Question

In the equation $A = \dfrac{B}{CD^2}$, if $B, C$ and $D$ have the dimensions of inductive reactance, capacitive reactance and angular frequency respectively, then the dimensions of $A$ are:

• A. $M^0L T^{-2}$
• B. $ML^0T^{-2}$
• C. $M^0L^0T^{2}$
• D. $M^{-1}L^0T^{-2}$
• E. $M^0L^0T^{-2}$

Hint:

Reactance has same dimensions as resistance.

Answer 1

The Correct Option is C

Concept:
• Inductive reactance $X_L = \omega L \Rightarrow ML^2T^{-3}A^{-2}$
• Capacitive reactance $X_C = \frac{1}{\omega C}$ has same dimension as resistance
• Angular frequency $\omega = T^{-1}$

Step 1: Dimensions
\[ [B] = [C] = ML^2T^{-3}A^{-2}, \quad [D] = T^{-1} \]

Step 2: Compute $A$
\[ [A] = \frac{B}{C D^2} = \frac{ML^2T^{-3}}{ML^2T^{-3}\cdot T^{-2}} = T^2 \] Final Conclusion:
\[ = M^0L^0T^2 \]