Problem 79 - Master Resource Book in JEE Main Physics

Question

The rays of sun are focussed on a piece of ice through a lens of diameter \(5 \mathrm{~cm}\), as a result of which \(10 \mathrm{~g}\) ice melts in 10 min. The amount of heat received from sun, per unit area per min is (a) \(4 \mathrm{cal} \mathrm{cm}^{-2} \min ^{-1}\) (b) \(40 \mathrm{cal} \mathrm{cm}^{-2} \mathrm{~min}^{-1}\) [c) \(4 \mathrm{Jm}^{-2} \mathrm{~min}\) (d) \(400 \mathrm{cal} \mathrm{cm}^{-2} \mathrm{~min}^{-1}\)

Step-by-Step Solution

Answer

(a) 4 \( \text{cal} \text{ cm}^{-2} \text{ min}^{-1} \)

Step 1: Calculate heat required to melt ice

To melt 10 g of ice, we need to calculate the amount of heat required using the formula: \( Q = mL_f \), where \( m = 10 \text{ g} \) is the mass of the ice and \( L_f = 80 \text{ cal/g} \) is the latent heat of fusion of ice. Thus, \( Q = 10 \times 80 = 800 \text{ cal} \).

Step 2: Determine time in minutes

The time given for 10 g of ice to melt is 10 minutes.

Step 3: Find the area of the lens

The diameter of the lens is 5 cm, so the radius is \(2.5 \text{ cm}\). The area \( A \) of the lens is calculated using the formula for the area of a circle: \( A = \pi r^2 = \pi \times (2.5)^2 = 19.625 \text{ cm}^2 \).

Step 4: Calculate heat received per unit area per minute

The total heat received in 10 minutes is 800 cal. To find the heat received per unit area per minute, use the formula: \( \text{Heat per unit area per minute} = \frac{Q}{A \times t} \), where \( Q \) is total heat, \( A \) is the area, and \( t \) is time in minutes. Thus: \[ \frac{800 \text{ cal}}{19.625 \text{ cm}^2 \times 10 \text{ min}} = 4.078 \approx 4 \text{ cal/cm}^2\underline{\phantom{xxx}}\text{min}. \]

Step 5: Select the correct answer

From our calculation, the heat received per unit area per minute is approximately \(4 \text{ cal/cm}^2 \underline{\phantom{xxx}}\text{min}\). Thus, the correct choice is (a) \(4 \text{ cal} \text{ cm}^{-2} \text{ min}^{-1}\).

Key Concepts

Heat Transfer

Heat transfer is a fascinating process essential in our everyday lives. It involves the exchange of thermal energy between physical systems, depending on temperature and pressure. The three primary modes of heat transfer are conduction, convection, and radiation.
Conduction occurs when heat travels through solid materials, like when a metal rod is heated. Convection takes place in fluids, such as air or water, where warmer areas of a liquid or gas rise and cooler areas sink. Radiation happens through waves, enabling heat transfer through space. Take, for example, the heat we feel from the sun; it comes via radiation.

• Conduction is direct contact heat transfer.
• Convection involves the movement of fluid (liquid or gas).
• Radiation requires no medium, transferring heat across space.

Understanding heat transfer helps explain natural phenomena and many technological processes, like heating systems or cooling mechanisms.
In the context of melting ice under the sun, the focus is on radiation. As sunlight, a form of radiant energy, is directed onto the ice, heat energy gets transferred, causing the ice to melt.

Latent Heat of Fusion

Latent heat of fusion is a critical concept in thermal physics. It refers to the amount of heat required to convert a solid into a liquid without changing its temperature. For water, this is the heat needed to melt ice into liquid water at 0°C.
In simple terms, latent heat of fusion is like the extra boost a substance needs to change its state. For example, when you melt ice, the temperature doesn't rise until all the ice has turned into liquid water. This is because the energy absorbed is used for changing the state rather than increasing temperature.
For ice, this value is often given as 80 calories per gram. So, for melting 10 grams of ice, we need:- Total heat required: \( Q = mL_f \)- Where \( m \) is mass (10 g) and \( L_f \) is the latent heat of fusion of ice.
This helps in calculations involving phase changes in processes like refrigeration, cooking, and weather systems.

Solar Energy

Solar energy is the clean, renewable energy generated from the sun. It reaches Earth through radiation, providing warmth and light that sustains life. Harnessing this energy can be done directly through solar panels, which convert sunlight into electricity.
The role of solar energy in ice melting experiments is particularly illuminating. When the sun's rays are focused on ice using a lens, it amplifies the amount of energy reaching the ice, accelerating the melting process. This is an excellent demonstration of solar energy's potential in generating significant amounts of heat for performing physical changes, like melting.

• Involves capturing the sun's energy.
• Can be used for electricity generation.
• Helps understand heat transfer processes.

The experiment reinforces how solar energy can be quantified by calculating the heat transferred to a given area over time, highlighting its power and applicability in various practical scenarios.