 #1
wcjy
 73
 10
 Homework Statement:

Consider 2 identical infinite currentcarrying wires which runs along the X and Y axes
respectively. They carry identical currents I and the currents flow in the directions of the
axes. Find the net magnetic field (in vector form) at the coordinate (x, y, z) = (4,4,5).
 Relevant Equations:
 Biot savar law
$$B = \frac {\mu_0 I}{2 \pi r} $$
By Righthand Grip Rule, the direction of the magnetic field by wire in yaxis is into the paper (z)
while the direction of the magnetic field by wire in Xaxis is upwards (+i)
The answer state the Magnetic field is in the (i  y) direction though.
Next calculating the magnitude,
Distance of point to the Y axis is $$\sqrt{4^2 + 4^2} = \sqrt{32}$$
Distance of point to the X axis is $$\sqrt{4^2 + 5^2} = \sqrt{41}$$
Therefore, Magnetic field = $$B = \frac {\mu_0 I}{2 \pi \sqrt{32}} +\frac {\mu_0 I}{2 \pi \sqrt{41}} $$
This will give some weird fraction which is wrong.
Correct answer is $$B = \frac {5\mu_0 I}{82 \pi } (\hat{i}  \hat{j}) $$
By Righthand Grip Rule, the direction of the magnetic field by wire in yaxis is into the paper (z)
while the direction of the magnetic field by wire in Xaxis is upwards (+i)
The answer state the Magnetic field is in the (i  y) direction though.
Next calculating the magnitude,
Distance of point to the Y axis is $$\sqrt{4^2 + 4^2} = \sqrt{32}$$
Distance of point to the X axis is $$\sqrt{4^2 + 5^2} = \sqrt{41}$$
Therefore, Magnetic field = $$B = \frac {\mu_0 I}{2 \pi \sqrt{32}} +\frac {\mu_0 I}{2 \pi \sqrt{41}} $$
This will give some weird fraction which is wrong.
Correct answer is $$B = \frac {5\mu_0 I}{82 \pi } (\hat{i}  \hat{j}) $$