Two circular coaxial coils of same radius and same number of turns having opposite current in the magnitude ratio 4 : 1 are placed at some separation apart. At how many points on the common axis magne

Question

Two circular coaxial coils of same radius and same number of turns having opposite current in the magnitude ratio 4 : 1 are placed at some separation apart. At how many points on the common axis magnetic field will be zero (exclude infinity)? Take separation as 2 2 ​ R where R is radius of coils.

TestHub · Accepted Answer

Field of a coil on its axis at distance x: $B = \frac{μ _{0} N I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}$

Currents are opposite (ratio 4 : 1), so the two fields point oppositely everywhere on the axis; zero field needs equal magnitudes: $\frac{4}{( R ^{2} + x ^{2} ) ^{3/2}} = \frac{1}{( R ^{2} + ( x - d ) ^{2} ) ^{3/2}}, d = 22 R$

Left of the first coil the nearer, stronger coil always wins — no zero there.

Solving between the coils gives one point (x ≈ 1.91 R from the stronger coil).

Beyond the weaker coil the fields can again balance, giving one more point (x ≈ 7.47 R).

Total points where B = 0: 2. Answer: 2

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