Question

The unit of \( \sqrt{LC} \) (where \( L \) is inductance and \( C \) is capacitance) is:

• A. second
• B. henry
• C. farad
• D. ampere

Hint:

Remember the LC-oscillation relation \( \omega = 1/\sqrt{LC} \) (or \( T = 2\pi\sqrt{LC} \)); \( \sqrt{LC} \) has the dimension of time.

Answer 1

The Correct Option is A

Step 1: Recall a relation containing \( LC \).
In an LC oscillating circuit the angular frequency of oscillation is
\[ \omega = \frac{1}{\sqrt{LC}} \]
The unit of angular frequency \( \omega \) is radian per second, i.e. \( \text{s}^{-1} \).

Step 2: Rearrange for \( \sqrt{LC} \).
\[ \sqrt{LC} = \frac{1}{\omega} \]

Step 3: Find the unit.
Since \( \omega \) has the unit \( \text{s}^{-1} \), its reciprocal \( \sqrt{LC} \) has the unit
\[ \frac{1}{\text{s}^{-1}} = \text{s (second)} \]

Step 4: Cross-check with base units.
Inductance: \( L \) has unit henry \( = \text{V}\,\text{s}\,\text{A}^{-1} = \Omega\cdot\text{s} \).
Capacitance: \( C \) has unit farad \( = \text{A}\,\text{s}\,\text{V}^{-1} = \text{s}\,\Omega^{-1} \).
Then \( LC \) has unit \( (\Omega\cdot\text{s})(\text{s}\,\Omega^{-1}) = \text{s}^2 \), so \( \sqrt{LC} \) has unit \( \sqrt{\text{s}^2} = \text{s} \).

Step 5: Choose the correct option.
Both methods give the second, option (i).

\[\boxed{\text{Unit of }\sqrt{LC} = \text{second}}\]