## PHYS208 Fundamentals of Physics II

### Answer for Ch. 30, 51P

a) Suppose the field is not parallel to the sheet, as shown:

Reverse the direction of the current. By the Biot-Savart law, **B** should reverse direction.

Now rotate the sheet by 180 degrees about a line perpendicular to the sheet.
**B** should now point down rather than up, for the same current configuratino as first shown --
in contradication to our initial assumption.

The only direction of **B** which is preserved under these transformations is one directed horizontally.

b) Evaluate the line integral of Ampere's law around the rectangular loop shown.
By the symmetry of the current sheet, **B** will have the same magnitude at the same
distance above and below the sheet; the upper and lower sides of the amperian loop are
constructed to be the same distance from the sheet, each having length *a*.

If the line integral is evaluated counterclockwise, the contribution of each of these sides
is *Ba*. There will be no contribution from the other two sides, since **B** is
perpendicular at all points. So the line integral around the amperian loop is 2*Ba*.

The enclosed current is the linear current density *lambda* times the length *a*
of the side of the loop; *i*_{enc} = *lambda a*.

Thus by Ampere's law, *B* = *mu*_{0} lambda/2,
directed as shown.

"http://www.physics.udel.edu/~watson/phys208/exercises/answers/30-51P.html"

Last updated Oct. 30, 1997.

Copyright George Watson, Univ. of Delaware, 1997.