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In a series LCR circuit, the the rms vol...

In a series LCR circuit, the the rms voltage supply is 170 V. If `V_(R), V_(L) and V_(C)` represent the rms voltage drop across resistor , inductor and capacitor, then

A

`V_(R) le 170V` (always)

B

`V_(L) ge 170 V` (always)

C

`V_(L) or V_(C)` may be greater than 170 V

D

`|V_(L)-V_(C)| lt 170`

Text Solution

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The correct Answer is:
To solve the problem regarding the series LCR circuit with an RMS voltage supply of 170 V, we will analyze the relationships between the RMS voltage drops across the resistor (V_R), inductor (V_L), and capacitor (V_C). ### Step-by-Step Solution: 1. **Understanding the Phasor Diagram**: In a series LCR circuit, the voltage across the resistor (V_R) is in phase with the current, while the voltages across the inductor (V_L) and capacitor (V_C) are out of phase with the current. The phasor diagram shows V_R along the horizontal axis, V_L upwards, and V_C downwards. 2. **Applying Kirchhoff's Voltage Law**: According to Kirchhoff's law, the total voltage in the circuit is given by: \[ V = V_R + (V_L - V_C) \] However, since V_L and V_C are out of phase, we need to consider their vector nature. 3. **Using the Pythagorean Theorem**: The effective voltage (V) in the circuit can be expressed as: \[ V^2 = V_R^2 + (V_L - V_C)^2 \] Given that the RMS voltage supply is 170 V, we have: \[ 170^2 = V_R^2 + (V_L - V_C)^2 \] 4. **Deriving Relationships**: - **For V_R**: Since \(V_R^2\) and \((V_L - V_C)^2\) are both positive, it follows that: \[ V_R^2 \leq 170^2 \implies V_R \leq 170 \text{ V} \] - **For V_L and V_C**: The difference \(V_L - V_C\) can also be analyzed. If we set \(V_R = 0\), we find: \[ 170^2 = (V_L - V_C)^2 \implies |V_L - V_C| = 170 \text{ V} \] This means that either \(V_L\) or \(V_C\) can be greater than 170 V, depending on their values. 5. **Conclusion**: - \(V_R \leq 170\) V (always true) - \(V_L\) or \(V_C\) can be greater than 170 V (possible) - The difference \(V_L - V_C\) can vary, but it can be less than or equal to 170 V depending on the values of \(V_L\) and \(V_C\). ### Summary of Relationships: 1. \(V_R \leq 170\) 2. \(V_L\) or \(V_C\) may be greater than 170. 3. \(V_L - V_C\) can be less than or equal to 170.
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