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There are two current-carrying planar coils made each

Question:

There are two current-carrying planar coils made each from identical wires of length L. C1 is circular and C2 is square. They are so constructed that they have the same frequency of oscillation when they are placed in the same uniform B and carry the same current. Find a in terms of R.

Solution:

C1 is the circular coil with radius R, length L, and no.of turns per unit length n1 = L/2πR

C2 is the square with side a, perimeter L, and no.of turns per unit length n2 = L/4a

Magnetic moment of C1 = m1 = n1iA1

Magnetic moment of C2 = m2 = n2iA2

m1 = LiR/2

m2 = Lia/4

Moment of inertia of C1 = I1 = MR2/2

Moment of inertia of C2 = Ma2/12

Solving the above, we get, a = 3R

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