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`

Options

A. Is dimensionless
B. Has dimensions `T^(-2)`
C. Has dimensions of P
D. Has dimensions `T^(2)`

Explanation

To determine the unit of the constant \( \alpha \) in the equation \( P = P_0 e^{-\alpha t^2} \), we will follow these steps:

### Step 1: Understand the equation
The equation \( P = P_0 e^{-\alpha t^2} \) describes how the physical quantity \( P \) changes over time \( t \). Here, \( P_0 \) is a constant, and \( \alpha \) is also a constant that we need to analyze.

### Step 2: Analyze the exponent
The term in the exponent, \( -\alpha t^2 \), must be dimensionless. This is because the exponent of an exponential function must not have any units; it must be a pure number.

### Step 3: Identify the dimensions
Let’s denote the dimensions of \( t \) (time) as \( [T] \). Therefore, the dimensions of \( t^2 \) will be \( [T^2] \).

### Step 4: Set up the dimensionless condition
Since \( -\alpha t^2 \) is dimensionless, we can express this mathematically:
\[ [\alpha] \cdot [T^2] = 1 \]
This implies that the dimensions of \( \alpha \) must cancel out the dimensions of \( t^2 \).

### Step 5: Solve for the dimensions of \( \alpha \)
To make \( [\alpha] \cdot [T^2] = 1 \), we can rearrange this to find the dimensions of \( \alpha \):
\[ [\alpha] = [T^{-2}] \]
This means that the unit of \( \alpha \) is the inverse of time squared.

### Step 6: Conclusion
Thus, the unit of \( \alpha \) is \( \text{s}^{-2} \) (per second squared).

### Final Answer:
The unit of the constant \( \alpha \) is \( \text{s}^{-2} \).

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