Q22P A series circuit containing indu... [FREE SOLUTION]

Chapter 31: Q22P (page 936)

A series circuit containing inductance L₁ and capacitance C₁ oscillates at angular frequency $蝇$ . A second series circuit, containing inductance L₂ and capacitance C₂, oscillates at the same angular frequency. In terms of $蝇$ , 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.)

Short Answer

Expert verified

The angular frequency of oscillation of a series circuit is $蝇$ .

Step by step solution

The given data

Two series LC circuits having components $L_{1}, C_{1}$ and $L_{2}, C_{2}$ are oscillating at the same angular frequency is given.

Understanding the concept of angular frequency of LC circuit

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, $蝇' = \frac{1}{\sqrt{L_{e q} C_{e q}}}$ (i)

The equivalent capacitance of a series connection, $C_{e q} = {鈭�}_{i}^{n} \frac{1}{C_{i}}$ (ii)

The equivalent inductance of a series connection, $L_{e q} = {鈭�}_{i}^{n} L_{i}$ (iii)

Calculation of the angular frequency

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

$\begin{matrix} 蝇 = \frac{1}{\sqrt{L_{1} C_{1}}} \\ = \frac{1}{\sqrt{L_{2} C_{2}}} . . . . . . . . . . . . . . . . . . . . . . (a) \end{matrix}$

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

$\begin{matrix} 蝇' = \frac{1}{\sqrt{(L_{1} + L_{2}) C_{1} C_{2} / (C_{1} + C_{2})}} \\ = \frac{1}{\sqrt{(L_{1} C_{1} C_{2} + (L_{2} C_{1} C_{2}) / (C_{1} + C_{2})}} \\ = \frac{1}{\sqrt{L_{1} C_{1}}} \frac{1}{\sqrt{(C_{1} + C_{2}) / (C_{1} + C_{2})}} (from equation (a)) \\ = 蝇 \end{matrix}$