A series circuit containing inductance L₁ and capacitance C₁ oscillates at angular frequency $\mathit{\omega }$. A second series circuit, containing inductance L₂ and capacitance C₂, oscillates at the same angular frequency. In terms of $\mathit{\omega }$, what is the angular frequency of oscillation of a series circuit containing all four of these elements? Neglect resistance. (Hint:Use the formulas for equivalent capacitance and equivalent inductance.)

Two series LC circuits having components $L_1,C_1$and$L_2,C_2$ are oscillating at the same angular frequency is given.

Using the concept of series connection and relation for angular frequency in terms of frequency, we can find the relationship for angular frequency.

Formulae:

The angular frequency of a circuit connection, $\omega \text{'}=\frac{1}{\sqrt{L_{eq}{C}_{eq}}}$ (i)

The equivalent capacitance of a series connection, $C_{eq}=\sum _i^n\frac{1}{C_i}$ (ii)

The equivalent inductance of a series connection, $L_{eq}=\sum _i^n{L}_i$ (iii)

Angular frequency for series circuit 1 and circuit 2 can be given using equation (i) as:

$\begin{array}{c}\omega =\frac{1}{\sqrt{L_1{C}_1}}\\ =\frac{1}{\sqrt{L_2{C}_2}}......................\left(a\right)\end{array}$

When two circuits are connected in series, the new frequency will be given using equations (ii) and (iii) in equation (i) as follows:

$\begin{array}{c}\omega \text{'}=\frac{1}{\sqrt{(L_1+L_2)C_1{C}_2/(C_1+C_2)}}\\ =\frac{1}{\sqrt{(L_1{C}_1{C}_2+(L_2{C}_1{C}_2)/(C_1+C_2)}}\\ =\frac{1}{\sqrt{L_1{C}_1}}\frac{1}{\sqrt{(C_1+C_2)/(C_1+C_2)}}(\mathrm{from}\mathrm{equation}(\mathrm{a}\left)\right)\\ =\omega \end{array}$

Hence, the value of the angular frequency is $\omega$.