Find the phase difference between voltage and current in L-C circuit.

Answer

To find the phase difference between voltage and current in an LC circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit Components**: In an LC circuit, we have an inductor (L) and a capacitor (C). The resistance (R) is considered to be zero since it is not mentioned in the question.

2. **Identify Reactances**: The inductive reactance (XL) and capacitive reactance (XC) are given by the formulas:
- \(X_L = \omega L\) (where \(\omega\) is the angular frequency)
- \(X_C = \frac{1}{\omega C}\)

3. **Phase Difference Formula**: The phase difference (\(\Phi\)) between the voltage across the inductor and the current in the circuit can be expressed as:
\[ \Phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right) \]

4. **Substituting R = 0**: Since we are dealing with an LC circuit, we set \(R = 0\). This gives us:
\[ \Phi = \tan^{-1}\left(\frac{X_L - X_C}{0}\right) \]

5. **Analyzing the Expression**: The expression \(\frac{X_L - X_C}{0}\) indicates that we are dividing by zero. This means that the value approaches infinity.

6. **Finding the Phase Difference**: The tangent function approaches infinity when its argument approaches \(\frac{\pi}{2}\). Therefore:
\[ \Phi = \tan^{-1}(\infty) = \frac{\pi}{2} \]

7. **Conclusion**: The phase difference between the voltage and the current in an LC circuit is:
\[ \Phi = \frac{\pi}{2} \text{ radians} \text{ or } 90^\circ \]