$L, C$ and $R$ represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula $M L^2 T^{-4} A^{-2}$ corresponds to _________.
Show Hint
Instead of recalling a resonance formula, try writing R, L and C from their own defining equations (V=IR, EMF=L dI/dt, Q=CV) and combine.
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 $\frac{R}{\sqrt{LC}}$.
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Approach Solution -2
Concept:
$R$, $L$ and $C$ each have a standard, well-known SI dimensional formula that most students memorise directly from the unit of resistance (ohm), inductance (henry) and capacitance (farad).
Recalling these standard results is much faster than deriving them every time, as long as you remember them correctly.
Step 1: Write down the standard dimensional formulas.
$[R] = [M L^2 T^{-3} A^{-2}]$
$[L] = [M L^2 T^{-2} A^{-2}]$
$[C] = [M^{-1} L^{-2} T^4 A^2]$
Step 3: Check each option against the target formula $[M L^2 T^{-4} A^{-2}]$.
Option $\dfrac{R}{\sqrt{LC}}$: $\dfrac{[M L^2 T^{-3} A^{-2}]}{[T]} = [M L^2 T^{-4} A^{-2}]$. This is an exact match.
The other three options either leave an extra $[T]$ power or mix $L$ and $C$ the wrong way, so they cannot match.
