Two concentric flat coils of radius r_1 = 2 cm and r_2 = 5 cm are lying in the s

Question

Two concentric flat coils of radius r_1 = 2 cm and r_2 = 5 cm are lying in the same plane. Each coil has N = 100 turns. The current l_1 in the smaller coil is (2 + 0.1N) A while the current I; in the larger coil is (5 + 0.1N) A. Find the magnitude of the magnetic field in the center of the coils if the currents in both coils flow in the same direction, and the currents in both coils flow in the opposite directions. At which radius r_1 of the smaller coil and at which directions of currents (same or opposite) the magnetic field in the center of the coils could be zero?

Explanation / Answer

Magnetic field at the center of a current carrying coil is given by B = oNI/2r

where N is the number of turns, I is the current and r is the radius of the coil.

and the direction will be given by applying the right-hand rule. However, for our problem it is sufficient if we understand that as long as the direction of current is same in both the coils, the magnetic field at the center due to both the coils will add up and the magnetic field at the center will be subtracted (or cancel each other) if they have opposite directions of current.

B1 = oNI1 /2r1  = 4 × 107 x 100 x 2.7 / (2x0.02) = 8.482300165 x 10-3 T

B2 = oNI2 /2r2  = 4 × 107 x 100 x 5.7 / (2x0.05) = 7.16283125 x 10-3 T

a.) If both coils have current in the same direction, then the magnetic fields will add up and hence net magnetic field will be B = B1 + B2 =  8.482300165 x 10-3 T + 7.16283125 x 10-3 T = 15.64513142 x 10-3 T

b.) If the coils have currents in the opposite direction, then the magnetic fields will subtract each other, so the net magnetic field will be B = B1 - B2 = 8.482300165 x 10-3 T - 7.16283125 x 10-3 T = 1.319468915 x 10-3 T

c.) Obviously, the current directions have to be opposite to cancel each other and give a resultant of zero at the centre. Also, to result in zero, they have to be equal in magnitude.

So, since we are asked about the radius of the first coil, we now have to find out at what radius of r1  will the magnitude of B1 equals that of B2

|B1 | = | B2 | = 7.16283125 x 10-3 T

|B1 | = 7.16283125 x 10-3 T

oNI1 /2r1  = 7.16283125 x 10-3 T

4 × 107 x 100 x 2.7 / (2x r1 ) =7.16283125 x 10-3 T

4 × 107 x 100 x 2.7 / (2x 7.16283125 x 10-3) = r1

0.02368421053 = r1

So, at the radius r1 = 2.37 cm and given the directions of the currents are opposite, the magnetic field at the centre will be zero .

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