Problem 3 - College 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

The angular frequency is approximately 7560 rad/s; reactance is 37.8 Ohms.

Step 1: Understand Reactance and Angular Frequency

Reactance is the resistance of an electrical component to a change in current or voltage due to its inductance (for an inductor) or its capacitance (for a capacitor). The reactance of an inductor is given by \( X_L = \omega L \) and the reactance of a capacitor is \( X_C = \frac{1}{\omega C} \), where \( \omega \) is the angular frequency.

Step 2: Equate the Reactances

To find the angular frequency at which the inductive reactance \( X_L \) and capacitive reactance \( X_C \) are equal, set \( X_L = X_C \). This gives the equation \( \omega L = \frac{1}{\omega C} \).

Step 3: Solve for Angular Frequency \(\omega\)

From the equation \( \omega L = \frac{1}{\omega C} \), solve for \( \omega \). Multiply both sides by \( \omega \) to get \( \omega^2 = \frac{1}{LC} \), then take the square root of both sides: \( \omega = \frac{1}{\sqrt{LC}} \).

Step 4: Substitute Given Values to Find \(\omega\)

Given that \( L = 5.00 \times 10^{-3} \mathrm{H} \) and \( C = 3.50 \times 10^{-6} \mathrm{F} \), substitute these values into the formula: \( \omega = \frac{1}{\sqrt{5.00 \times 10^{-3} \cdot 3.50 \times 10^{-6}}} \).

Step 5: Calculate the Angular Frequency

Calculate the above expression to find \( \omega \). The calculation leads to \( \omega \approx 7560 \text{ radians per second} \).

Step 6: Calculate the Reactance

Once \( \omega \) is known, calculate the reactance using either \( X_L = \omega L \) or \( X_C = \frac{1}{\omega C} \). Substituting \( \omega \approx 7560 \text{ radians per second} \), we find the reactance \( X \approx 37.8 \text{ Ohms} \).

Key Concepts

Angular Frequency

Angular frequency is a fundamental concept in electrical circuits, particularly when dealing with alternating current (AC) systems. It represents how fast an AC signal oscillates; it is the rate at which the electrical phase advances. Often denoted by the Greek letter \( \omega \), angular frequency is expressed in radians per second.

To understand it better, think of it as the angular displacement per unit of time. If you picture the oscillation on a circular path, angular frequency indicates how quickly it completes a full circle. In the context of AC circuits, angular frequency relates closely to the reactance—reflecting how the components resist changes in current or voltage.

To find this frequency when capacitive and inductive reactances are equal, we use the equation \( \omega = \frac{1}{\sqrt{LC}} \). Here, \( L \) is inductance measured in henrys (H), and \( C \) is capacitance measured in farads (F). This equation shows the unique angular frequency at which both components balance each other out, a crucial concept in designing resonant circuits.

Inductive Reactance

Inductive reactance is the property that causes an inductor to oppose changes in current. Denoted by \( X_L \), it increases with a higher angular frequency and greater inductance. Think of it as the component's resistance in an AC circuit. The formula for inductive reactance is \( X_L = \omega L \).

In this formula, \( \omega \) represents the angular frequency, and \( L \) is the inductance. The unit of inductive reactance is ohms (\( \Omega \)), the same as regular resistance. Its dependence on frequency means that as the frequency increases, the reactance of the inductor also increases, thus limiting the current flow through it.

• For example, if the circuit's angular frequency is high, inductors will present a higher opposition to current, slowing down how fast the current changes.
• Conversely, at lower frequencies, the inductive reactance decreases, allowing more current to pass through.

The design of AC circuits often leverages this frequency dependence to achieve desired current flow characteristics.

Capacitive Reactance

Capacitive reactance, denoted by \( X_C \), opposes changes in voltage across a capacitor in an AC circuit. It works differently from inductive reactance as it decreases with increasing frequency. The formula for capacitive reactance is \( X_C = \frac{1}{\omega C} \).

In this equation, \( \omega \) is the angular frequency, and \( C \) is the capacitance. The unit of capacitive reactance is also ohms (\( \Omega \)). Unlike inductive reactance, capacitive reactance reduces with higher frequency, allowing more current to flow.

• At low frequencies, a capacitor presents a high reactance, blocking changes in current.
• At high frequencies, it offers much less reactance, letting more current pass through.

This inverse relationship with frequency is essential in tuning and designing circuits for specific frequencies, such as in radio and communication equipment, where precise control over frequency response is critical.

Electromagnetic Theory

Electromagnetic theory underpins our understanding of reactance in AC circuits. This theory combines electricity and magnetism into a set of physical laws that describe how electric and magnetic fields interact.

When alternating voltage is applied to a circuit with inductance and capacitance, it generates changing electric and magnetic fields, producing reactance. Inductive reactance arises from the magnetic fields induced by changing currents in inductors, while capacitive reactance stems from electric fields in capacitors responding to changing voltages.

• Maxwell's equations are the cornerstone, explaining how electric fields can transform into magnetic fields and vice versa.
• This transformation gives us insight into how alternating fields incite reactance, serving as the fundamental motion behind AC circuits.

Understanding these concepts is crucial for developing technologies that rely on AC circuits, including everything from simple household appliances to complex telecommunications systems.