Problem 92 - Master Resource Book in JEE Main Physics

Question

A sphere and a cube of same material and same volume. One heated upto same temperature and allowed to cool in the same surroundings. The ratio of the amounts of radiation emitted will be (a) \(1: 1\) (b) \(\frac{4 \pi}{3}: 1\) (c) \(\left(\frac{\pi}{6}\right)^{1 / 3}: 1\) (d) \(\frac{1}{2}\left(\frac{4 \pi}{3}\right)^{2 / 3}: 1\)

Step-by-Step Solution

Answer

The ratio is \( \frac{1}{2} \left( \frac{4\pi}{3} \right)^{2/3} : 1 \); option (d).

Step 1: Understand the Problem

A sphere and a cube made of the same material have the same volume and are heated to the same temperature, then allowed to cool in similar surroundings. We need to find the ratio of the radiation emitted by them.

Step 2: Calculate Volume Equations

Since both objects have the same volume:- The volume of the sphere is given by \( V = \frac{4}{3} \pi r^3 \).- The volume of the cube is given by \( V = a^3 \), where \( a \) is the side length of the cube.

Step 3: Equate Volumes and Solve for Radius in Relation to Side

Equating the volumes, we have \( \frac{4}{3} \pi r^3 = a^3 \).Solving for \( r \), we have:\[ r = \left( \frac{3}{4\pi} \right)^{1/3} a \].

Step 4: Calculate Surface Area

Calculate the surface area for both shapes:- The sphere's surface area: \( A_s = 4 \pi r^2 \).- Substitute \( r \) from previous step, \( A_s = 4 \pi \left( \frac{3}{4\pi} \right)^{2/3} a^2 \).- The cube's surface area: \( A_c = 6a^2 \).

Step 5: Find Ratio of Surface Areas

The sphere emits radiation based on its surface area, which is proportional to \( A_s \) and the cube based on \( A_c \).Thus the ratio of emissions is \( \frac{A_s}{A_c} = \frac{4 \pi \left( \frac{3}{4\pi} \right)^{2/3} a^2}{6a^2} = \left( \frac{4\pi}{3} \right)^{2/3} \cdot \frac{2}{3} \).

Step 6: Simplify and Identify Ratio

Simplifying \( \left( \frac{4\pi}{3} \right)^{2/3} \cdot \frac{2}{3} \), we get the ratio as \( \frac{1}{2} \left( \frac{4\pi}{3} \right)^{2/3} : 1 \).

Key Concepts

Surface Area

The surface area of a geometric shape is the total area covered by the surface of the shape. It plays a crucial role in phenomena like radiation emission because the surface area determines how much heat or radiation an object can emit at a given time. For this reason, when comparing objects like a sphere and a cube, even if they have the same volume, their surface areas can be quite different. This directly influences how they cool down or emit radiation.
In general, the surface area formula for a sphere is given by:

• Sphere: \( A_s = 4 \pi r^2 \)

For a cube, it is calculated as:

• Cube: \( A_c = 6a^2 \)

In our exercise, these formulas help determine how different shapes, despite having the same volume, can have very different surface areas, thus affecting how much radiation each emits. This highlights the importance of understanding the surface area when analyzing heat transfer and energy emissions in physical objects.

Volume Calculation

Calculating the volume of a shape is about finding the space it occupies. Knowing the volume is essential in tasks involving the balance of space and materials. For example, calculating volumes allows us to determine how much of a material is needed to fill a shape.
In our exercise involving radiation emission, we consider two shapes with equal volume:

• Sphere: The formula is \( V = \frac{4}{3} \pi r^3 \).
• Cube: The formula is \( V = a^3 \), where \( a \) is the side of the cube.

Equating these volumes when they are equal helps us understand their relationship in terms of size and structure. It helps derive how changes in one dimension (like radius for a sphere) equate to changes in another (side length of the cube). This balance is key to linking volume to surface area, as shown by solving for \( r \) in terms of \( a \).
This comparison is instrumental in understanding the resulting physical properties of these shapes, such as radiation emission.

Geometric Shapes

Understanding geometric shapes is fundamental to solving many physical problems, including those involving radiation or heat emission. Different shapes like spheres and cubes have distinct properties that impact their physical behavior in varied environments.
The sphere is inherently symmetrical, with all points on its surface equidistant from its center. This symmetry leads to slightly smaller surface areas for large volumes compared to angular shapes like cubes.
The cube, with its equal sides and angular faces, tends to have a larger surface area relative to its volume. This is key in scenarios where surface contact or exposure is critical.
Both shapes serve as essential models in physics, engineering, and design, helping predict outcomes in practical scenarios like cooling rates, diffusion, and emission.
In our scenario, understanding these geometric principles helps us predict that even when volumes match, the emission rates can differ because of the surface area difference, reinforcing the knowledge that geometry matters greatly in real-world applications.