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 determine the dimensions of the constant \( \alpha \) in the equation \( P = P_0 e^{-\alpha t^2} \), we need to ensure that the exponent \( -\alpha t^2 \) is dimensionless. Here’s how we can solve it step by step:

### Step 1: Identify the requirement for the exponent
The expression \( e^{-\alpha t^2} \) must be dimensionless. This means that the quantity \( -\alpha t^2 \) must also be dimensionless.

### Step 2: Write down the dimensions of time
The dimension of time \( t \) is given by:
\[ [t] = T \]
where \( T \) represents the dimension of time.

### Step 3: Determine the dimensions of \( t^2 \)
Since \( t \) has the dimension \( T \), the dimension of \( t^2 \) is:
\[ [t^2] = T^2 \]

### Step 4: Set up the equation for \( \alpha \)
For \( -\alpha t^2 \) to be dimensionless, we can express this as:
\[ [\alpha] \cdot [t^2] = 1 \]
This implies:
\[ [\alpha] \cdot T^2 = 1 \]

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

### Conclusion
Thus, the dimension of the constant \( \alpha \) is:
\[ [\alpha] = T^{-2} \]