In an LCR circuit, which of the following expression has the dimension of frequency:

The correct answer is:

$$\begin{array}{l}\text{For RC circuits, time constant is,}\tau =RC\text{has unit of time,}\\ \text{so,}\frac{1}{RC}\text{has a unit of frequency,}\\ \text{similar for LC oscillations,}\tau =\sqrt{LC}\\ \text{for RL circuit,}\tau =\frac{L}{R}\\ \text{so,}\frac{R}{L}\text{and}\frac{1}{\sqrt{LC}}\text{has units of frequency.}\end{array}$$

Step-by-Step Explanation:

To determine which expressions in an LCR (Inductor-Capacitor-Resistor) circuit have the dimensions of frequency, we analyze the characteristic time constants associated with the components:

1. Resistor-Capacitor (RC) Combination:
For an RC circuit, the time constant $\tau$ is given by:
$$\tau =RC$$ Since the time constant $\tau$ has the dimension of time ([T]), its reciprocal $\frac{1}{\tau }$ must have the dimension of frequency ([T^-1]). Therefore:
$$\left[\frac{1}{RC}\right]=\left[\text{Frequency}\right]$$

2. Inductor-Resistor (RL) Combination:
For an RL circuit, the time constant $\tau$ is defined as:
$$\tau =\frac{L}{R}$$ Since this expression has the dimension of time, its reciprocal represents frequency:
$$\left[\frac{R}{L}\right]=\left[\text{Frequency}\right]$$

3. Inductor-Capacitor (LC) combination:
For LC oscillations, the natural time period/constant is proportional to:
$$\tau =\sqrt{LC}$$ Consequently, the reciprocal of this expression gives the angular frequency $\omega$, which shares the dimension of frequency:
$$\left[\frac{1}{\sqrt{LC}}\right]=\left[\text{Frequency}\right]$$

Thus, the expressions $\frac{1}{RC}$, $\frac{R}{L}$, and $\frac{1}{\sqrt{LC}}$ all carry the unit of frequency (hertz, or s^-1).