Question

The symbols L, C, and R represent inductance, capacitance and resistance respectively. Dimension of frequency are given by the combination

A. \[1/RC\]
B. \[R/L\]
C. \[\dfrac{1}{{\sqrt {LC} }}\]
D. \[C/L\]

Answer 1

Hint: We know that the reactance and inductive reactance has the unit of resistance that is \[\Omega \]. Recall the formula for reactance and inductive reactance to express the unit of inductance. We know the RC time factor is the product of resistance and capacitance.

Formula used:
\[{X_L} = 2\pi fL\]
\[\tau = RC\]
\[{f_r} = \dfrac{1}{{2\pi \sqrt {LC} }}\]
Here, f is the frequency, L is the inductance, R is the resistance and C is the capacitance.

Complete step by step answer:
We know the unit of time is second. The frequency is the inverse of time period. Therefore, the unit of frequency is per second.
We know that the reactance and inductive reactance has the unit of resistance that is \[\Omega \], unit of impedance Z is \[\Omega \].
We have the inductive reactance of the circuit is given as,
\[{X_L} = 2\pi fL\]
Therefore, the factor \[fL\] has a unit of resistance. Therefore, we can say the unit of inductance is \[\Omega \,{s^{ - 1}}\].
We know that formula for RC time factor,
\[\tau = RC\]
Here, R is the resistance and C is the capacitance.
Since time has units of seconds, the factor RC should have units of time. Therefore, we can say \[1/RC\] has a unit of frequency.
We know that the unit of inductance is \[\Omega \,{s^{ - 1}}\] and the unit of resistance is \[\Omega \]. Therefore, the unit of the term, \[R/L\] will be \[{s^{ - 1}}\]. Therefore, the unit of \[R/L\] has dimensions of frequency.
We know that the frequency in the series resonant circuit is given as,
\[{f_r} = \dfrac{1}{{2\pi \sqrt {LC} }}\]
Therefore, the term \[\dfrac{1}{{\sqrt {LC} }}\] has the dimensions of frequency.
The term \[C/L\] cannot have the dimensions of frequency since the unit of capacitance is Farad and not Ohm.

So, the correct answer is option (A), (B), and (C).

Note:
To answer these types of questions students need to remember all the formulae in AC circuits. Students should be able to convert the units depending on the condition given. If you know the units of inductive reactance, then you can use the formula for inductive reactance to determine the unit of inductance.