Problem 22 - Fundamentals of Physics

Question

A series circuit containing inductance \(L_{1}\) and capacitance \(C_{1}\) oscillates at angular frequency \(\omega .\) A second series circuit, containing inductance \(L_{2}\) and capacitance \(C_{2},\) oscillates at the same angular frequency. In terms of \(\omega,\) 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; see Module \(25-3\) and Problem 47 in Chapter \(30 .\)

Step-by-Step Solution

Answer

The angular frequency of the combined circuit is the same as \( \omega \).

Step 1: Understanding Angular Frequency in an LC Circuit

The angular frequency \( \omega \) of an LC circuit is given by the formula: \[ \omega = \frac{1}{\sqrt{LC}} \]Here, \( L \) and \( C \) refer to the inductance and capacitance of the circuit, respectively.

Step 2: Express Given Conditions Using Formulas

For the first circuit with \( L_1 \) and \( C_1 \), the angular frequency is \[ \omega = \frac{1}{\sqrt{L_1 C_1}} \] For the second circuit with \( L_2 \) and \( C_2 \), the angular frequency is \[ \omega = \frac{1}{\sqrt{L_2 C_2}} \] Since both frequencies are equal, we have \[ \frac{1}{\sqrt{L_1 C_1}} = \frac{1}{\sqrt{L_2 C_2}} = \omega \]

Step 3: Calculate Equivalent Inductance and Capacitance

For inductors in series, the equivalent inductance \( L_{eq} \) is the sum of the individual inductances: \[ L_{eq} = L_1 + L_2 \]For capacitors in series, the equivalent capacitance \( C_{eq} \) is given by the formula: \[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \]Thus, \[ C_{eq} = \frac{C_1 C_2}{C_1 + C_2} \]

Step 4: Determine the Angular Frequency of the Combined Circuit

Use the formula for angular frequency for the combined circuit: \[ \omega' = \frac{1}{\sqrt{L_{eq} C_{eq}}} \]Substitute the expressions for \( L_{eq} \) and \( C_{eq} \): \[ \omega' = \frac{1}{\sqrt{(L_1 + L_2) \times \frac{C_1 C_2}{C_1 + C_2}}} \]

Step 5: Relate \( \omega' \) to \( \omega \)

Using the expressions from Steps 2 and 4, and knowing \( \omega = \frac{1}{\sqrt{L_1 C_1}} = \frac{1}{\sqrt{L_2 C_2}} \), you relate them:\[ \left( \frac{1}{\omega} \right)^2 = L_1C_1 = L_2C_2 \]Insert these into the expression for \( \omega' \):\[ \omega' = \frac{1}{\sqrt{(L_1 + L_2) \times \frac{C_1 C_2}{C_1 + C_2}}} = \frac{1}{\sqrt{\frac{L_1L_2(C_1 + C_2)}{C_1C_2}}} \]

Key Concepts

Angular Frequency

When dealing with LC circuits, understanding angular frequency is crucial. Angular frequency, often represented as \( \omega \), defines how quickly the circuit oscillates, and it is measured in radians per second. This concept is a central feature of oscillatory systems like LC circuits, where inductors (L) and capacitors (C) work together.
Angular frequency in an LC circuit is calculated using the formula \( \omega = \frac{1}{\sqrt{LC}} \). This important formula signifies the relationship between the frequency and the inductance and capacitance of the circuit.
- The higher the inductance (L) or the capacitance (C), the lower the angular frequency. - Conversely, a smaller L or C results in a higher angular frequency.
This inverse relationship means that the oscillation depends heavily on how these two components interact. For specific setups where existing circuits oscillate at the same angular frequency, it indicates a balanced setup of L and C, maintaining the same heartbeat-like rhythm.

Equivalent Inductance

In circuits where multiple inductors are present, calculating the equivalent inductance is key to finding out how the circuit will behave as a whole. Inductors work by storing energy in a magnetic field when electrical current passes through them.
For inductors in series, like in the given exercise, the equivalent inductance \( L_{eq} \) is simply the sum of the individual inductances:\[ L_{eq} = L_1 + L_2 \]This is because the magnetic fields add up, increasing the total inductance.
Key points about inductors in series:

• The overall inductance increases as more inductors are added linearly.
• Each inductor contributes to the total inductance, directly affecting the angular frequency of the circuit.

This simple additive property makes calculating the equivalent inductance straightforward, thus playing a significant role in determining the circuit's angular frequency.

Equivalent Capacitance

Capacitors in a circuit store potential electrical energy in an electric field. To understand how a circuit with multiple capacitors behaves, we calculate the equivalent capacitance.
When capacitors are placed in series, the equivalent capacitance \( C_{eq} \) is calculated using:\[\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}\]
The result is:\[ C_{eq} = \frac{C_1 C_2}{C_1 + C_2} \]
Important points to note:

• Unlike inductors in series, the equivalent capacitance is less than the smallest capacitance in the series.
• Adding more capacitors in series decreases the overall capacitance, affecting the angular frequency inversely.

Understanding and calculating equivalent capacitance help in predicting the behavior of circuits with more than one capacitor and its impact on the overall oscillation and energy storage capability.