Question

If L is the inductance, R is the resistance and C is the capacitor, then which of the following does not represent the dimensional formula of time?

• A. $\left[\frac{L}{R}\right]$
• B. $\left[RC\right]$
• C. $\sqrt{LC}$
• D. $\frac{1}{\sqrt{L}C}$

Answer 1

The correct option is D $\frac{1}{\sqrt{L}C}$

(A) In an L-R circuit:

the time constant is R/L so t*(R/L)=[M°L°T°]

The equation comes in the form of e^(-t*R/L)

R/L=[T-¹]

So L/R will be [T].

(B) By Ohm’s law R =V/I, C = Q/v

on multiplying,

RC = (V/I)(Q/V) = Q/I

But Q = It

RC = It/I = t i.e dimesional formula would be [T].

So the unit of RC is second or time constant.

(C) Dimension of L is [ML^2I^-2T^-2] : Dimension of C is [M^-1L^2I^2T^4]

So the dimension of LC is [T^2]

So, the dimensional formula of root LC would be [T].

(D) Dimension of L is [ML^2I^-2T^-2] : Dimension of C is [M^-1L^2I^2T^4]

So the dimension of LC is [T^2]

=>Dimension of 1/root(LC) is [T^-1]

Alternatively, 1/root LC represents resonant frequency in series LCR circuit. So it's unit must be second inverse and hence dimension [T^-1].

Thus, the correct answer is 'D.'