Problem 78 - Master Resource Book in JEE Main Physics

Question

The rates of heat radiation from two patches of skin each of area \(A\), on a patient's chest differ by \(2 \%\). If the patch of the lower temperature is at \(300 \mathrm{~K}\) and emissivity of both the patches is assumed to be unity, the temperature of other patch would be (a) \(306 \mathrm{~K}\) (b) \(312 \mathrm{~K}\) (c) \(308.5 \mathrm{~K}\) (d) \(301.5 \mathrm{~K}\)

Step-by-Step Solution

Answer

The temperature of the other patch is approximately 301.5 K.

Step 1: Understanding Radiation Formula

The heat radiated per unit time per unit area by a black body is given by Stefan-Boltzmann law: \[ E = \sigma \cdot T^4 \] where \( E \) is the radiant energy, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the temperature in Kelvin. In our case, we are given two patches of skin with different temperatures causing a 2% difference in their heat radiation.

Step 2: Express Radiation Difference

Let the temperature of the second patch be \( T_2 \). The radiation difference between the two patches is \( 2\% \), so we can write: \[ E_2 = 1.02 \cdot E_1 \] where \( E_1 = \sigma \cdot T_1^4 \) for the first patch with \( T_1 = 300 \text{ K} \).

Step 3: Substitute Radiation Formulas

Substitute into the expression: \[ \sigma \cdot T_2^4 = 1.02 \cdot \sigma \cdot 300^4 \] Simplifying, since emissivity and area are constants and equal for both, \( \sigma \) terms cancel each other out.

Step 4: Calculate Temperature for 2% Increase

After cancellation and simplification, we have: \[ T_2^4 = 1.02 \cdot (300)^4 \] Taking the fourth root of both sides gives: \[ T_2 = (1.02^{1/4}) \cdot 300 \]

Step 5: Solve for Final Temperature

Calculate \( T_2 \):\[ T_2 = 1.0049385 \cdot 300 \approx 301.5 \text{ K}\] Thus, the temperature of the second patch is approximately 301.5 K.

Key Concepts

Heat Radiation

Heat radiation is a form of energy transfer through electromagnetic waves. It doesn't require a medium, which means it can occur in a vacuum, unlike conduction or convection that need a medium such as a solid or fluid. Heat radiation is an intrinsic process where infrared radiation is emitted from the surface due to its temperature.
All objects emit radiation, but the amount depends on their temperature and nature. The hotter an object is, the more radiation it emits. This radiation increases rapidly with temperature, governed by the Stefan-Boltzmann law.
Key points of heat radiation include:

• All objects with a temperature above absolute zero emit heat radiation.
• The intensity and frequency of emitted radiation can vary based on the object's temperature.
• Good emitters of radiation (like the sun) are also good absorbers.

Understanding heat radiation is essential because it is fundamental in fields like meteorology, astrophysics, and even in everyday technologies such as thermal cameras.

Temperature Difference

Temperature difference is crucial in understanding how heat transfers and distributes. When there is a temperature difference between two objects or areas, heat flows from the warmer to the cooler region.
In the context of the Stefan-Boltzmann law, even a small difference in temperature can lead to significant changes in emitted radiation. This is because the radiation heat emitted depends on the fourth power of the temperature (\( E = \sigma \cdot T^4 \).
Here’s why temperature difference matters:

• It determines the direction of heat flow—the greater the difference, the faster the transfer.
• In the exercise, a mere difference in radiation between the two skin patches signifies a temperature difference even when it is only 2%.
• Engineers and scientists must consider temperature differences to harness or control thermal energy effectively in design and problem-solving.

Balancing temperature differences is key in maintaining thermal comfort in environments and in preventing heat losses in systems.

Black Body Radiation

Black body radiation is a perfect model when studying heat and temperature because a black body is an object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It is a theoretical construct and is important in thermodynamics and quantum mechanics.
A real black body will emit radiation purely based on its temperature, described by the Stefan-Boltzmann Law: \( E = \sigma \cdot T^4 \), where \( \sigma \) (sigma) is the Stefan-Boltzmann constant.
Some key characteristics of black body radiation include:

• It is an ideal model used to study emission and absorption across different temperatures.
• The emitted spectrum and intensity depend solely on the temperature.
• Using the concept, scientists can approximate the thermal radiation emitted by any object given its effective emissivity and temperature.

Understanding black body radiation is foundational for studying stars, including our sun, and in designing efficient thermal systems.