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Question

A coil of radius R carries a current I. Another concentric coil of radius r ( r < < R) carries current I2. Initially planes of the two coils are mutually perpendicular and both the coils are free to rotate with M and m respectively (m < M). During the subsequent motion, let K1andK2 be the maximum kinetic energies of the two coils respectively and let U be the magnitude of maximum potential energy of magnetic interaction of the system of the coils. Choose the correct options :

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Solution

The correct options are

**B** K1=Umr2mr2+MR2,K2=UMR2mr2+MR2

**C** U=μ0πI2.r24R

**D** K2>>K1

Magnetic field at the centre due to circular ring of radius R carrying current I

B=μ0I22R. . . . .(1)

The dipole moment of circle of radius r carrying current I2 is,

μ=πr2.I2=πr2I2. . . . .(2)

Initial potential energy,

Ui=−μBcosθ=μBcos900=0

Final potential energy in magnetic field,

Uf=−μB=μ0πr2I24R (by putting the value of B and μ )

Change in potential energy in magnetic field,

ΔU=|Uf−Ui|=μB

ΔU=μ0πr2I24R=U which convert into kinetic energy.. . . . .(3)

In rotational mechanics,

by conservation of angular momentum,

i1ω1=i2ω2, where here, i1 and i2 are moment of inertia of radius of circle R and r respectively

ω1=i2ω2i2

Conservation of energy,

12i1ω21+12i2ω22=ΔU=U. . . . .(4)

put the value of ω1 in the above equation, we get,

12i2ω22=i1ΔUi1+i2=K2(Kinetic energy of small ring). . . . . .(5)

moment of inertia of ring of radius R is

i1=12MR2

moment of inertia of ring of radius r is,

i2=12mr2

Put the value of i1,i2 and ΔU in equation (4), we get

K2=UMR2MR2+mr2

Put K2 in the equation (4), we get

K1=12i1ω1=Umr2MR2+mr2

Given, M>m and R>r

Kinetic energy of ring is depended on mass and radius,

So, K2>>K1

Thus, the correct answer is B, C and D.

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