What do you mean by 'impedance' in an alternating circuit? Write its unit. Find the reading of the ammeter and voltmeter in the given circuit.
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Impedance in an AC circuit is similar to resistance in a DC circuit but includes both resistive and reactive components (inductive and capacitive reactances).
Step 1: Definition of Impedance.
Impedance (\( Z \)) is the total opposition that a circuit presents to the flow of alternating current. It is the combination of resistance (\( R \)) and reactance (\( X \)) in the circuit. Impedance is a complex quantity and is given by:
\[
Z = \sqrt{R^2 + (X_L - X_^2}
\]
Where:
- \( R \) is the resistance,
- \( X_L \) is the inductive reactance,
- \( X_C \) is the capacitive reactance.
The unit of impedance is \( \Omega \) (Ohms), the same as resistance.
Step 2: Given Data.
From the given circuit:
- \( R = 30 \, \Omega \) (Resistanc,
- \( X_L = 60 \, \Omega \) (Inductive reactanc,
- \( X_C = 20 \, \Omega \) (Capacitive reactanc,
- \( V = 200 \sqrt{2} \sin(\omega t) \, \text{V} \) (Voltag.
Step 3: Calculation of Impedance.
First, calculate the net reactance \( X = X_L - X_C \):
\[
X = 60 - 20 = 40 \, \Omega
\]
Now, calculate the impedance using the formula:
\[
Z = \sqrt{R^2 + X^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, \Omega
\]
Step 4: Finding the Ammeter and Voltmeter Readings.
The amplitude of the voltage is \( V_{\text{max}} = 200 \sqrt{2} \, \text{V} \). The RMS value of the voltage is:
\[
V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}} = \frac{200 \sqrt{2}}{\sqrt{2}} = 200 \, \text{V}
\]
The current in the circuit can be calculated using Ohm’s law:
\[
I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{200}{50} = 4 \, \text{A}
\]
Final Answer:
The impedance of the circuit is \( Z = 50 \, \Omega \), the reading of the ammeter is \( \boxed{4 \, \text{A}} \), and the reading of the voltmeter is \( \boxed{200 \, \text{V}} \).
