Physics

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?


ANSWER

$$\dfrac{1}{\sqrt LC}$$


SOLUTION
(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.'
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