Problem 42 - College Physics

Question

A 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? (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

Resonant frequency 5773.5 rad/s; max current 2.4 A; max voltage across capacitor and inductor 24,000 V; max energy stored in capacitor 28.8 J; in inductor 0.864 J.

Step 1: Identify Given Values

Let's first list all the given components in the circuit: - Resistor (R) = 100.0 Ω - Capacitor (C) = 0.100 μF = 0.100 × 10^(-6) F - Inductor (L) = 300.0 mH = 300.0 × 10^(-3) H - Voltage amplitude (V) = 240 V

Step 2: Calculate the Resonant Angular Frequency

The resonance angular frequency (ω₀) for an LC circuit is given by the formula:\[ω₀ = \frac{1}{\sqrt{LC}}\]Substitute the values for L and C:\[ω₀ = \frac{1}{\sqrt{300.0 \times 10^{-3} \times 0.100 \times 10^{-6}}}\]\[ω₀ = \frac{1}{\sqrt{3 \times 10^{-8}}} \]\[ω₀ = 5773.5 \text{ rad/s}\]

Step 3: Calculate Maximum Current at Resonance

At resonance, the impedance of the LC circuit equals the resistance R, so the maximum current (I₀) is given by Ohm's law:\[I₀ = \frac{V}{R}\]Substitute the known values:\[I₀ = \frac{240}{100} = 2.4 \text{ A}\]

Step 4: Maximum Voltage Across Capacitor at Resonance

At resonance, the voltage across the capacitor (V_C) can be expressed as:\[V_C = I₀ \times \frac{1}{ω₀C}\]Substitute the known values:\[V_C = 2.4 \times \frac{1}{5773.5 \times 0.100 \times 10^{-6}} = 24,000 \text{ V}\]

Step 5: Maximum Voltage Across Inductor at Resonance

At resonance, the voltage across the inductor (V_L) is equal to:\[V_L = I₀ \times ω₀L\]Substitute the known values:\[V_L = 2.4 \times 5773.5 \times 300.0 \times 10^{-3} = 24,000 \text{ V}\]

Step 6: Calculate Maximum Energy Stored in Capacitor at Resonance

The maximum energy (U_C) stored in the capacitor is given by:\[U_C = \frac{1}{2}C (V_C)^2\]Substitute the known values:\[U_C = \frac{1}{2} \times 0.100 \times 10^{-6} \times (24000)^2 = 28.8 \text{ J}\]

Step 7: Calculate Maximum Energy Stored in Inductor at Resonance

The maximum energy (U_L) stored in the inductor is given by:\[U_L = \frac{1}{2}L (I₀)^2\]Substitute the known values:\[U_L = \frac{1}{2} \times 300.0 \times 10^{-3} \times (2.4)^2 = 0.864 \text{ J}\]

Key Concepts

Resonant Angular Frequency

The concept of resonant angular frequency is crucial when dealing with RLC circuits arranged in series—where inductors, capacitors, and resistors are connected in a single loop. Resonant angular frequency, commonly represented as \( \omega_0 \), is the frequency at which the circuit naturally oscillates. This occurs when the inductive reactance equals the capacitive reactance, effectively cancelling each other out. As a result, the impedance of the circuit is minimized to just the resistance. The formula to calculate the resonant angular frequency is given by: \[ \omega_0 = \frac{1}{\sqrt{LC}}\]where \( L \) is the inductance measured in henrys, and \( C \) is the capacitance measured in farads. When you plug in the values for the given circuit with \( L = 300.0 \times 10^{-3} \) H and \( C = 0.100 \times 10^{-6} \) F, you will find that:- The resonant angular frequency \( \omega_0 \) is approximately 5773.5 rad/s.Understanding this frequency helps in tuning the circuit to its point of maximum efficiency, where it can oscillate without energy loss to reactance.

Maximum Current at Resonance

In an RLC circuit, achieving resonance means that the circuit can allow maximum current flow because the only opposition to the current is the resistance. The reason for this is that both the inductive and capacitive reactance are equal, and their effects cancel each other out. The maximum current at resonance can be derived using Ohm's law, expressed as:\[I_0 = \frac{V}{R}\]where \( I_0 \) is the maximum current, \( V \) is the voltage across the circuit, and \( R \) is the resistance. Using the values from our specific circuit, where \( V = 240 \) V and \( R = 100 \) Ω:- The maximum current \( I_0 \) is calculated to be 2.4 A.Understanding how the current behaves at resonance is essential when optimizing circuits for maximum power transfer and efficiency. Higher currents at resonance also imply that components must be rated to handle such current to avoid damage.

Energy Storage in Capacitor and Inductor

In resonant RLC circuits, both capacitors and inductors store energy. This energy storage plays an essential role in the oscillations that occur at resonance. The capacitor stores electrical energy in an electric field, while the inductor stores energy in a magnetic field.

• Energy in the Capacitor: The energy stored in the capacitor at resonance can be calculated using the formula:\[U_C = \frac{1}{2} C V_C^2\]where \( U_C \) is the stored energy, \( C \) is the capacitance, and \( V_C \) is the voltage across the capacitor. For this exercise:- With \( C = 0.100 \times 10^{-6} \) F and \( V_C = 24,000 \) V, the energy \( U_C \) turns out to be 28.8 Joules.
• Energy in the Inductor: The energy in the inductor is given by:\[U_L = \frac{1}{2} L I_0^2\]where \( U_L \) is the energy, \( L \) is the inductance, and \( I_0 \) is the maximum current. Here:- With \( L = 300.0 \times 10^{-3} \) H and \( I_0 = 2.4 \) A, the energy \( U_L \) is calculated to be 0.864 Joules.

At resonance, these energy exchanges happen without net loss, allowing the circuit to maintain oscillations, assuming no resistive losses. This swapping between magnetic and electric energy is a fundamental concept in AC circuits and affects how these circuits can be used effectively in various applications.