An `N` turn metallic ring of radius `a`, resistance `R`, and inductance `L` is held fixed with its axis along a spatially uniform magnetic field `vecB
An `N` turn metallic ring of radius `a`, resistance `R`, and inductance `L` is held fixed with its axis along a spatially uniform magnetic field `vecB` whose magnitude is given by `B_(0) sin (omegat)`.
(a)Set up the `emf` equation for the current `i` in the ring.
(b)Assuming that in the steady state `i`.oscillates with the same frequency `omega` as the magnetic field, obtain the expression for `i`.
( c)Obtain the force per unit length.Further obtain its oscillatory part and the time-averged compressional part.
(d)Calculate the time-averaged compressional force per unit length given that `B_(0)=1.00` tesla , `N=10`,`a=10.0 cm`, `omega=1000.0 rad s^(-1)`,`R=10.0 Omega`,`L=100.0 mH`.
(e)Answer the following two questions without providing rigourous justification :
(i)For `omega//2pi=60 Hz`,the ring emits a humming sound.What is the frequency of this sound?
(ii)A capacitor is included in the circuit.How does this affect the force on the ring?
1 Answers
Correct Answer - A
(a)`iR+L(di)/(dt)=-Npia^(2)B_(0)omega cos omegat`
(b)`i=(Npi a^(2) B_(0)omega (R cos omegat +omegaL sin omegat))/(R^(2)+omega^(2)L^(2))`
(c )`(dF)/(dl)=-(NB_(0)^(2))/(2(R^(2)+omega^(2)L^(2)))(Rsin omegat cos omegat+omega L sin^(2)omegat)`
`(dF)/(dl)|_(av) =-(NB_(0)^(2) pia^(2)omega^(2)L)/(2(R^(2)+omega^(2)L^(2)))=(dF)/(dl)|_(osc)=-(NB_(0)^(2) pia^(2)omega)/(2(R^(2)+omega^(2)L^(2)))(Rsin 2omegat cos omegat-omega L cos^(2)omegat)`
(d)`(dF)/(dl)|_(av) =1.55 N m^(-1)`
(e)`i`.The frequency of the sounds is `120 Hz`.