An AC of frequency f is flowing in a circuit containing a resistance R and capacitance C in series. The impedance of the circuit is equal to

Options

A. `R + 2pifL`
B. `sqrt(R^(2)+4pi^(2)f^(2)L^(2))`
C. `sqrt(R^(2)+L^(2))`
D. `sqrt(R^(2)+2pifL)`

Answer

To find the impedance of a circuit containing a resistance \( R \) and capacitance \( C \) in series with an alternating current (AC) of frequency \( f \), we can follow these steps:

### Step 1: Understand the Components
In an AC circuit with resistance \( R \) and capacitance \( C \) in series, the total impedance \( Z \) is influenced by both the resistance and the reactance of the capacitor.

### Step 2: Define Reactance of the Capacitor
The reactance \( X_C \) of the capacitor is given by the formula:
\[ X_C = \frac{1}{\omega C} \]
where \( \omega \) is the angular frequency defined as:
\[ \omega = 2\pi f \]
Thus, substituting for \( \omega \):
\[ X_C = \frac{1}{2\pi f C} \]

### Step 3: Calculate the Impedance
The impedance \( Z \) in an R-C series circuit can be calculated using the Pythagorean theorem, since the resistance and reactance are perpendicular to each other in the impedance triangle:
\[ Z = \sqrt{R^2 + X_C^2} \]

### Step 4: Substitute the Reactance into the Impedance Formula
Now, substituting \( X_C \) into the impedance formula:
\[ Z = \sqrt{R^2 + \left(\frac{1}{2\pi f C}\right)^2} \]

### Step 5: Final Expression
Thus, the final expression for the impedance \( Z \) is:
\[ Z = \sqrt{R^2 + \frac{1}{(2\pi f)^2 C^2}} \]

### Summary
The impedance of the circuit containing resistance \( R \) and capacitance \( C \) in series with an AC of frequency \( f \) is given by:
\[ Z = \sqrt{R^2 + \frac{1}{(2\pi f)^2 C^2}} \]

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