Problem 52 - Master Resource Book in JEE Main Physics

Question

Solar constant is defined as energy received by earth per \(\mathrm{cm}^{2}\) per minute. The dimensions of solar constant are (b) \(\left[\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-1}\right]\) (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]\) (c) \(\left[\mathrm{ML}^{0} \mathrm{~T}^{-3}\right]\) (d) \(\left[\mathrm{MLT}^{-2}\right]\)

Step-by-Step Solution

Answer

The dimensions of the solar constant are (c) \([ML^0T^{-3}]\).

Step 1: Understand the Problem

We need to find the dimensions of the solar constant, which is defined as energy received per unit time per unit area. This involves analyzing both the units of energy and the way it changes with time and area.

Step 2: Recall the Formula for Energy

Energy (such as that from solar radiation) is typically measured in joules. Therefore, we must recall the dimensional formula for energy, which is work done (force applied over a distance), and is given by \[[Energy] = [Force] \times [Distance]\].

Step 3: Find Dimensions of Force

Force is given by mass times acceleration. The dimensional formula for force is \[[Force] = [M][L][T^{-2}]\].

Step 4: Combine to Find Dimensions of Energy

Using the relationship from Step 2, the dimensional formula for energy becomes\[[Energy] = [M][L][T^{-2}] \times [L] = [M][L^2][T^{-2}]\].

Step 5: Account for Time in Solar Constant

Since the solar constant incorporates energy received **per unit time**, we divide the dimensional formula of energy by time, giving\[\frac{[M][L^2][T^{-2}]}{[T]} = [M][L^2][T^{-3}]\].

Step 6: Account for Area in Solar Constant

As the solar constant considers energy received **per unit area**, and area has dimensions \[[L^2]\], we divide again by the area,\[\frac{[M][L^2][T^{-3}]}{[L^2]} = [M][L^0][T^{-3}]\].

Step 7: Match the Final Dimensions to Options

The resulting dimensional formula \[[M][L^0][T^{-3}]\] matches option (c).

Key Concepts

Solar Constant

The solar constant is a measure of the amount of solar energy received by the Earth per unit area per unit time. It's an essential concept in understanding how energy from the sun reaches our planet. Imagine a square with an area of one square centimeter. The solar constant represents the energy that reaches this square every minute. When dealing with the solar constant, we focus on two major components:

• Energy: The total amount of energy received.
• Time and Area: Constraints related to time (how fast this energy is received) and area (how much space it is spread over).

Given the complexity of solar radiation, calculating the solar constant requires understanding its dimensional formula, which translates these concepts into manageable mathematical forms. To find the dimensions of the solar constant, we explore how energy interacts with time and area. By understanding the dimensions thoroughly, scientists can better forecast and understand climate change, solar energy harnessing, and more.

Energy Dimensions

Energy dimensions are critical in physics, as they allow us to understand the nature of various forces and motions. Primarily, energy is expressed in joules. A joule can be broken down as force applied across a distance.The dimensional formula for energy is derived from:

• Force: The push or pull on an object, represented dimensionally as \([M][L][T^{-2}]\).
• Distance: How far the force moves an object, with dimensions \([L]\).

Thus, we find that energy is represented dimensionally as \([M][L^2][T^{-2}]\). This formula captures how mass, distance, and time are connected in the concept of energy. By understanding this, students not only grasp abstract numbers but see the interplay of natural forces.

Force Dimensions

Force is one of the fundamental concepts in physics. It describes an interaction that changes the motion of an object. Understanding its dimensions helps us grasp how various physical phenomena affect objects in motion.Force can be measured through the formula force equals mass times acceleration. Its dimensional formula is derived from:

• Mass: The amount of matter in an object, annotated by \([M]\).
• Acceleration: The rate of change of velocity with time, noted as \([L][T^{-2}]\).

Therefore, force has dimensions as \([M][L][T^{-2}]\). Understanding this dimensional makeup equips students to dissect how changes in mass, distance, or time can affect an object's motion. It’s not just the numbers; it’s about the real-world applications they represent.