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 problem of finding the net impedance \( Z \) in a series LRC circuit at resonance, we can follow these steps:

### Step 1: Understand Resonance Condition
At resonance in a series LRC circuit, 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: Write the Impedance Formula
The total impedance \( Z \) in a series LRC circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance.

### Step 3: Substitute the Resonance Condition
Since at resonance \( X_L = X_C \), we can substitute this condition into the impedance formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (0)^2} \]

### Step 4: Simplify the Expression
This simplifies to:
\[ Z = \sqrt{R^2} = R \]

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