Problem 2132 - JEE Physics

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) \(\overline{\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3}}\)

Step-by-Step Solution

Answer

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

Step 1: 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: \( [P] = \dfrac{[E]}{[t]} \Rightarrow [P]=ML^2 T^{-3} \)

Step 2: 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: \( [A] = [L^2] \)

Step 3: 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) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\)

Key Concepts

Intensity of Radiation

The intensity of radiation is an essential concept in physics, particularly when discussing electromagnetic waves, light, and energy emission. It is defined as the power transferred per unit area, with the area oriented perpendicular to the direction of the energy flow.
Understanding this helps us measure how much power passes through a given surface area or how strong the radiation is.

In practical terms, the intensity tells us the concentration of energy being emitted by a source, like the sun. When sunlight hits a surface, the intensity gives us an idea of how much energy the surface receives.
This concept is vital in applications such as solar energy, where we aim to maximize the efficiency of solar panels by optimizing their placement and orientation to the sun.

• This measurement is crucial for safety, as high-intensity radiation can be harmful.
• It informs the design of systems meant to capture or utilize energy effectively, such as in solar technology.

By analyzing the intensity of radiation, engineers and scientists can predict how different materials and surfaces will respond to energy, influencing decisions in construction, material choice, and environmental physics.

Dimensions of Power

Power, a fundamental concept in physics, describes the rate at which work is performed or energy is transformed. To understand power at its core, we evaluate its dimensions, which are derived from the basic formula for power: work done divided by time.
In terms of dimensional analysis, power is expressed as \[ [P] = ML^2 T^{-3} \].
Here’s the breakdown:

• Mass (M): Represents the amount of matter in an object.
• Length (L): Often refers to the distance over which the force moves an object or the distance traveled.
• Time (T): Relates to the duration over which the force acts or the process occurs.

Understanding the dimensions of power helps in various ways, including designing mechanical systems, calculating electrical needs, or understanding the efficiency of different energy systems. It's a core metric used in diverse fields such as engineering and technology.

Dimensional Formula

A dimensional formula provides a way to understand physical quantities by expressing them in terms of basic dimensions: mass (M), length (L), and time (T).
This approach allows for the validation of physical equations and helps us understand relationships between different physical concepts.

In practice, using the dimensional formula:

• Allows for consistency checks in equations, ensuring that both sides have the same dimensions.
• Helps us convert units by understanding their relationship in terms of mass, length, and time, ensuring that we use compatible units in calculations.

For example, the dimensional formula for the intensity of radiation was shown to be \[ M^1 L^0 T^{-3} \], derived by dividing power by area. Such analyses are fundamental in cross-verifying the consistency and accuracy of physical laws.