To determine the ratio of the surface temperatures of the Sun and star X, we use Wien's displacement law: λₘₐₓ * T = b where: - λₘₐₓ is the wavelength at which the emission is maximum, - T is the temperature of the body, - b is Wien's constant (b = 2.898 × 10⁻³ m K). Thus, from this law, the temperaRead more
To determine the ratio of the surface temperatures of the Sun and star X, we use Wien’s displacement law:
λₘₐₓ * T = b
where:
– λₘₐₓ is the wavelength at which the emission is maximum,
– T is the temperature of the body,
– b is Wien’s constant (b = 2.898 × 10⁻³ m K).
Thus, from this law, the temperature of each star can be derived as follows:
T = b / λₘₐₓ
For the Sun:
Tₛᵤₙ = (2.898 × 10⁻³) / (510 × 10⁻⁹) = 5688 K
For star X:
Tₓ= (2.898 × 10⁻³) / (350 × 10⁻⁹) = 8271 K
Wien's displacement law is such that the absolute temperature of the black body, T, varies inversely as the wavelength at which the intensity of the emitted radiation is maximum, λₘₐₓ . This relationship can be put in the form of an equation: λₘₐₓ × T = constant. In this case, an increase in temperaRead more
Wien’s displacement law is such that the absolute temperature of the black body, T, varies inversely as the wavelength at which the intensity of the emitted radiation is maximum, λₘₐₓ . This relationship can be put in the form of an equation: λₘₐₓ × T = constant. In this case, an increase in temperature leads to a decrease in the wavelength of radiation emitted at its maximum intensity and vice versa.
In other words, if you increase the temperature of an object, the peak wavelength of the radiation emitted by that object shifts to shorter wavelengths. For example, a hotter object emits radiation with a peak wavelength in the ultraviolet region, while a cooler object emits radiation with a peak wavelength in the infrared region.
Thus, the correct relationship is:
λₘₐₓ * T = constant
This means that for any given black body, regardless of the size or the material composition, the product of the wavelength at maximum value λₘₐₓ and temperature T is a constant.
We use the Stefan-Boltzmann law to calculate the total energy emitted by the Sun after it expands: P = σ A T⁴ where: - P is the power (total energy emitted per unit time), - σ is the Stefan-Boltzmann constant, - A is the surface area of the Sun (which is proportional to the square of the radius), -Read more
We use the Stefan-Boltzmann law to calculate the total energy emitted by the Sun after it expands:
P = σ A T⁴
where:
– P is the power (total energy emitted per unit time),
– σ is the Stefan-Boltzmann constant,
– A is the surface area of the Sun (which is proportional to the square of the radius),
– T is the temperature of the Sun.
### Changes
– The radius of the Sun increases to 100 times its original radius. Thus surface area A increases by a factor of 100² = 10,000.
– The temperature of the Sun becomes half. Therefore T reduces by a factor of 2. Since the total energy emitted is proportional to T⁴, the power reduces by a factor of 2⁴ = 16.
Total change in energy:
The total energy emitted is proportional to the surface area and the fourth power of the temperature. Thus, the total energy emitted will change by:
To overcome this problem, we make use of the Stefan-Boltzmann law: that states that power radiated by any body is directly proportional to the fourth power of the absolute temperature as, P = σ A T⁴ P = the radiated power. σ is known as Stefan- Boltzmann constant, A is surface area of body, and T isRead more
To overcome this problem, we make use of the Stefan-Boltzmann law: that states that power radiated by any body is directly proportional to the fourth power of the absolute temperature as,
P = σ A T⁴
P = the radiated power.
σ is known as Stefan- Boltzmann constant,
A is surface area of body, and
T is temperature in Kelvin.
Since the surface area and the Stefan-Boltzmann constant do not change, we can compare the two powers at two different temperatures by using the ratio of their temperatures raised to the fourth power:
P₂ / P₁ = (T₂ / T₁)⁴
We are given:
– The initial temperature is T₁ = 150°C = 150 + 273 = 423 K
– The final temperature is T₂ = 300°C = 300 + 273 = 573 K
– The initial power is P₁ = 20 W
Applying the formula:
P₂ / 20 = (573 / 423)⁴
Now we can calculate P₂:
P₂ = 20 × (573 / 423)⁴ ≈ 67.34 W
Thus, the power radiated by the metal rod at 300°C will be approximately 67.34 W. The closest answer is: 68.3 W
We will use the Stefan-Boltzmann law, which states that the energy radiated by a black body is proportional to the fourth power of its temperature in Kelvin: E ∝ T⁴ Given: - Initial temperature T₁ = 27°C = 27 + 273 = 300 K - Final temperature T₂ = 127°C = 127 + 273 = 400 K The ratio of energies radiRead more
We will use the Stefan-Boltzmann law, which states that the energy radiated by a black body is proportional to the fourth power of its temperature in Kelvin:
E ∝ T⁴
Given:
– Initial temperature T₁ = 27°C = 27 + 273 = 300 K
– Final temperature T₂ = 127°C = 127 + 273 = 400 K
The sun emits a light with maximum wavelength 510 nm, while another star X emits a light with maximum wavelength of 350 nm. What is the ratio of surface temperature of the sun and the star X?
To determine the ratio of the surface temperatures of the Sun and star X, we use Wien's displacement law: λₘₐₓ * T = b where: - λₘₐₓ is the wavelength at which the emission is maximum, - T is the temperature of the body, - b is Wien's constant (b = 2.898 × 10⁻³ m K). Thus, from this law, the temperaRead more
To determine the ratio of the surface temperatures of the Sun and star X, we use Wien’s displacement law:
λₘₐₓ * T = b
where:
– λₘₐₓ is the wavelength at which the emission is maximum,
– T is the temperature of the body,
– b is Wien’s constant (b = 2.898 × 10⁻³ m K).
