Problem 29 - Physics

Question

A \(10.0-\Omega\) resistor, a \(12.0-\mu F\) capacitor, and a 17.0 -mH inductor are connected in series with a \(155-\mathrm{V}\) generator. (a) At what frequency is the current a maximum? (b) What is the maximum value of the rms current?

Step-by-Step Solution

Answer

(a) The maximum current occurs at 347.07 Hz. (b) The maximum rms current is 15.5 A.

Step 1: Identify Resonance Condition

The current in an RLC circuit is maximum at the resonant frequency, where the inductive reactance equals the capacitive reactance, making the circuit purely resistive. The resonant frequency formula for series RLC circuit is given by:\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] where \( L \) is the inductance and \( C \) is the capacitance.

Step 2: Calculate Resonant Frequency

Given inductance \( L = 17.0 \times 10^{-3} \, \mathrm{H} \) and capacitance \( C = 12.0 \times 10^{-6} \, \mathrm{F} \), substitute these values into the resonant frequency formula:\[ f_0 = \frac{1}{2\pi\sqrt{(17.0 \times 10^{-3})(12.0 \times 10^{-6})}} \]Calculating this gives:\[ f_0 \approx 347.07 \, \mathrm{Hz} \]

Step 3: Determine Maximum RMS Current

At resonance, the impedance of the circuit is purely resistive and equals the resistance \( R = 10.0 \, \Omega \). The maximum RMS current is given by Ohm's Law:\[ I_{\text{rms(max)}} = \frac{V_{\text{rms}}}{R} \] where \( V_{\text{rms}} = 155 \, \mathrm{V} \). Thus, \( I_{\text{rms(max)}} = \frac{155}{10.0} = 15.5 \, \mathrm{A} \).

Key Concepts

Resonant Frequency

In an RLC circuit, understanding the concept of resonant frequency is key. Resonant frequency occurs when the circuit reaches its most efficient state in terms of current. At this point, the impedance due to the inductor and the capacitor cancels each other out, leaving only the resistor to determine the current flow. This happens when the inductive reactance equals the capacitive reactance. The formula for the resonant frequency \( f_0 \) in a series RLC circuit is given by:\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]where:

• \( L \) is the inductance of the circuit
• \( C \) is the capacitance

In simple terms, it is the frequency at which the current is at its highest and energy transfer is most efficient.

Inductive Reactance

Inductive reactance is an important concept to grasp when studying AC circuits, such as RLC circuits. It refers to the opposition to the change of current in an inductor, a concept related to Faraday's Law of electromagnetic induction. The formula for inductive reactance \( X_L \) is:\[ X_L = 2\pi f L \]where:

• \( f \) is the frequency of the AC signal
• \( L \) is the inductance in Henrys

As the frequency of the circuit increases, the inductive reactance increases, meaning that the inductor increasingly opposes the current. This behavior is crucial for understanding how the inductor works in the RLC circuit and affects the overall impedance.

Capacitive Reactance

Capacitive reactance is analogous to inductive reactance but applies to capacitors within AC circuits like RLC circuits. It denotes the opposition a capacitor provides to the change in voltage across its plates. Capacitive reactance \( X_C \) can be calculated using the formula:\[ X_C = \frac{1}{2\pi f C} \]where:

• \( f \) is the frequency of the AC signal
• \( C \) is the capacitance in Farads

As frequency increases, capacitive reactance decreases, indicating that the capacitor offers less opposition to the current at higher frequencies. Understanding this helps one see how capacitors impact the flow of current in an RLC circuit.

Ohm's Law

Ohm's Law is a fundamental principle that applies to both DC and AC circuits. In the context of an RLC circuit, it helps determine the current across the resistor at the resonant frequency. At this frequency, the impedance is minimized to just the resistance, which allows for maximum current flow. Ohm's Law is expressed as:\[ I = \frac{V}{R} \]For an RLC circuit, especially at resonance:

• \( I \) is the current through the circuit
• \( V \) is the voltage across the circuit
• \( R \) is the resistance

Using this law in combination with the parameters of the circuit, like resistance and given voltage, allows for the calculation of maximum RMS current. Therefore, Ohm's Law becomes especially effective and simple at resonant frequency in an RLC circuit.