Two long, straight wires cross each other at right angles, as shown in the figur

Question

Two long, straight wires cross each other at right angles, as shown in the figure below. (Assume I1 = 5.05 A and I2 = 3.16 A.)

(a) Find the direction and magnitude of the magnetic field at point P, which is in the same plane as the two wires.

magnitude = ______ T

direction = out of the page

(b) Find the magnetic field at a point 30.0 cm above the point of intersection (30.0 cm out of the page, toward you).

Explanation / Answer

  I'm going to assume that the currents are flowing in the positive x and positive y directions...
a) Use the right hand rule. The x wire's field at point P goes out of the page, and the y-wire field goes into the page.
Field due to x wire:
B = ui/2r = 2E-7i/r = 2E-7*5.05/0.4 = 2.53E-6
Field due to y wire:
B = 2E-7*3.16/0.3 = 2.1E-6
x wins. Net magnitude: 2.53E-6 - 2.1E-6 = 4.3 E-7 T
Since x wins, the field goes out of the page.

b) I'm going to call that point Z
Distance from x wire to point Z: (0.30²+0.4²) = 0.5 m
Plug into formula
B = 2E-7*5.05/0.5 = 2.02E-6 T
The direction is a little bit tricky. It will be pointing out of the page and towards you. I'm going to call the positive x axis east, positive y axis north, and out of the page up. So the angle would be

arctan(0.4/0.30) = 53.13 [S 53.13 UP]

Do the same thing with y.
(0.30²+0.3²) = 0.424 m
B = 2E-7*3.16/0.424 = 1.49E-6 T
angle: arctan(0.3/0.3) = 45. [E 45 DOWN]
So now combine 2.02E-6 [S53.13 U] and 1.49E-6 [E45D] using components.
In the vertical direction: 5.3E-7 up

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