The richardson equaction is given by `I = AT^(2) e^(-B//kT)`. The dimensional formula for `AB^(2)` is

Options

A. 1. IT^(2)`
B. 2. `kT`
C. 3. `Ik^(2)`
D. 4. `Ik^(2)//T`

Explanation

To find the dimensional formula for \( AB^2 \) from the Richardson equation given by

\[ I = AT^2 e^{-\frac{B}{kT}} \]

we will follow these steps:

### Step 1: Understand the components of the equation
The equation consists of:
- \( I \): Current (with dimensions \([I]\))
- \( A \): A constant that we need to find the dimensions of
- \( T \): Temperature (with dimensions \([\Theta]\))
- \( B \): Another constant whose dimensions we need to find
- \( k \): Boltzmann's constant (with dimensions \([M^1 L^2 T^{-2} \Theta^{-1}]\))

### Step 2: Analyze the exponential term
The term \( e^{-\frac{B}{kT}} \) must be dimensionless. This means that the quantity \( \frac{B}{kT} \) must also be dimensionless. Therefore, we can write:

\[ [B] = [k][T] \]

### Step 3: Write down the dimensions of \( k \) and \( T \)
From the known dimensions:
- \( [k] = [M^1 L^2 T^{-2} \Theta^{-1}] \)
- \( [T] = [\Theta] \)

### Step 4: Substitute the dimensions into the equation
Substituting the dimensions of \( k \) and \( T \) into the equation for \( B \):

\[ [B] = [M^1 L^2 T^{-2} \Theta^{-1}][\Theta] = [M^1 L^2 T^{-2}] \]

### Step 5: Determine the dimensions of \( A \)
From the Richardson equation, we can rearrange it to find \( A \):

\[ I = A T^2 \implies [A] = \frac{[I]}{[T^2]} \]

Since the dimensions of current \( I \) are \([I]\), we can write:

\[ [A] = \frac{[I]}{[\Theta^2]} \]

### Step 6: Combine the dimensions to find \( AB^2 \)
Now we need to find the dimensions of \( AB^2 \):

\[ [AB^2] = [A][B^2] = \left(\frac{[I]}{[\Theta^2]}\right)[B]^2 \]

Substituting the dimensions of \( B \):

\[ [B^2] = [M^1 L^2 T^{-2}]^2 = [M^2 L^4 T^{-4}] \]

Thus,

\[ [AB^2] = \left(\frac{[I]}{[\Theta^2]}\right)[M^2 L^4 T^{-4}] = [I][M^2 L^4 T^{-4}][\Theta^{-2}] \]

### Final Result
The dimensional formula for \( AB^2 \) is:

\[ [AB^2] = [I][M^2 L^4 T^{-4}][\Theta^{-2}] \]