If $R , X _{ L }$, and $X _{ C }$ represent resistance, inductive reactance and capacitive reactance Then which of the following is dimensionless :
If $R , X _{ L }$, and $X _{ C }$ represent resistance, inductive reactance and capacitive reactance Then which of the following is dimensionless :
Show Hint
- $R X_L \quad X_C$
- $\frac{R}{\sqrt{X_L X_C}}$
- $\frac{R}{X_L X_C}$
- $R \frac{X_L}{X_C}$
The Correct Option is B
Solution and Explanation
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.
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Concepts Used:
Units and Measurement
Unit:
A unit of a physical quantity is an arbitrarily chosen standard that is broadly acknowledged by the society and in terms of which other quantities of similar nature may be measured.
Measurement:
The process of measurement is basically a comparison process. To measure a physical quantity, we have to find out how many times a standard amount of that physical quantity is present in the quantity being measured. The number thus obtained is known as the magnitude and the standard chosen is called the unit of the physical quantity.
Read More: Fundamental and Derived Units of Measurement
System of Units:
- CGS system
- FPS system
- MKS system
- SI units
Types of Units:
Fundamental Units -
The units defined for the fundamental quantities are called fundamental units.
Derived Units -
The units of all other physical quantities which are derived from the fundamental units are called the derived units.