(/30) Consider a sheet of width w and infinite length carrying a surface current
density K = Koy. We place an infinitely long wire carrying a current I in the
ŷ direction a distance h above the center of the sheet. The current in the wire
is in the same direction as the current in the sheet.
(a) (/5) Make a sketch of the problem, labelling the variables of interest.
Make sure to choose a right-handed coordinate system (so that you can
use the right-hand rule) that facilitates the use of symmetry.
(b) (/5) To calculate the force exerted on the wire by the magnetic field
produced by the sheet, it is easier to first find the force on the wire due to
a segment of the sheet of thickness say da at some distance z from say the
y-axis (the variables may change depending on your chosen coordinate
system). This segment of sheet carries a current Is = Kodr and can
be thought of as an infinitely long wire. What is the magnitude of the
magnetic field due to this segment of sheet at the position of the wire?
(c) (/10) Add to your sketch in a) a vector indicating the direction of the
force between a segment of sheet of thickness dr and the wire. Based
on symmetry, what is the magnitude and direction of the force per unit
length on the wire due to the magnetic field produced by two (symmetric)
segments of sheet?
Using this information, find the net force per unit length on the wire due
to the entire sheet. (Look up any integral you need).
(d) (/10) Using your result in c), what is the force per unit length on the wire
if the width of the sheet gets infinitely large (w → ∞)?
Check that this result agrees with the result you get for the force per
unit length on the wire if you use quasi-state Maxwell-Ampère law to
calculate the magnetic field of an infinite sheet of current (infinite in width
AND length, otherwise, you cannot use Ampère's law). Explain how
you can use symmetry to reduce the form of the magnetic field from
B(7) = B₂(x, y, z)î+By(x, y, z)ŷ + B₂(x, y, z)2 down to a form that you
can use with Ampère's law.
Fig: 1
