The time dependence of a physical quantity `P` is given by `P = P_(0)e^(-alpha t^(2))` , where `alpha` is a constant and `t` is time . Then constant `alpha` is//has

Options

A. Dimensionless
B. Dimensions of `T^(-2)`
C. Dimensions of `P`
D. Dimensions of `T^(2)`

Explanation

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

### Step 1: Understand the equation
The equation given is:
\[ P = P_0 e^{-\alpha t^2} \]
where \( P \) is a physical quantity that depends on time \( t \), and \( \alpha \) is a constant.

### Step 2: Analyze the exponential term
The term \( e^{-\alpha t^2} \) must be dimensionless because the exponent of an exponential function must be dimensionless. This means that the quantity \( -\alpha t^2 \) must also be dimensionless.

### Step 3: Identify the dimensions of \( t^2 \)
The dimension of time \( t \) is denoted as \( [T] \). Therefore, the dimension of \( t^2 \) is:
\[ [t^2] = [T^2] \]

### Step 4: Set up the dimensionless condition
Since \( -\alpha t^2 \) is dimensionless, we can express this as:
\[ [\alpha t^2] = 1 \]
This implies:
\[ [\alpha] \cdot [T^2] = 1 \]

### Step 5: Solve for the dimensions of \( \alpha \)
To isolate \( [\alpha] \), we rearrange the equation:
\[ [\alpha] = \frac{1}{[T^2]} \]
This means that the dimensions of \( \alpha \) are:
\[ [\alpha] = [T^{-2}] \]

### Conclusion
Thus, the constant \( \alpha \) has the dimensions of inverse time squared.

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