Question

Dimensional formula of intensity of radiation is (A) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (B) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\) (C) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3}\) (D) \(\overline{\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3}}\)

Short answer

The correct dimensional formula for the intensity of radiation is (B) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\).

Step-by-step solution

Determine the dimensions of power

Power (P) is defined as the work done (W) per unit time (t). The dimensions of work are the same as that of energy (E), which is given by mass (M) times acceleration (a) times distance (d). So the expression can be written as: \( P = \dfrac{W}{t} = \dfrac{E}{t} \) We know that the dimensions of mass, distance, and time are represented by M, L, and T, respectively. The dimensions of acceleration (a) can be obtained from the relation: \( a = \dfrac{d v}{d t} \) Where v is velocity. The dimensions of velocity are given by: \( v = \dfrac{d}{t} \Rightarrow [v] = LT^{-1} \) Thus, the dimensions of acceleration are: \( [a]=\dfrac{[v]}{[t]} \Rightarrow [a] = \dfrac{LT^{-1}}{T^{-1}} = LT^{-2} \) So the dimensions of energy can now be represented as: \( [E] = [M][L^2 T^{-2}] \) Now calculate the dimensions of power: \( [P] = \dfrac{[E]}{[t]} \Rightarrow [P]=ML^2 T^{-3} \)

Determine the dimensions of area

The intensity of radiation is defined as the power per unit area (A), so we need to find the dimensions of area. Since area is the product of two lengths, its dimensions are: \( [A] = [L^2] \)

Calculate the dimensions of the intensity of radiation

Now we can find the dimensions of the intensity of radiation (I) by dividing the dimensions of power by the dimensions of the area. It can be expressed as: \( [I] = \dfrac{[P]}{[A]} \) By substituting the dimensions of power and area we obtained in steps 1 and 2: \( [I] = \dfrac{ML^2 T^{-3}}{L^2} \Rightarrow [I]=M L^0 T^{-3} \) Comparing our result to the given options in the exercise, we can see that our derived dimensional formula matches option (B): \( \mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2} \) Thus, the correct dimensional formula for the intensity of radiation is: (B) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\)