JEE Physics 2023 Question with Solution

Given: $R$, $X_L$ and $X_C$ represent resistance, inductive reactance and capacitive reactance.

Find: Which expression is dimensionless.

All three quantities have the same dimensions because resistance and both reactances are measured in ohm.

$$[R] = [X_L] = [X_C] = [\Omega]$$

Now check the given expressions:

$$[R X_L X_C] = [\Omega^3]$$

So option A is not dimensionless.

$$\left[\frac{R}{\sqrt{X_L X_C}}\right] = \frac{[\Omega]}{\sqrt{[\Omega][\Omega]}} = \frac{[\Omega]}{[\Omega]} = 1$$

So option B is dimensionless.

$$\left[\frac{R}{X_L X_C}\right] = \frac{[\Omega]}{[\Omega^2]} = [\Omega^{-1}]$$

So option C is not dimensionless.

$$\left[\frac{X_L}{X_C}\right] = \frac{[\Omega]}{[\Omega]} = 1$$

This is also dimensionless.

The solution marks the correct option as B. Therefore, following the provided solution, the correct option is B.

Given: $R$, $X_L$ and $X_C$ have the same physical dimension.

Find: The dimensionless expression among the options.

Resistance, inductive reactance and capacitive reactance are all measured in ohms. Therefore they are dimensionally identical.

$$[R] = [X_L] = [X_C] = [\Omega]$$

Evaluate each option one by one:

1. For $R X_L X_C$,

$$[\Omega] [\Omega] [\Omega] = [\Omega^3]$$

Hence it is not dimensionless.

2. For $\frac{R}{\sqrt{X_L X_C}}$,

$$\sqrt{X_L X_C} \sim \sqrt{[\Omega][\Omega]} = [\Omega]$$

Therefore,

$$\left[\frac{R}{\sqrt{X_L X_C}}\right] = \frac{[\Omega]}{[\Omega]} = 1$$

So it is dimensionless.

3. For $\frac{R}{X_L X_C}$,

$$\left[\frac{R}{X_L X_C}\right] = \frac{[\Omega]}{[\Omega][\Omega]} = [\Omega^{-1}]$$

Hence it is not dimensionless.

4. For $\frac{X_L}{X_C}$,

$$\left[\frac{X_L}{X_C}\right] = \frac{[\Omega]}{[\Omega]} = 1$$

This is also dimensionless.

So the dimensional analysis shows that both B and D are dimensionless, but the provided solution explicitly concludes B. Therefore the extracted answer is B, and the source appears internally inconsistent.