Gauss' Law Application Question

[Pyroadept]

Homework Statement

A linear charge lambda = 10^-11 C/m is uniformly distributed along a thin nonconductive rod of length L = 0.5 m.
Use Gauss' Law to calculate the field at a distance of r = 0.1 m from the charged rod.

Homework Equations

∮E.da = Q/ε_0

The Attempt at a Solution

Hi everyone,

Firstly, I assumed the rod was a line charge (as opposed to a cylinder, as it's so thin, yes?).

Then I rewrote Gauss' Law as: E_x.2πx_0.dz, where x_0 = 0.1 (... I chose a cylinder perpendicular to the rod as my Gaussian surface)
and the right-hand side as: λdz/ε_0
And so: E_x = λ/2πx_0ε_0

But I don't think this can be correct, as I haven't taken the length of the rod into account.
Can anyone please point me in the right direction?

 

[LowlyPion]

I understand that the radial distance from the rod is .1m. I suppose that they want the field somewhere toward he middle so that you may presume that the field lines are ⊥ still to the surface? If there is no particular location along the rod they want, then I'd say you can ignore the length through symmetry and use the charge per unit length in your answer.

 

[atarr3]

It looks like you're doing it right. Length shouldn't matter in the problem. In your equation: E_x.2πx_0.dz = λdz/ε_0, your boundaries of integration should be 0 to the length of the Gaussian surface, L. You'll end up with E2πrL = λL/ε_0. Clearly, the L cancels out on both sides. Solving for E, E = λ/(2πrε_0)