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Obtain an expression for angular frequency of LC oscillations ? |
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Answer» Solution :i. The mechanical energy of the spring-mass system is given by `E=(1)/(2)mv^(2)+(1)/(2)kx^(2)` ii. The energy E remains constant for varying values of x and v. Differentiating E with respect to time, we get `(dE)/(dt)=(1)/(2)m(2v(dv)/(dt))+(1)/(2)k(2X(dx)/(dt))=0` `orm(d^(2)x)/(dt^(2))+kx=0` since `(dx)/(dt)=vand(dv)/(dt)=(d^(2)x)/(dt^(2))""...(1)` iii. This is the differential equation of the OSCILLATION of the spring-mass system. The general solution of equation (1) is of the form `x(t)=X_(m)cos(omegat+phi)` iv. where `X_(m)` is the maximum value of x (t), `omega` the angular frequency and `phi` the phase constant. Similarly, the electronmagnetic energy of the LC system is given by `U=(1)/(2)Li^(2)+(1)/(2)((1)/(C))q^(2)=" constant "` Differentiating U with respect to time, we get `U=(1)/(2)(2i(di)/(dt))+(1)/(2C)(2i(dq)/(dt))=0` `orL(d^(2)q)/(dt^(2))+(1)/(C)q=0""...(2)` since `i=(dq)/(dt)and(di)/(dt)=(d^(2)q)/(dt^(2))` v. The general solution of eqution (2) is of the form `q(t)=Q_(m)cos(omegat+phi)""(3)` where `Q_(m) is the maximum value of q (t), `omega` the angular frequency and `phi` the phase constant. Current in the LC circuit The current flowing in the LC circuit is obtained by differentiating q (t) with respect to time. `i(t)=(dq)/(dt)=(d)/(dt)[Q_(m)cos(omegat+phi)]` `=-Q_(m)omegasin(omegat+phi)""" since "I_(m)=Q_(m)omega` or `i(t)=-I_(m)sin(omegat+phi)""...(4)` The equation (4) clearly shows that current VARIES as a function of time t. In fact, it is a sinusoidally varying alternating current with angular frequency `omega.` Angular frequency of LC oscillations By differentiating equation (3) twice, we get `(d^(2)q)/(dt)=-Q_(m)omega^(2)cos(omegat+phi)""...(5)` SUBSTITUTING equations (3) and (5) in equation (2), we obtain `L[-Q_(m)omega^(2)cos(omegat+phi)+(1)/(C)Q_(m)cos(omegat+phi)=0` Rearranging the terms, the angular frequency of LC oscillations is given by `omega=(1)/(sqrt(LC))` This equation is the same as that obtained from qualitative analogy. |
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