Problem 61 - Master Resource Book in JEE Main Physics

Question

In AC series circuit, the resistance, inductive reactance and capacitive reactance are \(3 \Omega, 10 \Omega\) and \(14 \Omega\) respectively. The impedance of the circuit is [Orissa JEE 2011] (a) \(5 \Omega\) (b) \(4 \Omega\) (c) \(7 \Omega\) (d) \(10 \Omega\)

Step-by-Step Solution

Answer

The impedance of the circuit is \(5 \Omega\), option (a).

Step 1: Understand Impedance in an AC Circuit

In an AC circuit, the impedance (denoted as \( Z \)) is a combination of resistance (\( R \)) and reactance (either inductive or capacitive). The formula for calculating impedance in a series circuit is \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance.

Step 2: Substitute Given Values

Substitute the given values into the impedance formula. Here, \( R = 3 \Omega \), \( X_L = 10 \Omega \), and \( X_C = 14 \Omega \). The formula becomes: \[ Z = \sqrt{3^2 + (10 - 14)^2} \]

Step 3: Calculate the Reactive Difference

Calculate the difference between the inductive reactance and the capacitive reactance: \[ 10 - 14 = -4 \] Since the square of a negative value is positive, we proceed with \((-4)^2 = 16\).

Step 4: Calculate Impedance

Substituting back into the impedance formula, compute the value: \[ Z = \sqrt{3^2 + 16} = \sqrt{9 + 16} = \sqrt{25} \] Simplify the square root to find the impedance: \[ Z = 5 \Omega \].

Step 5: Select the Correct Option

Compare the computed impedance with the given options. The option that matches is (a) \(5 \Omega\).

Key Concepts

Impedance Calculation

In an alternating current (AC) circuit, impedance is a critical concept that describes the total opposition to the flow of current. Unlike direct current (DC) circuits, where resistance alone defines opposition, AC circuits involve both resistance and reactance (from inductors and capacitors). Impedance is expressed in ohms (\(\Omega\)) and represented by \(Z\). The general formula to calculate impedance in a series AC circuit is: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] Here, \(R\) is the resistance, \(X_L\) is the inductive reactance, and \(X_C\) is the capacitive reactance. The calculation involves squaring these components due to their vector nature, accounting for both magnitude and phase difference.

• Resistance \(R\) is straightforward.
• The impedance formula considers the net reactance \((X_L - X_C)\) to reflect how inductors and capacitors counteract each other.

Inductive Reactance

Inductive reactance is the opposition to the change in current by an inductor in an AC circuit. It depends on the frequency of the AC signal and the inductance of the coil. Symbolically, inductive reactance is represented by \(X_L\) and is calculated using the formula: \[ X_L = 2\pi f L \] where \(f\) is the frequency in hertz and \(L\) is the inductance in henrys. As the frequency or inductance increases, so does the inductive reactance.

• Inductive reactance increases with higher frequencies.
• It causes the current waveform to lag behind the voltage waveform by 90 degrees.

Understanding this concept helps in recognizing how inductors affect overall impedance in circuits.

Capacitive Reactance

Capacitive reactance is the measure of a capacitor's opposition to a change in voltage in an AC circuit. It is inversely related to both frequency and capacitance. Represented by \(X_C\), capacitive reactance can be calculated using the formula: \[ X_C = \frac{1}{2\pi f C} \] with \(f\) as the frequency in hertz and \(C\) as the capacitance in farads.

• Unlike inductive reactance, capacitive reactance decreases with increasing frequency.
• It causes the current waveform to lead the voltage waveform by 90 degrees.

This parameter is crucial for balancing and computing the impedance when capacitors are in a circuit, providing the opposite effect to inductors.

AC Series Circuit

An AC Series Circuit is one in which components like resistors, inductors, and capacitors are connected end-to-end, providing a single path for current flow. This configuration means the same current flows through each component, but the voltage across each can vary based on its impedance. The total impedance of a series circuit is computed by combining the resistance and net reactance, using the impedance formula.

• Resistors provide straightforward resistance, \(R\).
• Inductors contribute inductive reactance, \(X_L\).
• Capacitors contribute capacitive reactance, \(X_C\).

Analyzing an AC series circuit involves vectorially adding these components to determine the circuit's behavior under AC.

Resistance in AC Circuits

In AC circuits, resistance (symbolized by \(R\)) behaves the same as in DC circuits, opposing current flow and dissipating energy as heat. Resistance is a real component and doesn't change the phase of the current and voltage. It is measured in ohms (\(\Omega\)), and its value is independent of frequency or reactance from inductors and capacitors.

• Ohm's Law still holds: \(V = IR\).
• Power dissipation in resistors is calculated as \(P = I^2R\) in AC circuits.

Understanding resistance is important, as it combines with reactance to form the overall impedance in AC circuits, explaining how circuits behave and perform in real-life applications.