Problem 6 - University Physics with Modern Physics

Question

A capacitance \(C\) and an inductance \(L\) are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If \(L=5.00 \mathrm{mH}\) and \(C=3.50 \mu \mathrm{F}\) , what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Step-by-Step Solution

Answer

Angular frequency is \(7.55 \times 10^3\) rad/s, and reactance is 37.75 \(\Omega\).

Step 1: Understand Reactance

The reactance of an inductor (inductive reactance) is given by \( X_L = \omega L \), where \( \omega \) is the angular frequency and \( L \) is the inductance. The reactance of a capacitor (capacitive reactance) is \( X_C = \frac{1}{\omega C} \), where \( C \) is the capacitance.

Step 2: Equate Inductive and Capacitive Reactance

To find the angular frequency where the inductive and capacitive reactances are equal, set \( X_L = X_C \). This gives the equation: \[ \omega L = \frac{1}{\omega C} \].

Step 3: Solve for Angular Frequency

Rearrange the equation from Step 2 to solve for \( \omega \): \[ \omega^2 = \frac{1}{LC} \]. Then find \( \omega \) by \( \omega = \sqrt{\frac{1}{LC}} \).

Step 4: Calculate Angular Frequency

Substitute \( L = 5.00 \text{ mH} = 5.00 \times 10^{-3} \text{ H} \) and \( C = 3.50 \mu \text{F} = 3.50 \times 10^{-6} \text{ F} \) into the formula: \( \omega = \sqrt{\frac{1}{(5.00 \times 10^{-3})(3.50 \times 10^{-6})}} \).

Step 5: Compute Angular Frequency Value

Perform the calculation: \[ \omega = \sqrt{\frac{1}{(5.00 \times 10^{-3})(3.50 \times 10^{-6})}} \approx \sqrt{\frac{1}{1.75 \times 10^{-8}}} = \sqrt{5.71 \times 10^{7}} \approx 7.55 \times 10^3 \text{ rad/s} \].

Step 6: Find Reactance

Use the found \( \omega \) to calculate the reactance: \( X_L = \omega L = (7.55 \times 10^3)(5.00 \times 10^{-3}) \approx 37.75 \underline{\phantom{xxx}} \Omega \) and \( X_C = \frac{1}{\omega C} = \frac{1}{(7.55 \times 10^3)(3.50 \times 10^{-6})} \approx 37.75 \underline{\phantom{xxx}} \Omega \).

Key Concepts

Inductive Reactance

Inductive reactance is an important concept in understanding how inductors behave in an AC circuit. An inductor, typically a coil of wire, stores energy in a magnetic field when electric current flows through it. The opposition to the change in current is measured by inductive reactance.
- The formula for inductive reactance is given by \( X_L = \omega L \), where \( \omega \) is the angular frequency in radians per second and \( L \) is the inductance in henries.
- Inductive reactance increases with the frequency of the AC signal. This means that as the frequency increases, the inductor will oppose the change in current more strongly.
- It acts like a sort of "resistance" that varies with frequency but does not dissipate power like a resistor, instead storing energy in the magnetic field.
Understanding inductive reactance is crucial when designing circuits, especially when aiming for a certain behavior at specific frequencies.

Capacitive Reactance

Capacitive reactance is the measure of a capacitor's opposition to the change in voltage in an AC circuit. A capacitor stores energy in an electric field between its plates, which impacts how it reacts to varying frequencies.
- The formula for capacitive reactance is defined as \( X_C = \frac{1}{\omega C} \), where \( \omega \) is the angular frequency and \( C \) is the capacitance in farads.
- Contrary to inductive reactance, capacitive reactance decreases as the frequency increases. Thus, at higher frequencies, a capacitor allows the AC signal to pass more easily.
- Just like inductive reactance, capacitive reactance does not consume power but temporarily stores energy, influencing how circuits behave at different frequencies.
Understanding capacitive reactance is essential when managing the flow and storage of energy within AC circuits, especially when tuning circuits to particular frequencies.

LC Circuit

An LC circuit, or inductor-capacitor circuit, is a fundamental electrical circuit that consists of an inductor (L) and a capacitor (C) connected together.
- Such circuits are used to create resonant circuits, where the inductive reactance and capacitive reactance can balance each other out at a specific frequency known as the resonant frequency.
- The resonant frequency \( \omega_0 \) of an LC circuit is given by the formula \( \omega_0 = \sqrt{\frac{1}{LC}} \), showing the interplay between the inductance and capacitance.
- When an LC circuit reaches its resonance, the energy oscillates between the magnetic field of the inductor and the electric field of the capacitor.
This oscillating energy can produce very high currents and voltages, making LC circuits extremely useful in applications like radio transmitters and receivers, where specific frequencies need to be targeted.