What is the formula for radiant energy?
E = mc²
E = hf
E = qV
E = Fd
Radiant energy is the energy carried by electromagnetic waves, such as light and heat. The fundamental formula for radiant energy emitted by a black body in thermal equilibrium is given by
or
E=𝜎 𝑇⁴
This formula signifies that the radiant energy emitted by a black body per unit area is directly proportional to the fourth power of its absolute temperature.
The radiant energy formula was derived by Josef Stefan in 1879 based on experimental data, and later theoretically derived from thermodynamic principles by Ludwig Boltzmann in 1884. Hence, the law is known as the Stefan-Boltzmann Law. To derive this formula, consider a black body, an idealized physical body that absorbs all incident electromagnetic radiation. The total power per unit area (P) emitted by a black body is directly proportional to the fourth power of the absolute temperature, which is mathematically expressed as
This derivation involves assumptions of the black body as an ideal emitter and utilizes Planck’s law of black body radiation, integrating over all wavelengths of the emitted radiation.
Problem: Estimate the radiant energy emitted per square meter from the surface of the Sun, given its surface temperature is approximately 5,778 K.
Solution: Using the formula 𝐸 = 𝜎𝑇⁴
𝐸 = (5.67×10⁻⁸ W m⁻²K⁻⁴) x (5778 K)⁴
𝐸= 5.67×10⁻⁸×1.115×1014
𝐸 ≈ 6.33×10⁶ W m⁻²
Explanation: This calculation shows that the Sun emits about 6.33 million watts per square meter, demonstrating the immense power output of our star.
Problem: If the Earth and the Moon have average surface temperatures of 288 K and 220 K respectively, compare their radiant energies per unit area.
Solution: First, calculate for Earth:
Eₑₐᵣₜₕ= σ (288K)⁴ = 5.67 × 10⁻⁸ ×6.894×10⁹
Eₑₐᵣₜₕ ≈ 390 W m⁻²
Then, calculate for Moon:
Eₘₒₒₙ = σ (220K)⁴
𝐸ₘₒₒₙ = 5.67×10⁻⁸×2.336×109
𝐸ₘₒₒₙ ≈ 132 W m⁻²
Explanation: Earth emits about 390 watts per square meter, almost three times as much as the Moon, which emits about 132 watts per square meter. This difference is primarily due to Earth’s higher surface temperature.
Problem: A metal ball at 500 K loses energy by radiation. Calculate the radiant energy loss per square meter.
Solution: 𝐸 = 𝜎 (500 K)⁴
𝐸 = 5.67 × 10⁻⁸ × 6.25 × 10¹⁰
𝐸 ≈ 3545 W m⁻²
Explanation: The metal ball emits approximately 3545 watts per square meter due to its high temperature, showcasing how objects lose heat by emitting radiant energy.
Problem: If the temperature of an object increases from 300 K to 600 K, how does its radiant energy emission change?
Solution: Calculate the initial and final emissions:
𝐸ᵢₙₜᵢₐₗ = 𝜎 (300 K)⁴
𝐸բᵢₙₐₗ = 𝜎 (600 K)⁴
𝐸ᵢₙₜᵢₐₗ ≈ 459 W m⁻²
𝐸բᵢₙₐₗ ≈ 7344 W m⁻²
Explanation: By increasing the temperature, the radiant energy output increases by a factor of (600 / 300)⁴=16. This example illustrates the sensitivity of radiant energy emission to temperature changes.
Calculate radiant energy using the Stefan-Boltzmann law: 𝐸=𝜎𝑇⁴, where σ is the constant and T is temperature.
The general energy formula depends on context, but for radiant energy, it’s 𝐸=𝜎𝑇⁴, based on the Stefan-Boltzmann law.
The equation for radiant power is 𝑃=𝜎𝑇⁴, highlighting how power emitted relates to the fourth power of temperature.
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What is the formula for radiant energy?
E = mc²
E = hf
E = qV
E = Fd
What does 'h' represent in the formula E = hf?
Speed of light
Planck's constant
Boltzmann constant
Electric charge
What does 'f' represent in the formula E = hf?
Force
Frequency
Faraday's constant
Focal length
How does the radiant energy of a photon change if its frequency is doubled?
It remains the same
It is halved
It doubles
It becomes zero
What is the unit of frequency 'f' in the formula E = hf?
Hertz (Hz)
Joules (J)
Meters (m)
Seconds (s)
What is the wavelength of a photon with a frequency of 5 x 10¹⁴ Hz? (Speed of light, c = 3 x 10⁸ m/s)
600 nm
500 nm
400 nm
300 nm
How does the wavelength of a photon relate to its radiant energy?
Directly proportional
Inversely proportional
Exponentially proportional
No relation
Which phenomenon can be explained by the formula E = hf?
Reflection of light
Photoelectric effect
Refraction of light
Diffraction of light
How is the speed of light 'c' related to the wavelength 'λ' and frequency 'f' of a photon?
c = λf
c = λ/f
c = f/λ
c = λ²f
What happens to the radiant energy of a photon if its wavelength is halved?
It remains the same
It is halved
It doubles
It quadruples
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