The power radiated by a black body is proportional to the fourth power of its absolute temperature: P = σ A T⁴ where: - P: Power radiated - σ: Stefan-Boltzmann constant - A: Surface area of the body - T: Absolute temperature of the body If the body is in an environment at temperature Tₑ, the net radRead more
The power radiated by a black body is proportional to the fourth power of its absolute temperature:
P = σ A T⁴
where:
– P: Power radiated
– σ: Stefan-Boltzmann constant
– A: Surface area of the body
– T: Absolute temperature of the body
If the body is in an environment at temperature Tₑ, the net radiated power becomes:
P_net = σ A (T⁴ – Tₑ⁴)
Approximation for Small Temperature Differences:
For T ≈ Tₑ, expand T⁴ – Tₑ⁴ using the binomial approximation:
T⁴ – Tₑ⁴ ≈ 4 Tₑ³ (T – Tₑ)
Substituting into the net power equation:
P_net = σ A ⋅ 4 Tₑ³ (T – Tₑ)
Regarding Newton’s Law of Cooling:
Newton’s Law of Cooling reads: \dT/dt = -k (T – Tₑ) where k is a proportionality constant.
In the above equation:
P_net = σ A ⋅ 4 Tₑ³ (T – Tₑ)
If P_net is proportional to the rate of temperature change dT/dt:
k = (σ A ⋅ 4 Tₑ³) / (m c) where:
– m: Mass of the body .
-c: Specific heat capacity .
Therefore, in the limiting situation where temperature difference is negligible, the Stefan-Boltzmann Law becomes the Newton’s Law of Cooling.
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of its absolute temperature. Mathematically: P = σ A T⁴ Explanation: - P: Total power radiated by the black body (in watts, W) - σ: Stefan-BRead more
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of its absolute temperature.
Mathematically:
P = σ A T⁴
Explanation:
– P: Total power radiated by the black body (in watts, W)
– σ: Stefan-Boltzmann constant (5.67 × 10⁻⁸ W·m⁻²·K⁻⁴)
– A: Surface area of the black body (in square meters, m²)
– T: Absolute temperature of the black body (in kelvins, K)
The law describes the energy radiated by an ideal black body, which absorbs and emits all incident radiation perfectly. The radiative power increases significantly with temperature because of the T⁴ dependence, meaning small temperature changes cause large variations in emitted energy.
Applications:
1. Astrophysics: Calculating the energy output of stars and celestial objects.
2. Thermal Engineering: Designing heat radiators, furnaces, and thermal equipment.
3. Climate Science: Analyzing Earth’s energy balance with solar and terrestrial radiation.
The greenhouse effect is the process by which certain gases in the Earth's atmosphere, such as carbon dioxide (CO₂), methane (CH₄), and water vapor (H₂O), trap heat from the Sun. These gases allow sunlight to enter the atmosphere but prevent some of the outgoing infrared radiation from escaping backRead more
The greenhouse effect is the process by which certain gases in the Earth’s atmosphere, such as carbon dioxide (CO₂), methane (CH₄), and water vapor (H₂O), trap heat from the Sun. These gases allow sunlight to enter the atmosphere but prevent some of the outgoing infrared radiation from escaping back into space, keeping the planet warm.
Mechanism:
1. Sunlight enters the atmosphere, and the Earth’s surface absorbs it, warming up.
2. The heated surface radiates heat as infrared radiation. 3. Greenhouse gases absorb and re-emit some of this heat back towards the Earth’s surface.
This way, the temperature of the Earth is kept at a level which is favorable to life.
Significance:
1. It maintains the global temperature; otherwise, the Earth would be cold enough to not support life at an average temperature of about 15°C instead of -18°C without the greenhouse effect.
2. Supports ecosystems by providing thermal conditions necessary for biodiversity. 3. Enables liquid water to exist by preventing extreme cooling, which is essential for life.
Excessive greenhouse gases due to human activities can enhance this effect, causing global warming and significant environmental challenges.
When two rods are connected in parallel, both rods contribute equally to the heat transfer, which results in a higher overall rate of heat transfer (hence, higher melting rate of ice). When two rods are connected in series, the heat has to pass through both rods sequentially. This effectively reduceRead more
When two rods are connected in parallel, both rods contribute equally to the heat transfer, which results in a higher overall rate of heat transfer (hence, higher melting rate of ice).
When two rods are connected in series, the heat has to pass through both rods sequentially. This effectively reduces the rate of heat transfer compared to the parallel arrangement. The heat flow through the system will be limited by the rod with the lower heat transfer rate, reducing the overall rate of melting.
The rate of heat transfer is inversely proportional to the total resistance (or thermal resistance) in the system. The effective resistance is lower than that in the series case; thus, there will be a greater rate of heat transfer in the parallel case.
The ratio of the rates of melting q₂/q₁, when connected in series to parallel, is thus 1/2.
An ideal black body at room temperature absorbs all the incident radiations and therefore appears black. It begins emitting thermal radiation after heating in a furnace and then increases in brightness with increasing temperatures. An ideal black body would emit radiation dependent on its temperaturRead more
An ideal black body at room temperature absorbs all the incident radiations and therefore appears black. It begins emitting thermal radiation after heating in a furnace and then increases in brightness with increasing temperatures. An ideal black body would emit radiation dependent on its temperature and thus, be brightest when heated to its higher temperatures.
Derive Newton’s law of cooling from Stefan’s law.
