Dimensions: Practice Problem & Solution
The time dependence of a physical quantity (p) is given by \(p = p_0 \text{exp } (-\alpha t^2)\), where (alpha) is constant and (t) is the time. The constant \(\alpha\): [1993]
Solution Explained:
To solve this problem, we apply the core principles of Dimensions. Understanding the underlying formula is key to arriving at the correct answer below:
For \(\text{exp }(-\alpha t^2)\) to be dimensionless, \(\alpha t^2\) must be dimensionless. \([\alpha][t^2] = [M^0L^0T^0]\). Since \([t] = [T]\), \([\alpha][T^2] = [1]\). Thus, \([\alpha] = [T^{-2}]\).
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