Why Don't the Electric Fields Match at the Boundary in Gauss's Law Calculation?

[kingwinner]

I am very confused by this tough question. I hope some experts of Gauss's Law can help me out! Any help is greatly appreciated!

1) A long thin straight wire with linear charge ensity lambda runs down the centre of a thin hollow metal cylinder of radius R. The cylinder has a net linear charge density (2*lambda). Take lambda as positive. Find the electric field (strengh & direction)
a) inside the cylinder (r<R)
b) outside the cylinder (r>R)

For part b (r>R), I picked a coaxial cylinder with a radius r>R and of lengt L as the Gaussian surface

E=(Q_enclosed)/(A)(epsilon_o)
E=(lambda)L+2(lambda)L/(A)(epsilon_o)
E=3(lambda)L/(2pi*r*L)(epsilon_o)
E=3(lambda)/(2pi*r)(epsilon_o) [direction: radially outward]

Is this the correct answer Note that the radius of the hollow cylinder "R" is not used in any part of my calculation...did I do something wrong?

For part a, I got the electric field strength for r<R as [lambda/(2pi*epsilon_o*r)], and when I try to substitute r=R into the answers from part a & b, the electric fields DON'T match at the boundary...which further lowers my confidence of being right. But which part did I do it wrong? I can't find my error...Does anyone know how to solve this problem?

By the way, how come they use the term LINEAR charge density for a 3-dimensional hollow cylinder? Say, for example, if a certain hollow cylinder has a linear charge density of 2 C/m, what does it actually mean? A cylinder is definitely NOT a line...


Thank you again!

 

[kingwinner]

One more question, for part a, will the electric field also be pointing radially OUTWARD? There are many positive charges surrounding, so wouldn't the electric field be pointing IN?

 

[Dick]

There is no error. You did use R in the calculation. It tells you where one solution begins and another ends. There is a surface charge on the cylinder, so you shouldn't expect the electric field to be continuous across it. The phrase 'linear charge density' just means that they are giving charge per length rather than the charge per area. That's all.

 

[Dick]

One more question, for part a, will the electric field also be pointing radially OUTWARD? There are many positive charges surrounding, so wouldn't the electric field be pointing IN?

Positive charges? I thought charge density was measured in coulombs per something. And a coulomb was an amount of negative charge. So positive charge density actually mean negative charge. But I could be confused. One could always look it up, right?

 

[kingwinner]

according to my textbook and my notes, positive charge density is definitely used for positive charges