Question

The time dependence of a physical quantity \( P \) is given by \( P = P_0 e^{\alpha (-\alpha t^2)} \), where \( \alpha \) is a constant and \( t \) is time. The constant \( \alpha \) has dimensions of

• A. is dimensionless
• B. has dimensions of \( P \)
• C. has dimensions of \( T^2 \)
• D. has dimensions of \( T \)

Hint:

In exponential equations, the exponent must be dimensionless for consistency in physical units.

Answer 1

The Correct Option is C

Step 1: Analyzing the equation.
In the equation \( P = P_0 e^{\alpha (-\alpha t^2)} \), the exponential function must be dimensionless. Therefore, the argument of the exponential, \( -\alpha t^2 \), must also be dimensionless. This implies that \( \alpha \) has the dimension of \( T^{-2} \).

Step 2: Conclusion.
The constant \( \alpha \) must have the dimension of \( T^{-2} \). Therefore, the correct answer is (3).