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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? |
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Answer» (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`. |
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