If R, X_L, and X_C represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

Solution

Physics sir · 3 steps

R, X_L and X_C are all measured in ohms, so just count the powers of ohm in each option.

1. Same unit. Resistance and both reactances are oppositions to current, so all three have the unit ohm (Ω). Dimensionless means the ohms cancel completely, leaving Ω⁰.

   Option	Expression	Unit
   (a)	R/(X_L X_C)	Ω/Ω² = Ω⁻¹
   (b)	R X_L X_C	Ω³
   (c)	R/√(X_L X_C)	Ω/√Ω² = Ω⁰, dimensionless
   (d)	R X_L/X_C	Ω × Ω/Ω = Ω

2. Why it matters. Since X_L = ω L and X_C = 1/(ω C), the product X_L X_C = L/C. So option (c) is R√(C/L), which is 1/Q for a series LCR circuit, a pure number.
3. So. Only (c) has the ohms cancelling completely, so the answer is (c).

The idea to keep. Resistance, reactance and impedance all share the unit ohm. For dimension checks, treat each as one power of ohm and count until the powers add to zero.

Answer from the NTA final answer key. Solution by PrepWiser.