Gauss Law for finite line/plate

[henry3369]

Homework Statement

I just noticed that whenever I'm doing a problem involving Gauss Law, it always involves an infinite line/plate. I can't seem to figure out why it must be infinite large/long.

Here is an explanation I read, but don't quite understand:
http://physics.stackexchange.com/qu...nd-the-electric-field-of-a-finite-length-char

Homework Equations

∫E⋅A = Q_enc/ε

The Attempt at a Solution

I don't see why E becomes unpredictable and nonuniform on a finite line of charge. If the charge is still uniformly distributed, then the field should still be the same along the wire. Also, what causes the direction of E, to branch off into random directions rather than being radial like in an infinite line of charge? And, if the position of the Gaussian cylinder along the line of charge matters, why not just move it to the middle?

 

[henry3369]

The problem I was trying to solve with Gauss law is a finite line of charge Q distributed uniformly with length 2a, where one end is at +a, and one is at -a on the y-axis and I have to compute the electric field at point P, along the x-axis.

Why can't I just choose a Gaussian cylinder, centered at the origin, with length a, and compute it as if it was an infinite line of charge?

 

[henry3369]

Or even better, a Gaussian cylinder that encompasses the entire line of charge, so charge enclosed would be Q.

EA = Q/ε.
A = 4πr^2
E = Q/ε*4πr^2

Where r = x, the distance from the line of charge to the position P.

 

[haruspex]

In the OP you mentioned charge along a finite wire (a conductor). In a later post you wrote uniformly distributed along a finite wire. Which is it? The two are in conflict.

 

[henry3369]

In the OP you mentioned charge along a finite wire (a conductor). In a later post you wrote uniformly distributed along a finite wire. Which is it? The two are in conflict.

Don't they mean the same thing? Don't the electrons distribute themselves in a way to decrease interactions with each other along the wire?

 

[henry3369]

Here is the exact wording of the question:
Positive charge Q is distributed uniformly along the y-axis between y = -a and y = +a. Find the electric field at point P on the x-axis at a distance x from the origin.

 

[henry3369]

Either way, in a Gauss law problem, isn't the distribution irrelevant?

 

[SammyS]

Either way, in a Gauss law problem, isn't the distribution irrelevant?

Charge distribution is important anytime you want to determine the Electric field it produces.

 

[haruspex]

Don't they mean the same thing? Don't the electrons distribute themselves in a way to decrease interactions with each other along the wire?

Yes, the electrons move in a conductor so as to make the potential uniform, but that does not mean the charge will be uniform. In fact, the only finite case I can think of where the charge would be uniform is a thin spherical shell. As you should know, for any solid conductor all charge will be on the surface.
In the case of a finite wire, the charges (in their attempts to get away from each other) will tend to be comcentrated to wards the ends of the wire. As far as I know, the charge distribution in a finite wire is a hard problem. http://www.colorado.edu/physics/phys3320/phys3320_sp12/AJPPapers/AJP_E&MPapers_030612/Griffiths_ConductingNeedle.pdf
Anyway, it seems the problem at hand is for a uniform distribution, not a conductor. As SammyS says, the distribution is important for finding the field. The field from a single point charge 2q will be different from that from of two charges of q each a short distance apart.
Perform the integration.