A series LCR circuit is connected to a source of alternating voltage of angular frequency $\omega .$ Which of the following statements is/are TRUE if $\text{ω=}\frac{\text{1}}{\sqrt{\text{LC}}}$ and FALSE if $\text{ω}\ne \frac{\text{1}}{\sqrt{\text{LC}}}$?

The circuit is in resonance when $\omega =\frac{1}{\sqrt{LC}}.$

Therefore, we are being asked which statements are true ONLY when the circuit is in resonance and not otherwise. 

(A) The charge on the capacitor (or the potential difference across it) is always behind the current in the circuit by a phase angle $\frac{\pi }{2}$. This is true for any LCR circuit, for any frequency.

(B) The rms potential difference across the capacitor is ${\left(V_C\right)}_{RMS}=I_{RMS}{X}_C=\frac{I_{RMS}}{\omega C}$

The rms induced emf across the inductor is ${\left(V_L\right)}_{RMS}=I_{RMS}{X}_L=I_{RMS}\omega C$

We can see that these quantities are equal only when $\omega =\frac{1}{\sqrt{LC}}$ and not otherwise

(C) Since heat is only dissipated in the resistance, the rate of dissipation is $H-I^{2}R.$ This is always true, not only at resonance

(D) Let the instantaneous current be $I\left(t\right)=I_0\sin \left(\omega t\right)$

Here, $I_0$ is the maximum value of the current and it is a constant

Then, we know that the instantaneous potential difference across the capacitor, $V_C\left(t\right)=\frac{I_0}{\omega C}\sin (\omega t-\frac{\pi }{2})=-\frac{I_0}{\omega C}\cos \left(\omega t\right)$

The instantaneous energy stored in the capacitor, $U_C\left(t\right)=\frac{1}{2}C{V}_C^2=\frac{1}{2}\frac{I_0^2}{{\omega }^{2}C}{\cos }^2\left(\omega t\right)$

The instantaneous energy stored in the inductor, $U_L\left(t\right)=\frac{1}{2}L{I}^2==\frac{1}{2}L{I}_0^2{\sin }^2\left(\omega t\right)=\frac{1}{2}L{I}_0^2(1-{\cos }^2\left(\omega t\right))$

Hence, the total energy stored in both $U\left(t\right)=U_C\left(t\right)+U_L\left(t\right)=\frac{1}{2}L{I}_0^2+\frac{1}{2}{I}_0^2(\frac{1}{{\omega }^{2}C}-L){\cos }^2\left(\omega t\right)$

We can see that $U\left(t\right)$ remains constant with time only if $\frac{1}{{\omega }^{2}C}-L=0,$ i.e. at resonance.