For an LCR circuit driven at frequency `omega`, the equation reads `L (di)/(dt) + Ri + (q)/(C ) = upsilon_(i) = upsilon_(m) sin omega t`
(i) Multiply the equation by I and simplify where possible. (ii) Interpret each term physically.
(iii) Cast the equation in the form of a conservation of energy statement.
(iv) Intergrate the equation over one cycle to find that the phase difference between `upsilon` and i must be acute.

Explanation

For an LCR circuit driven at frequency `omega`, the given eqn. is
`L (di)/(dt) + Ri + (q)/(C ) = upsilon_(1) = upsilon_(m) sin omega t`
Multiplying both sides by I, we get `Li (di)/(dt) + Ri^(2) + (q i)/(C ) = upsilon i`
Now, `L I (di)/(dt) = (d)/(dt) ((1)/(2) Li^(2)) =` rate of change of energy stored in inductor, `R_(i)^(2)` = rate of Joule heating loss
`(q)/(C ) i = (d)/(dt) ((q^(2))/(2 C))` = rate of change of energy stroe in capacitor
`upsilon xx i` = rate at which driving force pours in energy Hence, eqn. (ii) is in the form of convervation of energy statement.
Integrating both sides of (ii) w.r.t. t over one full cycle `(0 rarr T)`, we may write
`int_(0)^(T) (d)/(dt) ((1)/(2) L i^(2) + (q^(2))/(2C)) dt + int_(0)^(T) R i^(2) dt = int_(0)^(T) upsilon i dt `
zero + positive = `int_(0)^(T) upsilon` i dt
`:. int_(0)^(T) upsilon i dt gt 0`, which is possible only when phase diff. between `upsilon` and i is an acute angle.