Find the odd one in terms of units.

Concept:

To find the odd one in terms of units, we need to analyze the dimensional units of each expression involving:

• L (Inductance): unit is henry (H) → \( [L] = \frac{V·s}{A} \)
• R (Resistance): unit is ohm (Ω) → \( [R] = \frac{V}{A} \)
• C (Capacitance): unit is farad (F) → \( [C] = \frac{A·s}{V} \)

Analyzing each option:

1. \(\frac{L}{R}\)

   Units: \(\frac{H}{\Omega} = \frac{V·s/A}{V/A} = s\)

   This has units of seconds (time).

2. RC

   Units: \(\Omega·F = \left(\frac{V}{A}\right)\left(\frac{A·s}{V}\right) = s\)

   This also has units of seconds (time).

3. \(\sqrt{LC}\)

   Units: \(\sqrt{H·F} = \sqrt{\left(\frac{V·s}{A}\right)\left(\frac{A·s}{V}\right)} = \sqrt{s^2} = s\)

   This also has units of seconds (time).

4. \(\frac{1}{\sqrt{LC}}\)

   Units: \(\frac{1}{\sqrt{H·F}} = \frac{1}{\sqrt{s^2}} = \frac{1}{s}\)

   This has units of 1/seconds (frequency).

Conclusion:

Options 1, 2, and 3 all have the same unit: seconds (s), meaning they represent time constants or durations.

Option 4 has units of 1/second (Hz), representing frequency — so it is dimensionally different.