Question

You want practically no magnetic field in a small region of space, not more than 10%...

You want practically no magnetic field in a small region of space, not more than
10% of Earth’s magnetic field at that location. The Earth’s magnetic field line is angled downward 72.7 degrees into the earth but practically due north.
The field strength is 55.66 microTesla
a. Take a single wire and make a loop with it with a radius of 80 cm. Put your small
apparatus right at the origin of this loop. This will work because it is just superposition
of magnetic fields from two different sources. How much current do you need to run
through it to cancel out the Earth’s magnetic field, and carefully draw how you would
orient the loop.
b. The field changes as you move away from the origin along the axis of the loop. How
far (in cm) can you go before the total field is 10% of Earth’s nominal field ? This tells
you the size of the region you created that meets your specification. For part b, your
solution should repeat the full derivation of the field on axis using the Biot Savart
method.

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Answer #1

Loop radius. r = 80 cm

magnetic field at the center of the loop.

B = oi/2r  

to have 0 magnetic field we must have a current i such that B = 55.66 micro T

i = 55.66E-6 *2*0.8 /1.26E-6 = 70.68 A

The magnetic field at the center of the loop is long the axis.

To compensate the earth magnetic field we must orient the loop it is due SOUTH and 72.7 deg out of earth.

The field in the direction of the axis, where the current looks CCW.

b) field at point away from the center is

B = oi/2r *1/( 1 + d2 /r2)3/2  

The field must not change more than 10% from that at center, so that the field due to earth is less than 10% of its value.

1/( 1 + d2 /r2)3/2 <0.9

d= +/- 0.27 r

= +/- 21.58 cm  

21.58 cm on either side of the center we all have 0 or less than 10% of earth magnetic field.

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