Ampere's Law and Infinite Current Sheets

[theBEAST]

Homework Statement

Here is the question:
http://dl.dropbox.com/u/64325990/HW%20Pictures/infinite%20current%20sheet.PNG

Homework Equations

∫B(dot)dl = μ_oI_encl

The Attempt at a Solution

Here is my attempt:
http://dl.dropbox.com/u/64325990/HW%20Pictures/Photo%202012-04-06%2010%2045%2008%20PM.jpg

According to the answer key my answer too large. The answer key has a coefficient of 1/2 in front of my answer... What did I do wrong? For my solution there is magnetic field in the sheet so the path is just L.

 

[tiny-tim]

hi theBEAST!

hint: what does "_enclosed" mean? ​

 

[theBEAST]

hi theBEAST!

hint: what does "_enclosed" mean? ​

OH so only half of the current is in that loop right?

Edit: Wait a second... What if my loop only enclosed 1/4 of the sheet so would my answer have a coefficient of 1/4 in front of it? Or would this not be legal as the amperian path is no longer symmetric?

 

[tiny-tim]

OH so only half of the current is in that loop right?

no!

the current has to be in the loop, so you need the loop to be both above and below the current

(you're probably thinking of electric cases where one side of a gaussian loop goes through a conductor )

 

[asteriatic]

Your attempt at a solution looks correct, but there may be a misunderstanding of the question. The question is asking for the magnetic field at a distance x from the sheet, not at the sheet itself. This means that the path for integration should be from 0 to x, not just L.

Also, the coefficient of 1/2 in the answer key may be due to the fact that the current sheet is infinite. In this case, the magnetic field would be the same on both sides of the sheet, so the integral from 0 to x would be equal to the integral from -x to 0. This would result in a factor of 1/2 in front of your answer.

I hope this helps!