An LCR series circuit is connected to an oscillator (AC supply) having RMS voltage 200 V. If the RMS voltages across resistor, inductor and capacitor are equal, then RMS voltage across the resistor would be

Options

A. `100 V`
B. `(200V)/(sqrt2)`
C. `200 V`
D. `(100V)/(sqrt2)`

Answer

To solve the problem, we need to analyze the LCR series circuit connected to an AC supply. The key points from the problem are:

1. The RMS voltage of the AC supply is given as 200 V.
2. The RMS voltages across the resistor (VR), inductor (VL), and capacitor (VC) are equal.

Let's denote the RMS voltage across the resistor, inductor, and capacitor as VR, VL, and VC respectively. Since they are all equal, we can write:

\[ VR = VL = VC = V_x \]

Where \( V_x \) is the common RMS voltage across each component.

### Step-by-Step Solution:

1. **Total RMS Voltage**: The total RMS voltage (V) across the LCR circuit is given as 200 V.

2. **Relationship Between Voltages**: In a series LCR circuit, the total voltage is the vector sum of the voltages across the resistor, inductor, and capacitor. Mathematically, this can be expressed as:
\[ V = \sqrt{VR^2 + (VC - VL)^2} \]

3. **Equal Voltages**: Since we know that \( VR = VL = VC = V_x \), we can substitute these into the equation. Thus, \( VC - VL = 0 \) because both voltages are equal.

4. **Substituting into the Equation**: The equation simplifies to:
\[ V = \sqrt{VR^2 + 0^2} = \sqrt{VR^2} \]
Therefore, we have:
\[ V = VR \]

5. **Final Calculation**: Since the total RMS voltage \( V \) is given as 200 V, we can conclude:
\[ VR = 200 \, \text{V} \]

Thus, the RMS voltage across the resistor \( VR \) is 200 V.

### Final Answer:
The RMS voltage across the resistor is **200 V**.

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