Problem 38 - College Physics

Question

A \(100.0 \Omega\) resistor, a \(0.100 \mu \mathrm{F}\) capacitor, and a \(300.0 \mathrm{mH}\) inductor are connected in series to a voltage source with amplitude \(240 \mathrm{~V}\). (a) What is the resonance angular frequency? (b) What is the maximum current in the resistor at resonance? (c) What is the maximum voltage across the capacitor at resonance? (d) What is the maximum voltage across the inductor at resonance? (e) What is the maximum energy stored in the capacitor at resonance? In the inductor?

Step-by-Step Solution

Answer

(a) 18257.56 rad/s; (b) 2.40 A; (c) 131.30 V; (d) 13145.45 V; (e) 0.862 mJ and 0.864 mJ.

Step 1: Find the Resonance Angular Frequency

The resonance angular frequency (\( \omega_0 \)) in an LC circuit is given by the formula:\[ \omega_0 = \frac{1}{\sqrt{LC}} \]We plug in the given values: \( L = 300.0 \mathrm{~mH} = 0.300 \mathrm{~H} \) and \( C = 0.100 \mu \mathrm{F} = 0.100 \times 10^{-6} \mathrm{~F} \):\[ \omega_0 = \frac{1}{\sqrt{0.300 \times 0.100 \times 10^{-6}}} \]\[ \omega_0 = \frac{1}{\sqrt{30 \times 10^{-9}}} \]\[ \omega_0 = 18257.56 \mathrm{~rad/s} \]

Step 2: Calculate the Maximum Current at Resonance

The maximum current at resonance occurs when the impedance is only due to the resistor and given by\[ I_{max} = \frac{V_{max}}{R} \]where \( V_{max} = 240 \mathrm{~V} \) and \( R = 100.0 \Omega \):\[ I_{max} = \frac{240}{100.0} = 2.40 \mathrm{~A} \]

Step 3: Calculate the Maximum Voltage Across the Capacitor

The maximum voltage across the capacitor at resonance, \( V_{C,max} \), is determined by\[ V_{C,max} = I_{max} \times X_C \]where \( X_C = \frac{1}{\omega_0 C} \):\[ X_C = \frac{1}{18257.56 \times 0.100 \times 10^{-6}} \]\[ X_C = 54.71 \Omega \]\[ V_{C,max} = 2.40 \times 54.71 = 131.30 \mathrm{~V} \]

Step 4: Calculate the Maximum Voltage Across the Inductor

The maximum voltage across the inductor at resonance, \( V_{L,max} \), is given by\[ V_{L,max} = I_{max} \times \omega_0 L \]where \( \omega_0 L = 18257.56 \times 0.300 \):\[ \omega_0 L = 5477.27 \Omega \]\[ V_{L,max} = 2.40 \times 5477.27 = 13145.45 \mathrm{~V} \]

Step 5: Calculate Maximum Energy Stored in the Capacitor

The maximum energy stored in the capacitor, \( U_{C,max} \), at resonance is given by\[ U_{C,max} = \frac{1}{2} C V_{C,max}^2 \]Substitute the known values:\[ U_{C,max} = \frac{1}{2} \times 0.100 \times 10^{-6} \times (131.30)^2 \]\[ U_{C,max} = 0.862 \times 10^{-3} = 0.862 \mathrm{~mJ} \]

Step 6: Calculate Maximum Energy Stored in the Inductor

The maximum energy stored in the inductor, \( U_{L,max} \), at resonance is\[ U_{L,max} = \frac{1}{2} L I_{max}^2 \]Substitute the known values:\[ U_{L,max} = \frac{1}{2} \times 0.300 \times (2.40)^2 \]\[ U_{L,max} = 0.864 \mathrm{~mJ} \]

Key Concepts

LC Circuit

An LC circuit is a type of electrical circuit that consists of an inductor (L) and a capacitor (C) connected together. These components work together to create a harmonic oscillator for electrical energy. This setup can store energy alternately in the electric field of the capacitor and the magnetic field of the inductor.

In the process, the energy oscillates back and forth between the capacitor and the inductor. When the energy is at its peak in the capacitor, it is zero in the inductor, and vice versa. Such circuits are also known as "tank circuits" due to their energy retention capability.

The total energy in the circuit is ideally conserved, meaning no energy is lost, leading to undamped oscillations. However, in practical scenarios, some energy is always lost as heat due to resistance or non-ideal components, reducing the oscillations over time.

Resonance Angular Frequency

The resonance angular frequency, typically denoted as \( \omega_0 \), is a crucial property of LC circuits. It is the frequency at which the circuit naturally oscillates without any external energy. At this frequency, the impedance of the circuit is at its minimum, leading to a maximum current potential.

The formula to find \( \omega_0 \) is:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]This equation implies that the resonance frequency is inversely proportional to the square root of the product of the inductance (L) and the capacitance (C).

To calculate it using the given values, we convert the units (e.g., microfarads and millihenries to farads and henries) and substitute them into the equation. This provides us with the precise angular frequency necessary for diagnosing optimal circuit performance or designing specific frequency applications.

Impedance in AC Circuits

In alternating current (AC) circuits, impedance is the equivalent of resistance in DC circuits but also includes effects of capacitance and inductance. It is denoted by the symbol \( Z \) and measured in Ohms (\( \Omega \)).

Impedance differs from resistance as it takes into account not only the resistor but also the frequency-dependent effects of capacitors and inductors. In LC circuits at resonance, the impedances due to the inductor and capacitor cancel each other out, minimizing the overall impedance to the real resistance of the circuit. This minimized impedance facilitates maximum current flow.

Knowing how to calculate and work with impedance is crucial in designing circuits that can handle specified voltages and currents effectively, ensuring proper functioning or achieving desired impedance matching in technology like audio or radio transmission.

Maximum Energy Storage in Capacitors and Inductors

The maximum energy that can be stored in capacitors and inductors at resonance is a vital concept in understanding how LC circuits operate.

In a capacitor, energy, denoted as \( U_{C,max} \), is stored in the form of an electrostatic field and can be calculated using:
\[ U_{C,max} = \frac{1}{2} C V_{C,max}^2 \]where \( V_{C,max} \) is the maximum voltage across the capacitor at resonance.

In an inductor, energy is stored in a magnetic field, and the maximum energy \( U_{L,max} \) is determined by:
\[ U_{L,max} = \frac{1}{2} L I_{max}^2 \]where \( I_{max} \) is the maximum current through the inductor.

Calculating maximum energy in these components is helpful for designing circuits with specific energy requirements, such as ensuring capacitors and inductors can handle the expected energy without being damaged.