In a series L, R, C, circuit which is connected to a.c. source. When resonance is obtained then net impedance Z will be

Options

A. Z=R
B. `Z=omegaL-(1)/(omegaC)`
C. `Z=omegaL`
D. `Z=(1)/(omegaC)`

Answer

To solve the question regarding the net impedance \( Z \) in a series LRC circuit at resonance, we will follow these steps:

### Step 1: Understand the Resonance Condition
In a series LRC circuit, resonance occurs when the inductive reactance \( X_L \) is equal to the capacitive reactance \( X_C \). This can be expressed mathematically as:
\[ X_L = X_C \]

### Step 2: Define Inductive and Capacitive Reactance
The inductive reactance \( X_L \) is given by:
\[ X_L = \omega L \]
where \( \omega \) is the angular frequency and \( L \) is the inductance.

The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} \]
where \( C \) is the capacitance.

### Step 3: Set Inductive Reactance Equal to Capacitive Reactance
At resonance, we set \( X_L \) equal to \( X_C \):
\[ \omega L = \frac{1}{\omega C} \]

### Step 4: Calculate the Impedance at Resonance
The total impedance \( Z \) in a series LRC circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
At resonance, since \( X_L = X_C \), we have:
\[ Z = \sqrt{R^2 + (X_L - X_L)^2} = \sqrt{R^2 + 0} = \sqrt{R^2} = R \]

### Step 5: Conclusion
Thus, the net impedance \( Z \) at resonance in a series LRC circuit is simply:
\[ Z = R \]

### Final Answer
The net impedance \( Z \) when resonance is obtained in a series LRC circuit is equal to the resistance \( R \).

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