The power radiated by a black body is proportional to the fourth power of its absolute temperature: P = σ A T⁴ where: - P: Power radiated - σ: Stefan-Boltzmann constant - A: Surface area of the body - T: Absolute temperature of the body If the body is in an environment at temperature Tₑ, the net radRead more
The power radiated by a black body is proportional to the fourth power of its absolute temperature:
P = σ A T⁴
where:
– P: Power radiated
– σ: Stefan-Boltzmann constant
– A: Surface area of the body
– T: Absolute temperature of the body
If the body is in an environment at temperature Tₑ, the net radiated power becomes:
P_net = σ A (T⁴ – Tₑ⁴)
Approximation for Small Temperature Differences:
For T ≈ Tₑ, expand T⁴ – Tₑ⁴ using the binomial approximation:
T⁴ – Tₑ⁴ ≈ 4 Tₑ³ (T – Tₑ)
Substituting into the net power equation:
P_net = σ A ⋅ 4 Tₑ³ (T – Tₑ)
Regarding Newton’s Law of Cooling:
Newton’s Law of Cooling reads: \dT/dt = -k (T – Tₑ) where k is a proportionality constant.
In the above equation:
P_net = σ A ⋅ 4 Tₑ³ (T – Tₑ)
If P_net is proportional to the rate of temperature change dT/dt:
k = (σ A ⋅ 4 Tₑ³) / (m c) where:
– m: Mass of the body .
-c: Specific heat capacity .
Therefore, in the limiting situation where temperature difference is negligible, the Stefan-Boltzmann Law becomes the Newton’s Law of Cooling.
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State and explain Stefan-Boltzmann law of black body radiation.
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of its absolute temperature. Mathematically: P = σ A T⁴ Explanation: - P: Total power radiated by the black body (in watts, W) - σ: Stefan-BRead more
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of its absolute temperature.
Mathematically:
P = σ A T⁴
Explanation:
– P: Total power radiated by the black body (in watts, W)
– σ: Stefan-Boltzmann constant (5.67 × 10⁻⁸ W·m⁻²·K⁻⁴)
– A: Surface area of the black body (in square meters, m²)
– T: Absolute temperature of the black body (in kelvins, K)
The law describes the energy radiated by an ideal black body, which absorbs and emits all incident radiation perfectly. The radiative power increases significantly with temperature because of the T⁴ dependence, meaning small temperature changes cause large variations in emitted energy.
Applications:
1. Astrophysics: Calculating the energy output of stars and celestial objects.
2. Thermal Engineering: Designing heat radiators, furnaces, and thermal equipment.
3. Climate Science: Analyzing Earth’s energy balance with solar and terrestrial radiation.
Click here for more:
See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-10/
What is Greenhouse effect for the atmosphere of the earth and what is its importance?
The greenhouse effect is the process by which certain gases in the Earth's atmosphere, such as carbon dioxide (CO₂), methane (CH₄), and water vapor (H₂O), trap heat from the Sun. These gases allow sunlight to enter the atmosphere but prevent some of the outgoing infrared radiation from escaping backRead more
The greenhouse effect is the process by which certain gases in the Earth’s atmosphere, such as carbon dioxide (CO₂), methane (CH₄), and water vapor (H₂O), trap heat from the Sun. These gases allow sunlight to enter the atmosphere but prevent some of the outgoing infrared radiation from escaping back into space, keeping the planet warm.
Mechanism:
1. Sunlight enters the atmosphere, and the Earth’s surface absorbs it, warming up.
2. The heated surface radiates heat as infrared radiation. 3. Greenhouse gases absorb and re-emit some of this heat back towards the Earth’s surface.
This way, the temperature of the Earth is kept at a level which is favorable to life.
Significance:
1. It maintains the global temperature; otherwise, the Earth would be cold enough to not support life at an average temperature of about 15°C instead of -18°C without the greenhouse effect.
2. Supports ecosystems by providing thermal conditions necessary for biodiversity. 3. Enables liquid water to exist by preventing extreme cooling, which is essential for life.
Excessive greenhouse gases due to human activities can enhance this effect, causing global warming and significant environmental challenges.
Click here for more:
See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-10/
Two identical rods are connected between two containers. One of them is at 100°C and another is at 0° C. If rods are connected in parallel then the rate of melting of ice is q₁g/sec. If they are connected in series then the rate is q₂. The ratio q₂/q₁ is
When two rods are connected in parallel, both rods contribute equally to the heat transfer, which results in a higher overall rate of heat transfer (hence, higher melting rate of ice). When two rods are connected in series, the heat has to pass through both rods sequentially. This effectively reduceRead more
When two rods are connected in parallel, both rods contribute equally to the heat transfer, which results in a higher overall rate of heat transfer (hence, higher melting rate of ice).
When two rods are connected in series, the heat has to pass through both rods sequentially. This effectively reduces the rate of heat transfer compared to the parallel arrangement. The heat flow through the system will be limited by the rod with the lower heat transfer rate, reducing the overall rate of melting.
The rate of heat transfer is inversely proportional to the total resistance (or thermal resistance) in the system. The effective resistance is lower than that in the series case; thus, there will be a greater rate of heat transfer in the parallel case.
The ratio of the rates of melting q₂/q₁, when connected in series to parallel, is thus 1/2.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-10/
An ideal black body at room temperature is thrown into furnace. It is observed that
An ideal black body at room temperature absorbs all the incident radiations and therefore appears black. It begins emitting thermal radiation after heating in a furnace and then increases in brightness with increasing temperatures. An ideal black body would emit radiation dependent on its temperaturRead more
An ideal black body at room temperature absorbs all the incident radiations and therefore appears black. It begins emitting thermal radiation after heating in a furnace and then increases in brightness with increasing temperatures. An ideal black body would emit radiation dependent on its temperature and thus, be brightest when heated to its higher temperatures.
Click here for more:
See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-10/