Question

An \(L-R-C\) series circuit consists of a source with voltage amplitude 120 V and angular frequency 50.0 rad/s, a resistor with R = 400 \(\Omega\), an inductor with \(L\) = 3.00 H, and a capacitor with capacitance \(C\). (a) For what value of C will the current amplitude in the circuit be a maximum? (b) When \(C\) has the value calculated in part (a), what is the amplitude of the voltage across the inductor?

Short answer

(a) C = 1.33 x 10^-4 F (b) Voltage across inductor = 45 V

Step-by-step solution

Identify the Condition for Maximum Current

The current in an RLC series circuit will be maximum when the circuit is at resonance. As resonance occurs when the inductive and capacitive reactances are equal, the condition for resonance is \( \omega L = \frac{1}{\omega C} \).

Solve for Capacitance C at Resonance

To find the capacitance that meets the resonance condition, use the formula: \( \omega L = \frac{1}{\omega C} \). Rearranging for \( C \), we get \( C = \frac{1}{\omega^2 L} \). Substituting the given values \( \omega = 50 \) rad/s and \( L = 3 \) H, we calculate:\[C = \frac{1}{(50)^2 \times 3}\]\[C = \frac{1}{7500} = 1.33 \times 10^{-4} \text{ F}\].

Calculate Voltage Across the Inductor

At resonance, the voltage across the inductor \( V_L \) is maximum and it equals \( I \times \omega L \), where \( I \) is the current. The current amplitude \( I \) at resonance is calculated using \( I = \frac{V}{R} \), where \( V = 120 \) V and \( R = 400 \) \(\Omega\). Thus,\[I = \frac{120}{400} = 0.3 \text{ A}\].Using \( V_L = I \times \omega L \), we get:\[V_L = 0.3 \times 50 \times 3 = 45 \text{ V}\].

Key concepts

Series Circuit

In a series circuit, all the components are arranged in a single path for the current to flow through. This means that the same electrical current passes through each component sequentially. In the case of an RLC series circuit, it includes a resistor (R), an inductor (L), and a capacitor (C) connected in series.

One key characteristic of a series circuit is that the total resistance or impedance is the sum of the individual resistances or impedances of each component. This results in current that is uniform across the circuit, though the voltage drop can vary across each part.

• Resistor (R) – opposes current flow, causing a voltage drop proportional to the current.
• Inductor (L) – opposes changes in current and creates a phase shift.
• Capacitor (C) – stores and releases energy, also creating a phase shift.

This uniform current flow is integral in analyzing series circuits, as it allows for the application of resonance conditions to optimize the circuit's performance.

Inductive Reactance

Inductive reactance (X_L) is a measure of how much an inductor resists changes in current. It is directly proportional to the frequency of the current and the inductance of the coil. The formula for inductive reactance is given by:

\[ X_L = \omega L \]

Where \( \omega \) is the angular frequency of the source and \( L \) is the inductance of the inductor.

Inductive reactance increases with an increase in frequency, making it an important factor in AC circuits. It causes the current to lag behind the voltage by 90 degrees, which can significantly affect circuit behavior. Understanding how inductive reactance works is essential in designing circuits that effectively handle high-frequency signals.

Capacitive Reactance

Capacitive reactance (X_C) is the opposition that a capacitor offers to the change of voltage across its plates. It is inversely related to both the frequency of the alternating current and the capacitance of the capacitor. The formula for capacitive reactance is:

\[ X_C = \frac{1}{\omega C} \]

This formula shows that as the frequency (\( \omega \)) increases, or the capacitance (\( C \)) decreases, the capacitive reactance reduces. Capacitive reactance leads to a 90-degree phase lead of the current over the voltage, which is the opposite effect of inductive reactance.

By balancing capacitive and inductive reactance, resonance can be achieved, which plays a key role in maximizing current in an RLC circuit.

Resonance Condition

The resonance condition in an RLC circuit is crucial for optimizing performance. It occurs when the inductive reactance equals the capacitive reactance, causing the total impedance to reach its minimum value.

Resonance is achieved under the condition:
\[ \omega L = \frac{1}{\omega C} \]

At resonance, the inductive and capacitive effects cancel each other out, and the circuit behaves purely resistive. This alignment of reactances results in maximum current flow through the circuit for a given voltage. To calculate the optimal capacitance at resonance, rearrange the formula to:
\[ C = \frac{1}{\omega^2 L} \]

By substituting known values into this equation, you can determine the specific capacitance needed to achieve resonance in a given RLC circuit. This allows the current amplitude to be at its peak, enhancing the circuit's efficiency and performance.