Question

A 2.00 - \(\mu\) F capacitor was fully charged by being connected to a 12.0 - \(V\) battery. The fully charged capacitor is then connected in series with a resistor and an inductor: \(R=50.0 \Omega\) and \(L=0.200 \mathrm{H}\). Calculate the damped frequency of the resulting circuit.

Short answer

Answer: To find the damped frequency (ω_d) of the RLC circuit, first calculate the angular frequency (ω_0) using the formula $$\omega_0 = \frac{1}{\sqrt{LC}}$$, where L = 0.200 H and C = 2.00 × 10^{-6} F. Next, use the formula for damped frequency, $$\omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2}$$, and plug in the values for R = 50.0 Ω, L = 0.200 H, and the calculated value of ω_0. Finally, calculate the damped frequency (ω_d) in rad/s.

Step-by-step solution

Write down the formula for damped frequency

The formula for damped frequency (ω_d) in an RLC circuit is given by: $$ \omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2} $$ where - ω_0 is the angular frequency of the circuit; - R is the resistance of the resistor; - L is the inductance of the inductor. Now, we need to calculate the angular frequency (ω_0).

Calculate the angular frequency (ω_0)

The angular frequency (ω_0) of an RLC circuit can be found using the formula: $$ \omega_0 = \frac{1}{\sqrt{LC}} $$ where - L is the inductance of the inductor; - C is the capacitance of the capacitor. We are given the values for L and C: L = 0.200 H; C = 2.00 × 10^{-6} F. Plug in the values into the formula and calculate ω_0: $$ \omega_0 = \frac{1}{\sqrt{(0.200)(2.00 × 10^{-6})}}\thinspace \mathrm{rad/s} $$

Calculate the damped frequency (ω_d)

Now that we have the value of ω_0, we can now calculate the damped frequency (ω_d) using the formula from Step 1: $$ \omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2} $$ We are given the values for R and L: R = 50.0 Ω; L = 0.200 H. Plug in the values and the calculated value of ω_0 into the formula and find ω_d: $$ \omega_d = \sqrt{(\frac{1}{\sqrt{(0.200)(2.00 × 10^{-6})}})^2 - (\frac{50.0}{2(0.200)})^2}\thinspace \mathrm{rad/s} $$ Calculate the damped frequency using the values provided. The result will be the damped frequency of the RLC circuit.

Key concepts

RLC Circuit

An RLC Circuit is an electrical circuit consisting of three main components: a resistor (R), an inductor (L), and a capacitor (C), all connected either in series or parallel. This type of circuit is known for its oscillatory behavior, which means it can create alternating current or voltage waves. These oscillations occur because energy is stored in the magnetic field of the inductor and the electric field of the capacitor. The resistor, on the other hand, dissipates energy, leading to a phenomenon known as damping. Key Characteristics:

• The presence of a resistor causes energy loss, reducing the oscillations over time, and resulting in a “damped” circuit.
• Inductors and capacitors store energy, but in different forms, contributing to the dynamic properties of the circuit.
• Combined, these components can filter or process signals in various electronic devices.

RLC circuits are fundamental in many electrical applications, such as in tuning radios and televisions, where specific frequency signals need to be selected or processed.

Angular Frequency

Angular Frequency, often denoted as \(\omega\), is a measure of how quickly the circuit oscillates as it resonates. It is related to the circuit's natural frequency and is crucial in analyzing the behavior of RLC circuits. The natural (or undamped) angular frequency of an RLC circuit is defined by the formula:\[\omega_0 = \frac{1}{\sqrt{LC}}\]where:

• \(L\) is the inductance, measured in henries (H).
• \(C\) is the capacitance, measured in farads (F).

This formula shows that the angular frequency is inversely proportional to the square root of the product of inductance and capacitance. Therefore, changes in these values will directly affect how rapidly the circuit can oscillate. In real-world applications, adjusting the inductance or capacitance alters the frequency at which the circuit naturally operates, enabling it to suit specific tasks, such as filtering or tuning.

Inductance

Inductance, represented by the symbol \(L\), is the property of an electrical conductor that opposes changes in the current flowing through it, due to the magnetic field created by the current. The standard unit of inductance is the henry (H). When current changes, the magnetic field also changes, inducing a voltage opposing the current change (according to Lenz's Law), a phenomenon known as self-inductance.Characteristics of Inductors:

• They are made from coils of wire, sometimes with a core material like iron to enhance their properties.
• Larger values of inductance mean greater opposition to changes in current.
• They play a crucial role in tuning circuits where frequency stability is needed.

In an RLC circuit, the inductor works with the capacitor to store energy temporarily in the form of a magnetic field, leading to oscillatory behavior when the circuit is activated.

Capacitance

Capacitance, denoted \(C\), is the ability of a component or circuit to store and hold electric charge. Measured in farads (F), it describes how much charge the capacitor can store per volt across its plates. Most commonly, capacitors are used to hold charge temporarily, which can then be released as needed to power electronic components.Key Points about Capacitors:

• They consist of two conductive plates separated by an insulator (dielectric).
• The larger the capacitance, the more charge a capacitor can store.
• They are integral in smoothing out voltage fluctuations in power supplies and in timing circuits.

Within an RLC circuit, capacitors work alongside inductors to create oscillations. As they charge and discharge, they aid in defining the timing characteristics and energy exchange processes that occur during oscillations.

Resistance

Resistance, symbolized by \(R\), is a fundamental property of materials that quantifies how strongly they oppose the flow of electric current. Measured in ohms (\(\Omega\)), resistance results in the conversion of electrical energy into heat. In RLC circuits, the resistor predominantly determines the degree of damping in the oscillations.Importance and Effects:

• A higher resistance leads to more energy being dissipated as heat, thus increasing the damping effect and reducing the amplitude of oscillations over time.
• Commonly used to control current within circuits, ensuring safe operation of components.
• They are constructed from materials like carbon or metal, each providing varying resistance characteristics.

Resistance is a pivotal element in practical applications, influencing the stability and efficiency of electronic circuits by limiting the amount of current that can pass through at any given time.