You have two single turn circular loops of wire oriented such that they have a c

Question

You have two single turn circular loops of wire oriented such that they have a common center and the planes of the loops are perpendicular. As shown in the figure below, loop #1 lies in the in the xy-plane and has a counterclockwise current when viewed from the positive z-axis, while loop #2 lies in the xz-plane and has a counterclockwise current when viewed from the positive y-axis.

(a) If the loops each have a radius of 2.9 cm and they each carry a current of 2.1 A, determine the magnitude of the net magnetic field at the common center.
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(b) The direction of a vector in three dimensions is often given by the angle it makes with the positive x, y, and z-axes. What is the angle between the net magnetic field at the center and the positive x-axis?

Explanation / Answer

First consider loop 1

Magnetic field due to this loop at the centre = u0I / 2R in the +z direction.

Or in vector notation, B1 = u0I / 2R k

Now consider loop 2 :

Magnetic field due to this loop at the centre = u0I / 2R int the +y direction.

Or in vector notation, B1 = u0I / 2R j

So net magnetic field at the centre = u0I / 2R j + u0I / 2R k

= (4 * pi * 10-7 * 2.1) / (2 * 2.9 * 10-2) j +  (4 * pi * 10-7 * 2.1) / (2 * 2.9 * 10-2) k

= 4.549 * 10-5j + 4.549 * 10-5k

Magnitude = [ (4.549 * 10-5)2 + (4.549 * 10-5)2 ]1/2 = 6.434 * 10-5

The net magnetic field at the centre is a vector lying in the yz plane which has two equal components along the y-axis and z-axis.

Therefore,

Angle between the net magnetic field at the center and the positive x-axis = 90 degrees

Angle between the net magnetic field at the center and the positive y-axis = 45 degrees

Angle between the net magnetic field at the center and the positive z-axis = 45 degrees

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