To determine which group of electrical quantities has similar units, we need to perform a dimensional analysis for each expression involving resistance (R), capacitance (C), and inductance (L).

Understanding Electrical Component Units

First, let's recall the standard units for resistance (R), capacitance (C), and inductance (L):

Now, let's analyze the units of each combination given in the options.

Dimensional Analysis of Combinations

Units of RC

The unit of RC is found by multiplying the unit of R by the unit of C:

• Unit of R = \(\Omega\)
• Unit of C = \(\rm \frac{s}{\Omega}\)
• Unit of RC = \(\rm \Omega \cdot \frac{s}{\Omega} = s\) (seconds)

So, RC has the unit of Time.

Units of \(\rm \frac{1}{RC}\)

Since the unit of RC is seconds, the unit of \(\rm \frac{1}{RC}\) will be the reciprocal:

• Unit of \(\rm \frac{1}{RC} = \frac{1}{s}\) (per second)

So, \(\rm \frac{1}{RC}\) has the unit of Frequency.

Units of \(\rm \sqrt{LC}\)

The unit of \(\rm \sqrt{LC}\) is found by multiplying the unit of L by the unit of C and then taking the square root:

• Unit of L = \(\rm \Omega \cdot s\)
• Unit of C = \(\rm \frac{s}{\Omega}\)
• Unit of LC = \(\rm (\Omega \cdot s) \cdot (\frac{s}{\Omega}) = s^2\)
• Unit of \(\rm \sqrt{LC} = \sqrt{s^2} = s\) (seconds)

So, \(\rm \sqrt{LC}\) has the unit of Time.

Units of \(\rm \frac{1}{\sqrt{LC}}\)

Since the unit of \(\rm \sqrt{LC}\) is seconds, the unit of \(\rm \frac{1}{\sqrt{LC}}\) will be the reciprocal:

• Unit of \(\rm \frac{1}{\sqrt{LC}} = \frac{1}{s}\) (per second)

So, \(\rm \frac{1}{\sqrt{LC}}\) has the unit of Frequency. This term often represents the resonant angular frequency.

Units of \(\rm \frac{R}{L}\)

The unit of \(\rm \frac{R}{L}\) is found by dividing the unit of R by the unit of L:

• Unit of R = \(\Omega\)
• Unit of L = \(\rm \Omega \cdot s\)
• Unit of \(\rm \frac{R}{L} = \frac{\Omega}{\Omega \cdot s} = \frac{1}{s}\) (per second)

So, \(\rm \frac{R}{L}\) has the unit of Frequency.

Units of \(\rm \frac{L}{R}\)

The unit of \(\rm \frac{L}{R}\) is found by dividing the unit of L by the unit of R:

• Unit of L = \(\rm \Omega \cdot s\)
• Unit of R = \(\Omega\)
• Unit of \(\rm \frac{L}{R} = \frac{\Omega \cdot s}{\Omega} = s\) (seconds)

So, \(\rm \frac{L}{R}\) has the unit of Time.

Evaluating the Options for Similar Units

Let's check each option based on the unit analysis:

Option 1: RC, \(\rm \sqrt{LC}\), \(\rm \frac{R}{L}\)

• Unit of RC: seconds (Time)
• Unit of \(\rm \sqrt{LC}\): seconds (Time)
• Unit of \(\rm \frac{R}{L}\): \(\rm \frac{1}{seconds}\) (Frequency)

This group does not have similar units.

Option 2: \(\rm \frac{R}{L}\), \(\rm \frac{1}{\sqrt{LC}}\), \(\rm \frac{1}{RC}\)

• Unit of \(\rm \frac{R}{L}\): \(\rm \frac{1}{seconds}\) (Frequency)
• Unit of \(\rm \frac{1}{\sqrt{LC}}\): \(\rm \frac{1}{seconds}\) (Frequency)
• Unit of \(\rm \frac{1}{RC}\): \(\rm \frac{1}{seconds}\) (Frequency)

This group has similar units (Frequency).

Option 3: \(\rm \frac{1}{\sqrt{LC}}\), RC, \(\rm \frac{R}{L}\)

• Unit of \(\rm \frac{1}{\sqrt{LC}}\): \(\rm \frac{1}{seconds}\) (Frequency)
• Unit of RC: seconds (Time)
• Unit of \(\rm \frac{R}{L}\): \(\rm \frac{1}{seconds}\) (Frequency)

This group does not have similar units.

Option 4: RC, \(\rm \frac{1}{\sqrt{LC}}\), \(\rm \frac{L}{R}\)

• Unit of RC: seconds (Time)
• Unit of \(\rm \frac{1}{\sqrt{LC}}\): \(\rm \frac{1}{seconds}\) (Frequency)
• Unit of \(\rm \frac{L}{R}\): seconds (Time)

This group does not have similar units.

Conclusion

Based on the dimensional analysis, the group that has similar units is \(\rm \frac{R}{L}\), \(\rm \frac{1}{\sqrt{LC}}\), and \(\rm \frac{1}{RC}\). All three expressions have the unit of frequency (\(\rm \frac{1}{seconds}\) or Hertz).