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Show that in the free oscillations of an...

Show that in the free oscillations of an LC circuit, the sum of energies stored in the capacitor and the inductor is constant in time.

Text Solution

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Energy stored in the capacitor at time t,
`U_E=q^2/(2C)=(q_m^2)/(2C)cos^2omegat`

Energy stored in the inductor at time t,
`U_B=1/2LI^2=L/2[q_momegasinomegat]^2`
`=(q_m^2)/2Lomega^2sin^2omegat=(q_m^2)/(2LC)Lsin^2omegat=(q_m^2)/(2C)sin^2omegat`
Therefore, total energy
`U=U_E+U_B=(q_m^2)/(2C)(cos^2omegat+sin^2omegat)=(q_m^2)/(2C)`
Therefore, for free oscillations of an LC circuit, the net energy remains constant at all times.
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