Question

Time dependence of quantity \(P=P_0 e^{-a t^2}\), dimension of a is

• A. dimension less
• B. dimension of \(t^{-2}\)
• C. dimensions of P
• D. dimension of \(t^2\)

Hint:

Always equate exponent term to dimension 1.

Answer 1

The Correct Option is B

Step 1: Recall a fundamental principle of dimensional analysis: the exponent of an exponential function must be dimensionless. Since the exponent is given as \(a t^2\), its dimensions must satisfy \[ [a t^2] = 1. \]
Step 2: Writing this in terms of dimensions, we have \[ [a][t^2] = 1. \] Hence, \[ [a] = [t^{-2}]. \]
Step 3: Therefore, the dimension of the constant \(a\) is the inverse of time squared. Hence → (B).