In LCR circuit, the voltage across the terminals of a resistance, inductance & capacitance are 40V, 30V & 60V, then the voltage across the main source will be

Options

A. 130 volt
B. 100 volt
C. 70 volt
D. 50 volt

Answer

To find the voltage across the main source in an LCR circuit given the voltages across the resistance, inductance, and capacitance, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Voltage across the resistor (Vr) = 40V
- Voltage across the inductor (Vl) = 30V
- Voltage across the capacitor (Vc) = 60V

2. **Understand the Phase Relationships**:
- In an LCR circuit, the voltage across the resistor (Vr) is in phase with the current.
- The voltage across the inductor (Vl) leads the current by 90 degrees.
- The voltage across the capacitor (Vc) lags the current by 90 degrees.

3. **Apply the Formula for Total Voltage**:
The formula to find the voltage across the main source (V) in an LCR circuit is given by:
\[ V = \sqrt{V_r^2 + (V_l - V_c)^2} \]
Here, \(V_l - V_c\) accounts for the phase difference between the inductor and capacitor.

4. **Substitute the Values into the Formula**:
- Substitute Vr = 40V, Vl = 30V, and Vc = 60V into the formula:
\[ V = \sqrt{40^2 + (30 - 60)^2} \]

5. **Calculate the Squares**:
- Calculate \(40^2 = 1600\)
- Calculate \(30 - 60 = -30\) and then \((-30)^2 = 900\)

6. **Add the Results**:
- Add the results from the squares:
\[ V = \sqrt{1600 + 900} = \sqrt{2500} \]

7. **Find the Square Root**:
- Calculate the square root:
\[ V = 50V \]

8. **Conclusion**:
- The voltage across the main source is 50V.

### Final Answer:
The voltage across the main source will be **50V**.

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