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Two circular coils are concentric and lie in the same plane. The inner coil contains 120 turns of wire, has a radius of 0.012 m, and carries a current of 5.0 A. The outer coil contains 165 turns and has a radius of 0.017 m. What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, such that the net magnetic field at the common center of the two coils is zero

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Answer:

5.152 A

Opposite to the inner coil

Explanation:

[tex]\mu_0[/tex] = Vacuum permeability = [tex]4\pi \times 10^{-7}\ H/m[/tex]

[tex]I_o[/tex] = Outer coil current

[tex]I_i[/tex] = Inner coil current = 5 A

[tex]N_i[/tex] = Turns in inner coil = 120

[tex]N_o[/tex] = Turns in outer coil = 165

[tex]R_i[/tex] = Inner radius = 0.012 m

[tex]R_o[/tex] = Outer radius = 0.017 m

Putting the magnetic field as equal we have

[tex]\dfrac{\mu_0I_iN_i}{2R_i}=\dfrac{\mu_0I_oN_o}{2R_o}\\\Rightarrow I_o=I_i\dfrac{N_i}{N_o}\dfrac{R_o}{R_i}\\\Rightarrow I_o=5\times \dfrac{120}{165}\times \dfrac{0.017}{0.012}\\\Rightarrow I_o=5.152\ A[/tex]

The magnitude of current is 5.152 A,

The direction of the current is opposite to that of the inner coil as they have opposing magnetic fields