Problem 49 - Master Resource Book in JEE Main Physics

Question

The amount of heat conducted out per second through a window, when inside temperature is \(10^{\circ} \mathrm{C}\) and outside temperature is \(-10^{\circ} \mathrm{C}\), is \(1000 \mathrm{~J} .\) Same heat will be conducted in through the window, when outside temperature is \(-23^{\circ} \mathrm{C}\) and inside temperature is (a) \(23^{\circ} \mathrm{C}\) (b) \(230 \mathrm{~K}\) (c) \(270 \mathrm{~K}\) (d) \(296 \mathrm{~K}\)

Step-by-Step Solution

Answer

The inside temperature should be 270 K, which matches option (c).

Step 1: Understand the Problem

We need to determine the inside temperature when the outside temperature is -23°C such that the heat conducted through the window remains the same as before when the outside temperature was -10°C and the inside temperature was 10°C.

Step 2: Recall the Heat Conduction Formula

The heat conducted per second, according to Fourier's law, is given by the formula: \[ q = kA \frac{(T_{inside} - T_{outside})}{d} \]where, - \(q\) is the heat conducted per second, - \(k\) is the thermal conductivity, - \(A\) is the area of the window, - \(d\) is the thickness of the window, - \(T_{inside}\) and \(T_{outside}\) are the inside and outside temperatures respectively.

Step 3: Equate Heat Conducted for Both Conditions

Since the heat conducted in both scenarios is the same and the conductivity parameters \(k\), \(A\), and \(d\) are constant, we set the temperature differences equal: \[ (T_{inside1} - T_{outside1}) = (T_{inside2} - T_{outside2}) \]where, - \(T_{inside1} = 10°C, \ T_{outside1} = -10°C \) for the first scenario, and - \(T_{inside2}\) unknown, \(T_{outside2} = -23°C \) for the second.

Step 4: Solve for the Unknown Inside Temperature

Substitute the given values into the equation derived in Step 3:\[ (10 - (-10)) = (T_{inside2} - (-23)) \]Solve for \(T_{inside2}\):\[ 20 = T_{inside2} + 23 \]\[ T_{inside2} = 20 - 23 = -3°C \]

Step 5: Convert to Kelvin

Since one of the options is in Kelvin, convert -3°C to Kelvin. The conversion from Celsius to Kelvin is given by:\[ K = °C + 273.15 \]Thus:\[ -3°C = -3 + 273.15 = 270.15K \]Approximately, this is 270K.

Key Concepts

Fourier's Law

Heat conduction through a material can be understood through Fourier's Law. This principle is foundational in the study of heat conduction, describing how heat transfer occurs across a material due to a temperature difference. The heat transfer rate is proportional to this temperature difference and the properties of the material.

• Fourier's Law is expressed mathematically as: \[ q = kA \frac{(T_{inside} - T_{outside})}{d} \]
• Where:
  • \(q\) is the rate of heat transfer (in Joules per second).
  • \(k\) represents the thermal conductivity of the material.
  • \(A\) is the cross-sectional area through which heat is being conducted.
  • \(d\) is the thickness of the material.
  • \(T_{inside}\) and \(T_{outside}\) are the inside and outside temperatures respectively, indicating the temperature gradient.

This law explains why heat moves from areas of higher temperature to lower temperature, striving to reach thermal equilibrium.

Thermal Conductivity

The term 'Thermal Conductivity' describes a material's ability to conduct heat. Different materials have varying thermal conductivities, which determine how effectively they can transfer thermal energy.

• Metals like copper and aluminum typically have high thermal conductivity; they transfer heat quickly.
• Materials like wood and rubber have low thermal conductivity; they transfer heat slowly.
• The value of thermal conductivity is crucial in many practical applications, such as building insulation, heat exchangers, and electronic cooling systems.

Thermal conductivity is a constant value \(k\) in Fourier's Law, influencing the amount of heat transferred across a material given a specific temperature difference and geometric setup of the material. This means that even if the temperature differences are the same, the material with higher thermal conductivity will transfer more heat.

Temperature Difference

Understanding the concept of temperature difference is essential in solving any heat conduction problem. It represents the driving force for heat flow between two areas or objects.

• When there's a larger temperature difference, more heat is conducted across the material.
• The significance of this can be seen directly in Fourier's Law, as the temperature difference \((T_{inside} - T_{outside})\) directly affects the rate of heat transfer \(q\).
• In real-world scenarios, maintaining or manipulating temperature differences can be crucial, such as in climate control and heating systems.

By equating the temperature differences in the given exercise, we could solve for the unknown temperature, illustrating how pivotal this concept is in practical applications.

Celsius to Kelvin Conversion

Temperature measurements can be tricky when different scales are involved, such as Celsius and Kelvin. Celsius is often used for everyday temperature reporting, while Kelvin is essential in scientific calculations due to its absolute nature.

• To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature.
  • For instance, \(-3^{\circ}C\) becomes:
  • \(-3 + 273.15 = 270.15K\)
• Kelvin is an absolute scale, starting at absolute zero, where molecular motion ceases.
• This conversion is critical in scientific measurements as it ensures calculations start from a universally recognized base.

In the exercise, converting the temperature between these scales allowed for straightforward comparison, reinforcing the need for fluency in converting between Celsius and Kelvin.