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

To determine which expression among the given options is dimensionless, we need to analyze each one in terms of their dimensional formulas. The dimensional formula represents the physical dimensions (like length, time, mass, etc.) of a quantity.

Here, we're given:

• \(R\): Resistance, with dimensional formula \([M^1L^2T^{-3}I^{-2}]\).
• \(X_L\): Inductive reactance, with the same dimensional formula as resistance, \([M^1L^2T^{-3}I^{-2}]\).
• \(X_C\): Capacitive reactance, also with the same dimensional formula, \([M^1L^2T^{-3}I^{-2}]\).

We will analyze each expression:

1. \(RX_L \quad X_C\): This is a product of either two or three terms of the same dimension.
   • Dimensional formula: \({[M^1L^2T^{-3}I^{-2}]}^2 = [M^2L^4T^{-6}I^{-4}]\).
   • Conclusion: This is not dimensionless.
2. \(\frac{R}{\sqrt{X_L X_C}}\):
   • Dimensional formula for \(\sqrt{X_L X_C}\): \(\sqrt{[M^1L^2T^{-3}I^{-2}][M^1L^2T^{-3}I^{-2}]} = [M^1L^2T^{-3}I^{-2}]\).
   • The expression becomes \(\frac{[M^1L^2T^{-3}I^{-2}]}{[M^1L^2T^{-3}I^{-2}]} = [1]\).
   • Conclusion: This is dimensionless.
3. \(\frac{R}{X_L X_C}\):
   • Dimensional formula for \(X_L X_C\): \({[M^1L^2T^{-3}I^{-2}]}^2 = [M^2L^4T^{-6}I^{-4}]\).
   • The expression becomes \(\frac{[M^1L^2T^{-3}I^{-2}]}{[M^2L^4T^{-6}I^{-4}]} = [M^{-1}L^{-2}T^{3}I^{2}]\).
   • Conclusion: This is not dimensionless.
4. \(R \frac{X_L}{X_C}\):
   • Dimensional formula for \(\frac{X_L}{X_C}\): \([M^0L^0T^0I^0]\) (since \(X_L\) and \(X_C\) are dimensionally equivalent).
   • The expression effectively is \([M^1L^2T^{-3}I^{-2}][1] = [M^1L^2T^{-3}I^{-2}]\).
   • Conclusion: This is not dimensionless.

Therefore, the only dimensionless expression is \(\frac{R}{\sqrt{X_L X_C}}\), which is the correct answer.