Write dimensional formula for the intensity of radiation

Options

A. `M^(1)L^(0) T^3`
B. `M^(1)L^(0) T^(-3)`
C. `M^(1)L^(-2) T^(-2)`
D. `M^(1)L^(2) T^(-3)`

Explanation

To find the dimensional formula for the intensity of radiation, we will follow these steps:

### Step 1: Understand the Definition of Intensity
Intensity (I) of radiation is defined as the amount of energy (E) passing through a unit area (A) in a unit time (t). Mathematically, it can be expressed as:
\[ I = \frac{E}{A \cdot t} \]

### Step 2: Identify the Dimensions of Energy
Energy can be expressed in terms of work done, which is given by the formula:
\[ \text{Energy} = \text{Force} \times \text{Displacement} \]
The dimensional formula for force (F) is:
\[ F = \text{mass} \times \text{acceleration} = M \cdot L \cdot T^{-2} \]
Thus, the dimensional formula for energy (E) becomes:
\[ E = F \cdot L = (M \cdot L \cdot T^{-2}) \cdot L = M \cdot L^2 \cdot T^{-2} \]

### Step 3: Identify the Dimensions of Area
Area (A) is given by:
\[ A = \text{Length}^2 = L^2 \]

### Step 4: Substitute into the Intensity Formula
Now substituting the dimensions of energy and area into the intensity formula:
\[ I = \frac{E}{A \cdot t} = \frac{M \cdot L^2 \cdot T^{-2}}{L^2 \cdot T} \]

### Step 5: Simplify the Expression
Now we simplify the expression:
\[ I = \frac{M \cdot L^2 \cdot T^{-2}}{L^2 \cdot T} = M \cdot L^{2-2} \cdot T^{-2-1} = M \cdot L^0 \cdot T^{-3} \]

### Step 6: Write the Final Dimensional Formula
Thus, the dimensional formula for the intensity of radiation is:
\[ [I] = M^1 \cdot L^0 \cdot T^{-3} \]

### Conclusion
The dimensional formula for the intensity of radiation is:
\[ [I] = M^1 \cdot L^0 \cdot T^{-3} \]

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