Question

A very small circular loop of radius `r` is initially coplanar and concentric with a much larger circular loop of radius `R(gt gtr)` . A constant current `i` is passed in the larger loop which is kept fixed in space and the small loop is rotated with angular velocity `omega` about a diameter. The resistance of the small loop is `R_(0)` and the inductance is negligible.
(a) Find the current in the small loop as a function of time.
(b) Calculate how much troque must be exerted on the small loop to rotate it.

Answer 1

Magnetic field due to large loop at centre
`B=(mu_(0)i)/(2R)ox`
Let at any time `t` , normal to small loop makes angle `theta(=omegat)` with magnetic field
(a) Flux throuth smaller loop
`phi=Bacosthe`
`=(mu_(0)i)/(2R).pir^(2)cosomegat`
`e=-(dphi)/(dt)=(mu_(0)ipir^(2)omegasinomegat)/(2R)`
Induced current
`I=(e)/(R_(0))=(mu_(0)ipir^(2)omegasinomegat)/(2RR_(0))`
(b) Torque on smaller loop
`tau=MBsintheta`
`=Ipir^(2)Bsinomegat`
`=(mu_(0)ipir^(2)omega.pir^(2))/(2R_(0)).(mu_(0)i)/(2R)sin^(2)omegat`
`=(mu_(0)^(2)i^(2)pi^(2)r^(2)omega)/(4R^(2)R_(0))sin^(2)omegat`