Question

$L, C$ and $R$ represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula $M L^2 T^{-4} A^{-2}$ corresponds to _________.

• A. $\frac{R}{\sqrt{LC}}$
• B. $\frac{R}{LC}$
• C. $\frac{C}{\sqrt{LR}}$
• D. $\frac{1}{R} \sqrt{\frac{L}{C}}$

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find which combination of inductance (L), capacitance (C), and resistance (R) has the given dimensional formula $[M L^2 T^{-4} A^{-2}]$.
Step 2: Key Formula or Approach:
The most efficient way is to use known dimensional relationships from physics formulas rather than deriving each from base units.
- Resistance $[R]$ from Power $P=I^2R$.
- The time constant of an LC circuit, related to its resonant frequency $\omega = 1/\sqrt{LC}$, gives the dimension of $\sqrt{LC}$.
Step 3: Detailed Explanation:
Let's find the dimensions of the fundamental quantities involved.
Dimension of R:
Using the formula for power, $P = I^2R$, we get $R = P/I^2$. The dimension of Power (Work/Time) is $[P] = [M L^2 T^{-3}]$. The dimension of Current is the base unit $[A]$.
So, $[R] = \frac{[M L^2 T^{-3}]}{[A^2]} = [M L^2 T^{-3} A^{-2}]$.
Dimension of $\sqrt{LC$:}
The angular frequency of resonance in an LC circuit is $\omega = \frac{1}{\sqrt{LC}}$.
The dimension of angular frequency is $[\omega] = [T^{-1}]$.
Therefore, $[\frac{1}{\sqrt{LC}}] = [T^{-1}]$, which means $[\sqrt{LC}] = [T]$.
Dimension of the Target Expression:
Now let's check the dimensions of the expression in option (A), $\frac{R}{\sqrt{LC}}$:
\[ \left[\frac{R}{\sqrt{LC}}\right] = \frac{[R]}{[\sqrt{LC}]} = \frac{[M L^2 T^{-3} A^{-2}]}{[T]} = [M L^2 T^{-4} A^{-2}] \] This matches the dimensional formula given in the question.
Step 4: Final Answer:
The dimensional formula corresponds to $\frac{R}{\sqrt{LC}}$.