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. We will derive the expression for the phase difference when all three components are connected in series.

### Step-by-Step Solution:

1. **Understanding Phase Difference**:
The phase difference (φ) between voltage and current in an AC circuit is determined by the relationship between the inductive reactance (X_L), capacitive reactance (X_C), and resistance (R). The formula for the phase difference is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]

2. **Phase Difference with Resistor and Inductor**:
When the circuit contains only a resistor and an inductor, the phase difference is denoted as φ₁. In this case, the capacitive reactance (X_C) is zero:
\[ \tan(\phi_1) = \frac{X_L}{R} \]

3. **Phase Difference with Resistor and Capacitor**:
When the inductor is replaced by a capacitor, the phase difference becomes φ₂. Here, the inductive reactance (X_L) is zero:
\[ \tan(\phi_2) = \frac{X_C}{R} \]

4. **Combining Resistor, Inductor, and Capacitor**:
Now, when all three components (R, L, and C) are connected in series, the total phase difference φ can be expressed as:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]

5. **Substituting Known Values**:
From the previous steps, we have:
- \(X_L = R \tan(\phi_1)\) (from step 2)
- \(X_C = R \tan(\phi_2)\) (from step 3)

Substituting these into the equation for φ gives:
\[ \tan(\phi) = \frac{R \tan(\phi_1) - R \tan(\phi_2)}{R} \]
Simplifying this, we find:
\[ \tan(\phi) = \tan(\phi_1) - \tan(\phi_2) \]

6. **Final Expression for Phase Difference**:
The phase difference when all three elements are connected in series can be expressed as:
\[ \phi = \tan^{-1}(\tan(\phi_1) - \tan(\phi_2)) \]

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