Question

If $R , X _{ L }$, and $X _{ C }$ represent resistance, inductive reactance and capacitive reactance Then which of the following is dimensionless :

• A. $R X_L \quad X_C$
• B. $\frac{R}{\sqrt{X_L X_C}}$
• C. $\frac{R}{X_L X_C}$
• D. $R \frac{X_L}{X_C}$

Hint:

For an impedance problem, a ratio of resistance and reactance terms can be dimensionless if the dimensions of the terms cancel out.

Answer 1

The Correct Option is B

All three quantities, \( R \), \( X_L \), and \( X_C \), have the same dimension, which is ohms (\( \Omega \)). This implies that \( \frac{R}{\sqrt{X_L X_C}} \) is dimensionless.

In electrical circuits, \( R \) represents resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance. These are the three primary components that determine the impedance in an AC circuit. The relationship between these quantities is crucial for understanding the behavior of series RLC circuits.

The formula \( \frac{R}{\sqrt{X_L X_C}} \) appears when analyzing the resonant behavior of series RLC circuits. At resonance, the inductive reactance (\( X_L \)) and the capacitive reactance (\( X_C \)) cancel each other out, and the impedance is purely resistive. In such a case, the ratio of \( R \) to the square root of the product of \( X_L \) and \( X_C \) becomes dimensionless, meaning the quantities involved in this ratio are directly proportional without any physical unit attached to the result.

This dimensionless ratio can be helpful in analyzing the quality factor (Q-factor) of the circuit, which is a measure of the sharpness of the resonance. The Q-factor is defined as the ratio of the energy stored in the system to the energy dissipated per cycle.

The dimensionless nature of the ratio \( \frac{R}{\sqrt{X_L X_C}} \) simplifies calculations in circuit analysis and allows for easier comparison of different systems' resonant properties.