Question

A resistor \(30 \Omega\), inductor of reactance \(10 \Omega\) and the capacitor of reactance \(10 \Omega\) are connected in series to an ac voltage source \(\mathrm{e}=300 \sqrt{2} \sin (\omega \mathrm{t})\) The current in the circuit is (a) \(10 \sqrt{2 \mathrm{~A}}\) (b) \(10 \mathrm{~A}\) (c) \(30 \sqrt{11 \mathrm{~A}}\) (d) \((30 / \sqrt{11}) \mathrm{A}\)

Short answer

The current in the circuit is (a) \(10\sqrt{2}\)A.

Step-by-step solution

Find the impedance of the circuit

: The impedance (Z) of a series RLC circuit is given by the formula: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] with R, X_L, and X_C representing the resistance, inductive reactance, and capacitive reactance respectively. Use the given values to calculate the impedance.

Plug in the given values and calculate the impedance

: Now, substitute the given values into the impedance formula: \[ Z = \sqrt{(30\Omega)^2 + (10\Omega - 10\Omega)^2} \] \[ Z = \sqrt{(30\Omega)^2 + (0\Omega)^2} \] \[ Z = 30\Omega \] The impedance of the circuit is 30Ω.

Use Ohm's Law to find the current

: Ohm's Law for an AC circuit is given by the formula: \[ I = \frac{V}{Z} \] where I is the current, V is the voltage, and Z is the impedance. Use the given voltage value, and the calculated impedance value to find the current.

Plug in the given values and calculate the current

: Now, substitute the given voltage value and the calculated impedance value into Ohm's Law: \[ I = \frac{300\sqrt{2}V}{30\Omega} \] \[ I = 10\sqrt{2}A \] So, the current in the circuit is \(10\sqrt{2}A\). Therefore, the correct answer is (a) \(10\sqrt{2}\)A.

Key concepts

Impedance

In an AC circuit, impedance is a crucial concept as it represents the opposition to the flow of alternating current, much like resistance in a DC circuit. While resistance considers only resistors, impedance accounts for resistors, capacitors, and inductors. It is a comprehensive measure combining resistance (R), inductive reactance (X_L), and capacitive reactance (X_C). Impedance is denoted by Z and is measured in Ohms (Ω).
In a series RLC circuit like the one in our exercise, the impedance is calculated using the formula:

• \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

The impedance formula shows how inductive and capacitive reactances can offset each other, influencing the total opposition in the circuit. Solving the impedance can help you find how much the circuit resists the current flow.

Resistor

The resistor in an AC circuit serves as a component that restricts the flow of electric current, turning electrical energy into heat. It's measured in Ohms (Ω) and typically affects both AC and DC circuits equally. In our example, the resistor has a value of 30 Ω, which remains unchanged whether the current is alternating or direct.
Resistors have the simplest role in an AC circuit—unlike reactance forms—considering they resist both the immediate current and the potential voltage changes. They are crucial for controlling current and reducing the risk of circuit overloads. Understanding the properties and role of resistors helps in designing circuits that operate efficiently and safely.

Inductive Reactance

Inductive reactance (X_L) occurs in an AC circuit due to inductors, which oppose changes in current flow. This reactance causes the current to lag behind the voltage. It's calculated with the expression:

• \[ X_L = 2\pi f L \]

where \( f \) is the frequency and \( L \) is the inductance. Measured in Ohms, the inductive reactance increases with both frequency and inductance value.
In the given exercise, the inductive reactance is 10 Ω. This reactance is a vital consideration for predicting how the circuit will react to different AC frequencies, influencing the phase relationship between voltage and current.

Capacitive Reactance

Capacitive reactance, represented as \( X_C \), emerges in an AC circuit due to capacitors, which oppose changes in voltage. Unlike inductive reactance, capacitive reactance causes the current to lead the voltage, calculated with the formula:

• \[ X_C = \frac{1}{2\pi f C} \]

where \( f \) is the frequency and \( C \) is the capacitance. It is inversely proportional to frequency and capacitance, meaning it decreases as either increases.
In our specific problem, the capacitive reactance is also 10 Ω, perfectly balancing the inductive reactance. Balancing these elements is crucial in tuning the circuit, allowing for resonant frequency equalization.

Ohm's Law

Ohm's Law is a fundamental principle in both AC and DC circuits, relating voltage (V), current (I), and impedance (Z) in an AC context. For AC circuits, the law is expressed as:

• \[ I = \frac{V}{Z} \]

This formula helps find the current given the voltage and total impedance of the circuit. It's crucial for analyzing circuit behavior and ensuring components operate within their limits.
In the exercise, using Ohm’s Law allowed us to find the current of \( 10\sqrt{2} A \). Applying this law benefits the understanding of dynamic changes in voltage and current, no matter the complexity of the circuit components.