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Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temp...
Given information:
- Radius of body A, rA = 6 cm
- Radius of body B, rB = 18 cm
- Maximum intensity wavelength for body A, λA = 500 nm = 500 x 10^-9 m
- Maximum intensity wavelength for body B, λB = 1500 nm = 1500 x 10^-9 m
Formula:
The rate of total energy radiated by a black body is given by the Stefan-Boltzmann Law:
E = σAT^4
where,
E = rate of total energy radiated
σ = Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A = surface area of the body
T = temperature of the body
Calculating the surface area of the bodies:
Surface area of a sphere is given by the formula:
A = 4πr^2
Using this formula, we can calculate the surface areas of bodies A and B.
Surface area of body A, SA = 4π(6 cm)^2 = 4π(36 cm^2) = 144π cm^2
Surface area of body B, SB = 4π(18 cm)^2 = 4π(324 cm^2) = 1296π cm^2
Calculating the rate of total energy radiated by the bodies:
Rate of total energy radiated by body A, EA = σA(T1)^4
Rate of total energy radiated by body B, EB = σA(T2)^4
Calculating the ratio of the rates of total energy radiated:
Ratio of the rates of total energy radiated = EA / EB
= (σA(T1)^4) / (σA(T2)^4)
= (T1/T2)^4
Calculating the temperatures in Kelvin:
Temperature in Kelvin is given by the formula:
T (in K) = T (in °C) + 273
Converting the temperatures T1 and T2 to Kelvin:
T1 (in K) = T1 (in °C) + 273
T2 (in K) = T2 (in °C) + 273
Substituting the values and calculating the ratio:
Ratio of the rates of total energy radiated = (T1/T2)^4
= ((T1 + 273)/(T2 + 273))^4
Since the temperatures T1 and T2 are not given, we cannot calculate the exact ratio. However, we can see that the ratio will be (T1/T2)^4, which suggests that the ratio is inversely proportional to the fourth power of the temperatures. Therefore, if T1 is three times smaller than T2, the ratio will be (1/3)^4 = 1/81. If T1 is two times smaller than T2, the ratio will be (1/2)^4 = 1/16. However, if T1 is three times larger than T2, the ratio will be (3/1)^4 = 81.
Therefore, the ratio of the rate of total energy radiated by body A to that by body B is 81, which is the given
- Radius of body A, rA = 6 cm
- Radius of body B, rB = 18 cm
- Maximum intensity wavelength for body A, λA = 500 nm = 500 x 10^-9 m
- Maximum intensity wavelength for body B, λB = 1500 nm = 1500 x 10^-9 m
Formula:
The rate of total energy radiated by a black body is given by the Stefan-Boltzmann Law:
E = σAT^4
where,
E = rate of total energy radiated
σ = Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A = surface area of the body
T = temperature of the body
Calculating the surface area of the bodies:
Surface area of a sphere is given by the formula:
A = 4πr^2
Using this formula, we can calculate the surface areas of bodies A and B.
Surface area of body A, SA = 4π(6 cm)^2 = 4π(36 cm^2) = 144π cm^2
Surface area of body B, SB = 4π(18 cm)^2 = 4π(324 cm^2) = 1296π cm^2
Calculating the rate of total energy radiated by the bodies:
Rate of total energy radiated by body A, EA = σA(T1)^4
Rate of total energy radiated by body B, EB = σA(T2)^4
Calculating the ratio of the rates of total energy radiated:
Ratio of the rates of total energy radiated = EA / EB
= (σA(T1)^4) / (σA(T2)^4)
= (T1/T2)^4
Calculating the temperatures in Kelvin:
Temperature in Kelvin is given by the formula:
T (in K) = T (in °C) + 273
Converting the temperatures T1 and T2 to Kelvin:
T1 (in K) = T1 (in °C) + 273
T2 (in K) = T2 (in °C) + 273
Substituting the values and calculating the ratio:
Ratio of the rates of total energy radiated = (T1/T2)^4
= ((T1 + 273)/(T2 + 273))^4
Since the temperatures T1 and T2 are not given, we cannot calculate the exact ratio. However, we can see that the ratio will be (T1/T2)^4, which suggests that the ratio is inversely proportional to the fourth power of the temperatures. Therefore, if T1 is three times smaller than T2, the ratio will be (1/3)^4 = 1/81. If T1 is two times smaller than T2, the ratio will be (1/2)^4 = 1/16. However, if T1 is three times larger than T2, the ratio will be (3/1)^4 = 81.
Therefore, the ratio of the rate of total energy radiated by body A to that by body B is 81, which is the given
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Community Answer
Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temp...
According to Wien's displacement law,
According to Stefan Boltzmann law, rate of energy radiated by a black body is
[Here, A = 4πR2]
(Using (i))= 9
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Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by B?Correct answer is '9'. Can you explain this answer?.
Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by 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 temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 mm 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 by B?Correct answer is '9'. Can you explain this answer? theory, EduRev gives you an
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