Problem 43 - College Physics

Question

A series RLC circuit has a resistance of \(25 \Omega\), a capacitance of \(0.80 \mu \mathrm{F},\) and an inductance of \(250 \mathrm{mH}\). The circuit is connected to a variable-frequency source with a fixed rms voltage output of \(12 \mathrm{~V}\). If the frequency that is supplied is set at the circuit's resonance frequency, what is the rms voltage across each of the circuit elements?

Step-by-Step Solution

Answer

The rms voltage across the resistor is 12 V, and each reactive component (inductor and capacitor) has maximum but equal and opposite voltages.

Step 1: Identify Resonance Frequency

The resonance frequency of an RLC series circuit is given by the formula \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]where \( L = 250 \text{ mH} \) and \( C = 0.80 \mu F \). We need to convert units to henries and farads, so \( L = 0.250 \text{ H} \) and \( C = 0.80 \times 10^{-6} \text{ F} \).

Step 2: Calculate Resonance Frequency

Substitute the values of \( L \) and \( C \) into the resonance frequency formula:\[f_0 = \frac{1}{2\pi\sqrt{0.250 \times 0.80 \times 10^{-6}}}\]Solving this gives the resonance frequency \( f_0 \).

Step 3: Resonance Condition in Circuit

At resonance frequency, the inductive reactance \( X_L \) and capacitive reactance \( X_C \) cancel each other out (i.e., \( X_L = X_C \)). Therefore, the total impedance in the circuit is equal to the resistance \( R = 25 \Omega \).

Step 4: Apply Ohm’s Law at Resonance

The rms current \( I \) in the circuit at resonance can be calculated by Ohm’s Law: \[ I = \frac{V}{R} = \frac{12 \text{ V}}{25 \Omega} = 0.48 \text{ A} \]

Step 5: Calculate Voltage Across Each Element

Since \( X_L = X_C \) and they cancel out, the entire applied rms voltage of \( 12 \text{ V} \) appears across the resistor at resonance. The voltage across the inductor \( V_L \) and capacitor \( V_C \) can still be calculated using the reactance formulas:\[V_L = I \times X_L\]\[V_C = I \times X_C\]However, at resonance, since \( X_L = X_C \) and they are zero, voltage across individual reactive components is at maximum but equal to the voltage drop that adds to zero net across the series connection in terms of vector sum.

Key Concepts

Resonance Frequency

In a series RLC circuit, the resonance frequency is a key concept because it represents the frequency at which the inductive reactance equals the capacitive reactance. At this specific frequency, the circuit offers minimal impedance to the alternating current (AC), resulting in the maximum possible current flow. The resonance frequency is determined using the formula: \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] where \( L \) is the inductance in henries and \( C \) is the capacitance in farads. It's important to convert the given values into these units for accurate calculation. The occurrence of resonance frequency signifies a balance in the circuit, leading to many fascinating real-world applications like in tuning radios and televisions to specific frequencies.

Ohm's Law

Ohm's Law is fundamental in electronics and is defined by the formula: \[ V = IR \] where \( V \) is the voltage across the circuit, \( I \) is the current through the circuit, and \( R \) is the resistance in the circuit. At the resonance frequency of a series RLC circuit, the impedance is minimal and purely resistive, meaning the circuit behaves like a simple resistor. This allows us to easily calculate the rms current during resonance by using the applied voltage and total resistance:

• Given: \( V = 12 \text{ V rms} \) (root mean square voltage)
• \( R = 25 \Omega \)
• Calculate \( I \): \( I = \frac{V}{R} = \frac{12}{25} = 0.48 \text{ A} \)

This current is crucial because it circulates through each element in the series circuit, and its consistency aids in the calculations of voltages across individual components.

Inductive Reactance

Inductive reactance is the opposition to the change of current by an inductor in an AC circuit. It is expressed in ohms and increases with frequency, calculated by the formula: \[ X_L = 2\pi f L \] where \( f \) is the frequency in hertz and \( L \) is the inductance. At resonance frequency in an RLC circuit, even though the inductive reactance reaches its point where it would normally hinder current flow, it exactly cancels out the capacitive reactance. This means that at resonance, the total reactance is zero, but individually, the voltage across the inductor can still be calculated as \[ V_L = I \times X_L \] However, the net effect at resonance is that the applied voltage accounts for the ohmic voltage drop, highlighting the beauty of phase relation between reactance terms.

Capacitive Reactance

Capacitive reactance is the resistance that a capacitor offers to alternating current, calculated by: \[ X_C = \frac{1}{2\pi f C} \] where \( f \) is the frequency, and \( C \) is the capacitance. In an RLC circuit, as the frequency increases, capacitive reactance decreases, opposing the changes in voltage. At resonance, capacitive reactance precisely counterbalances inductive reactance. This means that their effects negate each other, allowing the circuit to have minimal impedance---only the resistance remains active. It's a fascinating interplay where although the voltage across the capacitor is maximum---since it cancels with the inductor---the net reactive voltage drop is zero, focusing the circuit dynamics on resistive behavior primarily at resonance.