The correct answer is 1/√(LC).

In an LCR (Inductor-Capacitor-Resistor) circuit, the resonant (natural) angular frequency is given by ω = 1/√(LC), and therefore the expression 1/√(LC) carries the dimension of frequency. Let us verify this through a careful dimensional analysis.

Step 1: Identify the SI units and dimensions of L and C

The SI unit of Inductance (L) is the Henry (H).
Dimensionally, from the relation $V=L\frac{dI}{dt}$, we get:

$\left[L\right]=\frac{\left[V\right]·\left[t\right]}{\left[I\right]}=\frac{M{L}^2{T}^{-3}{A}^{-1}·T}{A}=M{L}^2{T}^{-2}{A}^{-2}$

The SI unit of Capacitance (C) is the Farad (F).
Dimensionally, from the relation $C=\frac{Q}{V}$, we get:

$\left[C\right]=\frac{\left[Q\right]}{\left[V\right]}=\frac{A·T}{M{L}^2{T}^{-3}{A}^{-1}}=M^{-1}{L}^{-2}{T}^4{A}^2$

Step 2: Find the dimension of the product LC

$\left[LC\right]=\left(M{L}^2{T}^{-2}{A}^{-2}\right)\times \left(M^{-1}{L}^{-2}{T}^4{A}^2\right)=T^2$

So, [LC] = T², meaning the product LC has the dimension of time squared.

Step 3: Find the dimension of 1/√(LC)

$\left[\frac{1}{\sqrt{LC}}\right]=\frac{1}{\sqrt{T^2}}=\frac{1}{T}=T^{-1}$

Step 4: Compare with the dimension of frequency

Frequency (f) is defined as the number of cycles per second. Its SI unit is the Hertz (Hz), and its dimension is simply:

$\left[f\right]=T^{-1}$

Since $\left[\frac{1}{\sqrt{LC}}\right]=T^{-1}$, it perfectly matches the dimension of frequency.

Why the other options are incorrect:

• √(L/C): $[L/C]=M^2{L}^4{T}^{-6}{A}^{-4}$, whose square root is NOT T^-1. This is actually the dimension of impedance (Ohms), not frequency.

• C/R: $[C/R]=M^{-1}{L}^{-2}{T}^3{A}^2$, which is a time constant (T), not T^-1.

• 1/√(RC): $\left[RC\right]=T$ (also a time constant), so $1/\sqrt{RC}$ gives dimension T^-1/2, which is not the dimension of frequency (T^-1).

Conclusion: Only $\frac{1}{\sqrt{LC}}$ yields the dimension T^-1, which is the dimension of frequency. This expression also corresponds to the physical resonant frequency of an LCR circuit, confirming both dimensionally and physically that it is the correct answer.