Question

The unit of $\sqrt{LC}$ is
A. Henry
B. Farad
C. Second
D. Ampere

Answer 1

L here refers to the inductance of an inductor and C here refers to the capacitance of a capacitor. We will find the expression for impedance for the inductor and capacitor and use it to find the units for the square root of the product of the inductance and capacitance.

Formula Used:
Impedance of an inductor
${{X}_{L}}=\omega L$
Impedance of a capacitor
${{X}_{C}}=\dfrac{1}{\omega C}$

Complete step-by-step answer :
The quantity for which both the inductor and capacitor can be used to express must be used to calculate the units of the quantity given. One such quantity is impedance. First, we will write the formulas for the impedance due to a capacitor and an inductor in a circuit in alternating current.
Impedance due to an inductor in a circuit with alternating current of frequency $\omega$.
${{X}_{L}}=\omega L$
Impedance due to a capacitor in a circuit with alternating current of frequency $\omega$.
${{X}_{C}}=\dfrac{1}{\omega C}$
Now both of them are impedances and have the same unit as resistance, as they both provide the resistance to the flow of an alternating current. So, they can be exchanged with each other if we are using only their units.
$\begin{aligned} & units\left( \omega L \right)=units\left( \dfrac{1}{\omega C} \right) \\\ & units\left( LC \right)=units\left( \dfrac{1}{{{\omega }^{2}}} \right) \\\ & units\left( \sqrt{LC} \right)=units\left( \dfrac{1}{\omega } \right) \\\ \end{aligned}$
The units of the required quantity are the same as the inverse of frequency. The units of frequency is $\sec on{{d}^{-1}}$. So, the units of the quantity given i.e. $\sqrt{LC}$ will be second. Hence, the correct option is C, i.e. second.

Note : Students must observe that for a resonance LC circuit we got the frequency as $\dfrac{1}{\sqrt{LC}}$and this quantity is inverse of that. We can directly use this result if we recall it and don’t have to calculate the units of the quantity given i.e. $\sqrt{LC}$. This happens when the impedances of both the inductor and capacitance is the same.