A capacitor with capacitance `C=1.0 muF` and a coil with active resitance `R=0.10 Omega` and inductance `L=1.0 m H` are connected in parallel to a source of sinusoidal voltage `V=31 V`. Find `:`
`(a)` the frequency `omega` at which the resonance sets in ,
`(b)` the effective value of the fed current in resonance , as well as the corresponding currents flowing through the coil and through the capacitor.

Explanation

When the coil and the condenser are in parallel, the equation is
`L(dI_(1))/(dt)+RI_(1)=(intI_(2)dt)/( C)=V_(m) cos omegat`
`I=I_(1)+I_(2)`
Using complex voltages
`I_(1)=(V_(m)e^(iomegat))/( R+iomegaL), I_(2)=iomegaCV_(m) e^(iomegat)`
and
`I=((1)/(R+iomegaL)+iomegaC)V_(m) e^(iomegat)=[(R-iomegaL+iomegaC(R^(2))+omega^(2)L^(2))/(R^(2)+omega^(2)L^(2))]V_(m)e^(iomegat)`
Thus, taking real parts `I=(V_(m))/( |cancel(Z)(omega)|)=cos ( omegat- varphi)`
where `(1)/(|cancel(Z)(omega)|)=([R^(2){ omegaC(R^(2)+omega^(2)L^(2))-omegaL}^(2)])/((R^(2) + omega^(2)L^(2))^(1//2))`
and `tan varphi=(omegaL-omegaC(R^(2)+omega^(2)L^(2)))/(R)`
`(a)` To get the frequency of resonance we must define what we mean by resouance. Once definition requires the extremum `(` maximum or minimum `)` of current amplitude. The other definition requires rapid change of phase with `varphi` passing through zero at resonance. For the series circuit.
`I_(m)=(V_(m))/({R^(2)+(omegaL-(1)/(omegaC))}^(1//2))` and `tanvarphi=(omegaL-(1)/(omegaC))/(R)`
both definitions give `omega^(2)=(1)/( LC)` at resonance. In the present case the two definitions do not agree `(` except when `R=0)`. The definition that has been adopted in the answer given in the book is the vanishing of phase. This requires.
`C(R^(2)=omega^(2)L^(2))=L`
or `omega^(2)=(1)/( LC)-( R^(2))/( L^(2))=omega_(res)^(2), omega_(res)31.6xx10^(3)rad//s`
Note that for `R, varphi` rapidly changes from `~~-(pi)/(2)` to `+(pi)/( 2)` as `omega` passes through `omega_(res)` from `lt omega_(res)` to `gt omega_(res)`.
`(b)` At resonance `I_(m)=(V_(m)R)/(L//C)=V_(M)(CR)/( L)`
`I=` effective value of total current `=V(CR)/(L) =3.1 mA.`
similarly ` I_(L)=(V)/( sqrt(L//C))=Vsqrt((C)/(L))=0.98A.`
`I_(C)=omegaCV=Vsqrt((C)/(L)-(R^(2)C^(2))/(L^(2)))~=0.98 A`
Note `:-` The vanishing of phase `(` its passing through zero `)` is considered a more basic definition of resonance .