point charges, q and -q apart a distance d. taking a closed surface enclosing both charges, we d get that the enclosed charge Equals zero, however the Electric Field inside the surface wouldn t have to be zero. Clearly there is something wrong in my argumentation regarding the use and interpretation of gauss Law.
Now, back to the actual problem:
I read about the following approach which argues for that there has to be a nonzero e Field along the surface, namely:
There is a current and a resistance, thus some sort of friction force that tries to slow Down the charged particles in motion. Assuming constant I, we have to have a force that opposed the friction force due to R, namely qE, where E is the e Field inside the conductor. Now, assuming the e Field i due to an electrostatic charge distribution, the e Field has to be purely conservative, meaning that the potential difference from a to b along the conductor has to be Equal to, say, V no matter what path we take (There is a nonzero potential difference since there is a resistance R). If we now take a path C1 to be a straight line from a to b an another path C2 that would start in a orthogonal to C1 until we Reach the surface of the conductor, then parallel to C1 and then, when we are "above" or "below" b, straight Down to b such that C2 would intersect C1 orthogonally. The first and the Third piece of C2 are orthogonal to the E Field inside the conductor, thus we get 0 from E dot dl, so the potential difference from a b along C1 has to Equal the potential difference from the parallel line along the surface. Thus the e Field along the surface has to be the same as the e Field inside the conductor.
Assuming we'd have a timevarying current, we d get an induced nonconversative e Field that would be orthogonal to the parallel line from a to be, thus we'd get the same answer, namely, that there has to be an e Field along the surface of the conductor.