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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 :

A
K1K2=Mm(Rr)2
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B
K1=Umr2mr2+MR2,K2=UMR2mr2+MR2
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C
U=μ0πI2.r24R
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D
K2>>K1
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Solution

## The correct options are B K1=Umr2mr2+MR2,K2=UMR2mr2+MR2 C U=μ0πI2.r24R D K2>>K1Magnetic field at the centre due to circular ring of radius R carrying current IB=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=0Final 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|=μBΔU=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 respectivelyω1=i2ω2i2Conservation 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 isi1=12MR2moment of inertia of ring of radius r is,i2=12mr2Put the value of i1,i2 and ΔU in equation (4), we getK2=UMR2MR2+mr2Put K2 in the equation (4), we getK1=12i1ω1=Umr2MR2+mr2Given, M>m and R>rKinetic energy of ring is depended on mass and radius,So, K2>>K1Thus, the correct answer is B, C and D.

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