Thus, from this law, the temperature of each star can be derived as follows:
T = b / λₘₐₓ
For the Sun:
Tₛᵤₙ = (2.898 × 10⁻³) / (510 × 10⁻⁹) = 5688 K
For star X:
Tₓ= (2.898 × 10⁻³) / (350 × 10⁻⁹) = 8271 K
Now, the ratio of the surface temperatures is :
Tₛᵤₙ / Tₓ = 5688 / 8271 ≈ 0.688
Hence, the correct answer is: 0.68
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According to Wien’s displacement law
Wien's displacement law is such that the absolute temperature of the black body, T, varies inversely as the wavelength at which the intensity of the emitted radiation is maximum, λₘₐₓ . This relationship can be put in the form of an equation: λₘₐₓ × T = constant. In this case, an increase in temperaRead more
Wien’s displacement law is such that the absolute temperature of the black body, T, varies inversely as the wavelength at which the intensity of the emitted radiation is maximum, λₘₐₓ . This relationship can be put in the form of an equation: λₘₐₓ × T = constant. In this case, an increase in temperature leads to a decrease in the wavelength of radiation emitted at its maximum intensity and vice versa.
In other words, if you increase the temperature of an object, the peak wavelength of the radiation emitted by that object shifts to shorter wavelengths. For example, a hotter object emits radiation with a peak wavelength in the ultraviolet region, while a cooler object emits radiation with a peak wavelength in the infrared region.
Thus, the correct relationship is:
λₘₐₓ * T = constant
This means that for any given black body, regardless of the size or the material composition, the product of the wavelength at maximum value λₘₐₓ and temperature T is a constant.
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Suppose the sun expands so that its radius becomes 100 times its present radius and its surface temperature becomes half of its present value. The total energy emitted by it then will increases by a factor of
We use the Stefan-Boltzmann law to calculate the total energy emitted by the Sun after it expands: P = σ A T⁴ where: - P is the power (total energy emitted per unit time), - σ is the Stefan-Boltzmann constant, - A is the surface area of the Sun (which is proportional to the square of the radius), -Read more
We use the Stefan-Boltzmann law to calculate the total energy emitted by the Sun after it expands:
P = σ A T⁴
where:
– P is the power (total energy emitted per unit time),
– σ is the Stefan-Boltzmann constant,
– A is the surface area of the Sun (which is proportional to the square of the radius),
– T is the temperature of the Sun.
### Changes
– The radius of the Sun increases to 100 times its original radius. Thus surface area A increases by a factor of 100² = 10,000.
– The temperature of the Sun becomes half. Therefore T reduces by a factor of 2. Since the total energy emitted is proportional to T⁴, the power reduces by a factor of 2⁴ = 16.
Total change in energy:
The total energy emitted is proportional to the surface area and the fourth power of the temperature. Thus, the total energy emitted will change by:
Change in energy = (Aₙₑʷ * Tₙₑʷ⁴) / (Aₒˡᵈ * Tₒˡᵈ⁴) = (100² * (1/2)⁴) / 1 = (10,000 * 1) / 16 = 625
Thus, the total energy emitted will increase by a factor of 625.
So, the correct answer is: 625
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A metal rod at a temperature of 150°C, radiates energy at a rate of 20W. If its temperature is increased to 300°C, then it will radiate at the rate of
To overcome this problem, we make use of the Stefan-Boltzmann law: that states that power radiated by any body is directly proportional to the fourth power of the absolute temperature as, P = σ A T⁴ P = the radiated power. σ is known as Stefan- Boltzmann constant, A is surface area of body, and T isRead more
To overcome this problem, we make use of the Stefan-Boltzmann law: that states that power radiated by any body is directly proportional to the fourth power of the absolute temperature as,
P = σ A T⁴
P = the radiated power.
σ is known as Stefan- Boltzmann constant,
A is surface area of body, and
T is temperature in Kelvin.
Since the surface area and the Stefan-Boltzmann constant do not change, we can compare the two powers at two different temperatures by using the ratio of their temperatures raised to the fourth power:
P₂ / P₁ = (T₂ / T₁)⁴
We are given:
– The initial temperature is T₁ = 150°C = 150 + 273 = 423 K
– The final temperature is T₂ = 300°C = 300 + 273 = 573 K
– The initial power is P₁ = 20 W
Applying the formula:
P₂ / 20 = (573 / 423)⁴
Now we can calculate P₂:
P₂ = 20 × (573 / 423)⁴ ≈ 67.34 W
Thus, the power radiated by the metal rod at 300°C will be approximately 67.34 W. The closest answer is: 68.3 W
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A black body is heated from 27°C to 127°C. The ratio of their energies of radiations emitted will be
We will use the Stefan-Boltzmann law, which states that the energy radiated by a black body is proportional to the fourth power of its temperature in Kelvin: E ∝ T⁴ Given: - Initial temperature T₁ = 27°C = 27 + 273 = 300 K - Final temperature T₂ = 127°C = 127 + 273 = 400 K The ratio of energies radiRead more
We will use the Stefan-Boltzmann law, which states that the energy radiated by a black body is proportional to the fourth power of its temperature in Kelvin:
E ∝ T⁴
Given:
– Initial temperature T₁ = 27°C = 27 + 273 = 300 K
– Final temperature T₂ = 127°C = 127 + 273 = 400 K
The ratio of energies radiated:
E₂ / E₁ = (T₂ / T₁)⁴
Now calculate the ratio:
E₂ / E₁ = (400 / 300)⁴ ≈ 3.16
Hence, the nearest answer is: 3 : 4
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