Question

The time dependence of a physical quantity \(P\) is given by \[ P = P_0 \exp(-\alpha t^2) \] where \(\alpha\) is a constant and \(t\) is time. The constant \(\alpha\) will:

• A. Have dimensions as that of \(P\)
• B. Have dimensions equal to that of \(P t^2\)
• C. Have no dimensions
• D. Have dimensions of \(t^{-2}\)

Hint:

Whenever you have an exponential function, remember that the exponent must be dimensionless. This will help you determine the dimensions of the constants involve(D)

Answer 1

The Correct Option is D

The given equation for \(P\) is: \[ P = P_0 \exp(-\alpha t^2) \] In this equation, \(P\) is a physical quantity, and \(P_0\) is the initial value of \(P\). The exponential function \(\exp(-\alpha t^2)\) is dimensionless, as the exponent of an exponential function must be dimensionless. This means that the term inside the exponential must be dimensionless: \[ -\alpha t^2 \] Therefore, the dimensions of \(\alpha\) must cancel out the dimensions of \(t^2\), which means: \[ [\alpha] = \frac{1}{[t^2]} \] So, \(\alpha\) has dimensions of \(t^{-2}\). Thus, the correct answer is that \(\alpha\) has dimensions of \(t^{-2}\).