Can we use gauss' law to find the e field of a finite line of charge?

[scrappy]

 

[stevenb]

can we use gauss' law to find the e field of a finite line of charge?

No, you have to use Coulomb's law.

 

[scrappy]

really? there isn't any way in the world of using gauss' law to solve for the e field of a finite line of charge? it was given as an assignment and i doubt that our teacher gave us this to make fun of us.

thank you for the response sir.

 

[stevenb]

really? there isn't any way in the world of using gauss' law to solve for the e field of a finite line of charge? it was given as an assignment and i doubt that our teacher gave us this to make fun of us.

thank you for the response sir.

Well, I'll wait and see if anyone posts a way to do it. If you find a way on your own, please post it. I'd like to learn if there is a way.

There is one sense in which you can do it. Since Gauss's Law basically contains Coulomb's Law in it, you can argue that the use of Coulomb's law is essentially the same as using Gauss' Law. However, I don't see any way to use Guass's law directly on a finite length of charge because there is not enough symmetry to get an exact solution this way. You can do it for an infinite length of charge.

 

[scrappy]

yeah, me too. i can't find any way. its because our last lesson was about finding one for the infinite line of charge. and he gave us this as a follow up homework. well i wish someone knows a way though.

anyway, thank you so much for your help sir. i wish someone could tell us how to.

 

[Apphysicist]

You cannot (effectively) because a finite length line of charge does not have the symmetry required in order to use it effectively. You will have "edge-effects" of the field around the endpoints, that is, the E-field will not be nice and perpendicular to the line of charge and will kind of bend in toward the endpoint. Your Gaussian surface needs to be either parallel or orthogonal to the E-field at all points (for a Gaussian sphere around a point charge, it's perfectly orthogonal...for your Gaussian cylinder around an infinite line of charge, the bases of the cylinder are parallel while the rectangular surface area is orthogonal)...the edge-effects complicate that significantly.

Gauss's law is a law, so it will always "work." For a finite line of charge, it is just not useful.

 

[scrappy]

so it means no one uses gauss law to solve for it? omg, what am i going to do with my homework? anyway sir, thank you so much for the insight.
are there any solutions online using gauss law to solve this? if anybody knows, please inform me. your help will be very much appreciated. ^_^

 

[stevenb]

so it means no one uses gauss law to solve for it? omg, what am i going to do with my homework? anyway sir, thank you so much for the insight.
are there any solutions online using gauss law to solve this? if anybody knows, please inform me. your help will be very much appreciated. ^_^

The following reference contains the derivation of several charge distributions including the finite line of charge. Here they use Coulomb's Law, which is the way most people approach solving this problem.

http://iweb.tntech.edu/murdock/books/v4chap2.pdf

Note that it is trivial to derive Coulomb's Law directly from Gauss' Law by applying it to a point charge using spherical symmetry. It is also possible to derive Gauss' Law from Coulomb's Law, although this is not quite as easy to do. For example, see Jackson's well known EM book for a very elegant derivation; or, Schwartz provides a step by step mathematical derivation.

So, it seems to me that applying Coulomb's Law is an acceptable way to solve the given problem. First derive Coulomb's Law from Gauss' Law, and then apply Coulomb's Law to the given problem using integration and the principle of superposition.