The phase difference between voltage and current in an AC circuit containing a resistor and an inductor in series is `phi_1`. When the inductor is replaced by a capacitor , the phase difference is changed to `phi_2` . The phase difference when all the three elements are connected in series with the same AC source will be

Options

A. `tan^(-1)(tanphi_1+tanphi_2)`
B. `tan^(-1)(tanphi_2-tanphi_1)`
C. `cos^(1)(cosphi_1-cosphi_2)`
D. `cos^(1)(cosphi_2-cosphi_1)`

Answer

To solve the problem, we need to analyze the phase differences in an AC circuit containing a resistor (R), an inductor (L), and a capacitor (C) in series.

### Step-by-Step Solution:

1. **Understanding Phase Difference**:
- In an AC circuit with a resistor and an inductor in series, the phase difference between the voltage and current is denoted as \( \phi_1 \).
- When the inductor is replaced by a capacitor, the phase difference becomes \( \phi_2 \).

2. **Phase Difference Formulas**:
- The phase difference \( \phi_1 \) for the resistor and inductor can be expressed using the formula:
\[ \tan \phi_1 = \frac{X_L}{R} \]
where \( X_L \) is the inductive reactance.
- The phase difference \( \phi_2 \) for the resistor and capacitor can be expressed as:
\[ \tan \phi_2 = \frac{X_C}{R} \]
where \( X_C \) is the capacitive reactance.

3. **Combining Inductor and Capacitor**:
- When both the inductor and capacitor are connected in series with the resistor, the total reactance \( X \) is given by:
\[ X = X_L - X_C \]
- The phase difference \( \phi \) for the entire circuit (R, L, and C in series) can be expressed as:
\[ \tan \phi = \frac{X_L - X_C}{R} \]

4. **Substituting Values**:
- We can substitute the expressions for \( X_L \) and \( X_C \) in terms of \( R \) and the phase differences:
\[ \tan \phi = \frac{R \tan \phi_1 - R \tan \phi_2}{R} \]
- This simplifies to:
\[ \tan \phi = \tan \phi_1 - \tan \phi_2 \]

5. **Final Expression**:
- Therefore, the phase difference when all three elements are connected in series is:
\[ \phi = \tan^{-1}(\tan \phi_1 - \tan \phi_2) \]

### Conclusion:
The phase difference when all three elements (R, L, and C) are connected in series with the same AC source is given by:
\[ \phi = \tan^{-1}(\tan \phi_1 - \tan \phi_2) \]