A long currentcarrying wire oriented NorthSouth lies on a ta

A long current-carrying wire, oriented North-South, lies on a table (it is connected to batteries which are not shown). A compass lies on top of the wire, with the compass needle about 3 mm above the wire. With the current running, the compass deflects 13 degrees to the West. At this location, the horizontal component of the Earth\'s magnetic field is about 2e-5 tesla. What is the magnitude of the magnetic field at location A, on the table top, a distance 2.7 cm to the East of the wire, due only to the current in the wire?

Solution

You have Bearth and you know Bnet so make a right triangle to find Bwire.
Bwire = tan(13)*2e-5 = 4.617e-6

You first have to use the approximation of the magnetic field of a straight wire.
[ Bwire ~ (MUnot/4*Pi) (2*I)/r ] *NOTE* MUnot/4*Pi is a constant equal to 1e-7

Since you have Bwire, MUnot/4*Pi, and r (r=0.003) you can now solve for I
[ I = (Bwire*r)/(2*1e-7) ] so I = 0.06924

Now you have to go back to the approximation of the magnetic field of a straight wire.
[ Bwire ~ (MUnot/4*Pi) (2*I)/r ]
BUT now we have a different r value & you now have a I value
your r value is 0.027
Bwire ~ (1e-7) (2*0.06924)/(0.027) = 5.128e-7

The answer is 5.128e-7