At the equator, the earth’s field is essentially horizontal; near the north pole

Question

At the equator, the earth’s field is essentially horizontal; near the north pole, it is nearly vertical. In between, the angle varies. As you move farther north, the dip angle, the angle of the earth’s field below horizontal, steadily increases. Green turtles seem to use this dip angle to determine their latitude. Suppose you are a researcher wanting to test this idea. You have gathered green turtle hatchlings from a beach where the magnetic field strength is 50 T and the dip angle is 56. You then put the turtles in a 1.3 m diameter circular tank and monitor the direction in which they swim as you vary the magnetic field in the tank. You change the field by passing a current through a 100-turn horizontal coil wrapped around the tank. This creates a field that adds to that of the earth.
What current should you pass through the coil, to produce a net field in the center of the tank that has a dip angle of 62 ? Express your answer to two significant figures and include the appropriate units. At the equator, the earth’s field is essentially horizontal; near the north pole, it is nearly vertical. In between, the angle varies. As you move farther north, the dip angle, the angle of the earth’s field below horizontal, steadily increases. Green turtles seem to use this dip angle to determine their latitude. Suppose you are a researcher wanting to test this idea. You have gathered green turtle hatchlings from a beach where the magnetic field strength is 50 T and the dip angle is 56. You then put the turtles in a 1.3 m diameter circular tank and monitor the direction in which they swim as you vary the magnetic field in the tank. You change the field by passing a current through a 100-turn horizontal coil wrapped around the tank. This creates a field that adds to that of the earth.
What current should you pass through the coil, to produce a net field in the center of the tank that has a dip angle of 62 ? Express your answer to two significant figures and include the appropriate units. What current should you pass through the coil, to produce a net field in the center of the tank that has a dip angle of 62 ? Express your answer to two significant figures and include the appropriate units. What current should you pass through the coil, to produce a net field in the center of the tank that has a dip angle of 62 ? Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

Magnetic field of earth, magnitude, Be = 50*10^-6 T [ average value]
now, initial dip angle, theta-i = 56 deg [ the angle Be makes with horizontal]
lets add a vertical magnetic field B to this
then if initial magnetic field could be written as Be(cos(theta-i)i + sin(theta-i)j)
then final magnetic field is Be(cos(theta-i)i + sin(theta-i)j) + Bj
so, new dip angle, theta-f = arctan([B + Besin(THETA-I)]/Becos(theta-i)) = 62 deg
1.8807 = [B + 5*10^-5sin(56)]/5*10^-5cos(56)
B = 1.1132*10^-5 T

now magnetic field due to a n turn loop of radius r = n*mu*I/2R [ mu is permeability of free space, ]
1.1132*10^-5 = 100*4*pi*10^(-7)*I/1.3
I = 0.115167 A

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