The time dependance of a physical quantity 'P' is given by `P=P_(0)exp(-at^(2))`, where a is a constant and 't' is time . The constant a is

Options

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

Answer

To find the dimension of the constant \( a \) in the equation \( P = P_0 \exp(-at^2) \), we can follow these steps:

### Step 1: Understand the Exponential Function
The term inside the exponential function must be dimensionless. This means that the expression \( -at^2 \) must have no dimensions.

### Step 2: Identify the Dimensions
Let’s denote the dimensions of \( P \) (and \( P_0 \)) as \( [P] \). Since both \( P \) and \( P_0 \) have the same dimensions, we can say that:

\[ [P] = [P_0] \]

### Step 3: Set Up the Dimensionless Condition
Since \( -at^2 \) must be dimensionless, we can write:

\[ [a][t^2] = 1 \]

Here, \( [t] \) is the dimension of time.

### Step 4: Rearranging the Equation
From the equation \( [a][t^2] = 1 \), we can rearrange it to find the dimension of \( a \):

\[ [a] = \frac{1}{[t^2]} \]

### Step 5: Express the Dimension of \( a \)
The dimension of time \( [t] \) is denoted as \( T \). Therefore, we can express the dimension of \( a \) as:

\[ [a] = T^{-2} \]

### Conclusion
Thus, the dimension of the constant \( a \) is \( T^{-2} \).