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Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.
The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.
Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that
of B?
The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.
Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that
of B?
Correct answer is '9'. Can you explain this answer?
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The Stefan-Boltzmann Law
The rate of energy radiated by a black body is given by the Stefan-Boltzmann Law, which states that the rate of energy radiated by a black body is proportional to the fourth power of its absolute temperature.
Calculating the Intensity
The intensity of the emission spectrum of a black body is inversely proportional to the wavelength at which the maximum intensity occurs. Therefore, the ratio of the maximum intensities of bodies A and B can be calculated as the ratio of the wavelengths at which the maximum intensities occur.
Given:
Radius of body A (r1) = 6 cm
Radius of body B (r2) = 18 cm
Temperature of body A (T1) = ?
Temperature of body B (T2) = ?
Calculating the Temperatures
To calculate the temperatures of bodies A and B, we can use the Wien's displacement law, which states that the wavelength at which the maximum intensity occurs is inversely proportional to the temperature.
Using this law, we can write:
λ1/T1 = λ2/T2
where λ1 = 500 nm = 500 × 10^-9 m
and λ2 = 1500 nm = 1500 × 10^-9 m
Solving for T1:
T1 = (λ1/λ2) × T2
= (500 × 10^-9 m)/(1500 × 10^-9 m) × T2
= 1/3 × T2
Calculating the Ratio of Energy Radiated
Now, let's calculate the ratio of the rate of total energy radiated by A to that of B.
Using the Stefan-Boltzmann Law, we can write:
Rate of energy radiated by A/Rate of energy radiated by B = (T1^4)/(T2^4)
Substituting the value of T1 in terms of T2:
Rate of energy radiated by A/Rate of energy radiated by B = (1/3 × T2)^4/T2^4
= (1/81) × T2^4/T2^4
= 1/81
Therefore, the ratio of the rate of total energy radiated by A to that of B is 1/81, which simplifies to 1/9.
Hence, the correct answer is 9.
The rate of energy radiated by a black body is given by the Stefan-Boltzmann Law, which states that the rate of energy radiated by a black body is proportional to the fourth power of its absolute temperature.
Calculating the Intensity
The intensity of the emission spectrum of a black body is inversely proportional to the wavelength at which the maximum intensity occurs. Therefore, the ratio of the maximum intensities of bodies A and B can be calculated as the ratio of the wavelengths at which the maximum intensities occur.
Given:
Radius of body A (r1) = 6 cm
Radius of body B (r2) = 18 cm
Temperature of body A (T1) = ?
Temperature of body B (T2) = ?
Calculating the Temperatures
To calculate the temperatures of bodies A and B, we can use the Wien's displacement law, which states that the wavelength at which the maximum intensity occurs is inversely proportional to the temperature.
Using this law, we can write:
λ1/T1 = λ2/T2
where λ1 = 500 nm = 500 × 10^-9 m
and λ2 = 1500 nm = 1500 × 10^-9 m
Solving for T1:
T1 = (λ1/λ2) × T2
= (500 × 10^-9 m)/(1500 × 10^-9 m) × T2
= 1/3 × T2
Calculating the Ratio of Energy Radiated
Now, let's calculate the ratio of the rate of total energy radiated by A to that of B.
Using the Stefan-Boltzmann Law, we can write:
Rate of energy radiated by A/Rate of energy radiated by B = (T1^4)/(T2^4)
Substituting the value of T1 in terms of T2:
Rate of energy radiated by A/Rate of energy radiated by B = (1/3 × T2)^4/T2^4
= (1/81) × T2^4/T2^4
= 1/81
Therefore, the ratio of the rate of total energy radiated by A to that of B is 1/81, which simplifies to 1/9.
Hence, the correct answer is 9.
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Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer?
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Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer?.
Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer?.
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Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer?, a detailed solution for Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? has been provided alongside types of Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to thatof B?Correct answer is '9'. Can you explain this answer? theory, EduRev gives you an
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