Question

If L and C are inductance and capacitance respectively, then the dimensional formula of $(LC)^{-\frac{1}{2}}$ is

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

Hint:

Memorize that the time constant $\tau = RC = L/R = \sqrt{LC}$ has dimensions of Time $[T]$. Therefore, $1/\sqrt{LC}$ is Frequency $[T^{-1}]$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The expression $(LC)^{-1/2}$ corresponds to the resonance angular frequency ($\omega$) of an LC circuit. The formula for resonance frequency is: \[ \omega = \frac{1}{\sqrt{LC}} = (LC)^{-\frac{1}{2}} \]
Step 2: Determine Dimensions:
The dimension of angular frequency $\omega$ is the inverse of time ($T^{-1}$), as frequency is cycles per unit time. \[ [\omega] = [T^{-1}] \] Therefore, the dimensional formula for $(LC)^{-1/2}$ is $[M^0 L^0 T^{-1}]$. Alternative Method:
$L = [M L^2 T^{-2} A^{-2}]$ $C = [M^{-1} L^{-2} T^4 A^2]$ $LC = [T^2]$ $(LC)^{-1/2} = (T^2)^{-1/2} = [T^{-1}]$.