Laws of Radiation MCQ Quiz - Objective Question with Answer for Laws of Radiation - Download Free PDF

Explanation:

Stefan-Boltzmann Constant

• The Stefan-Boltzmann constant, denoted by the symbol σ, is a physical constant that denotes the total energy radiated per unit surface area of a black body in unit time. This radiation is proportional to the fourth power of the black body's thermodynamic temperature.

Stefan-Boltzmann Law: The Stefan-Boltzmann law describes the power radiated from a black body in terms of its temperature. The law is mathematically expressed as:

P = σT⁴

Where:

• P is the power radiated per unit area.
• σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m²K⁴).
• T is the absolute temperature in Kelvin (K).

Importance:

• The Stefan-Boltzmann constant plays a crucial role in thermodynamics and quantum mechanics, particularly in the study of black body radiation.
• It helps in determining the radiant energy emitted by stars, including our Sun, which is essential in astrophysics.
• This constant is used to calculate the radiation heat transfer between objects and is pivotal in various engineering applications like furnace design, radiative cooling of spacecraft, and climate modeling.

Explanation:

Stefan-Boltzmann Law

• The Stefan-Boltzmann law is a fundamental principle in thermal radiation that describes the power radiated from a black body in terms of its temperature. The law is named after physicist Josef Stefan, who experimentally established the law in 1879, and Ludwig Boltzmann, who derived it theoretically.
• The law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's absolute temperature. This implies that even a small increase in temperature results in a significant increase in the amount of radiation emitted.

Mathematical Expression: The Stefan-Boltzmann law is mathematically expressed as:

J = σT⁴

Where:

• J is the total power radiated per unit surface area of the black body.
• σ is the Stefan-Boltzmann constant, approximately equal to 5.67 × 10^-8 W/m²K⁴.
• T is the absolute temperature of the black body in Kelvin (K).

Option 2: Black bodies

• This option correctly identifies the type of bodies to which the Stefan-Boltzmann law primarily applies. A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Because of this property, a black body is also the best emitter of thermal radiation. The Stefan-Boltzmann law applies specifically to black bodies, which radiate energy according to the formula J = σT⁴.

Calculation:

We are given that the maximum energy in the thermal radiation from a hot body A occurs at a wavelength of 130 μm. For a second hot body B, the maximum energy in the thermal radiation occurs at a wavelength of 65 μm.

According to Wien's displacement law, the wavelength of the maximum energy emission is inversely proportional to the temperature of the black body, which is given by:

λ_max * T = constant

For body A:

λ_maxA * T_A = constant

For body B:

λ_maxB * T_B = constant

Since the constant is the same for both bodies, we can write:

λ_maxA * T_A = λ_maxB * T_B

Substituting the given wavelengths:

130 μm * T_A = 65 μm * T_B

Dividing both sides by 65 μm:

2 * T_A = T_B

Therefore, the relationship between the temperatures is:

Final Answer: T_B = 2T_A

Towards the end of the life, filament will become thinner. Resistance will increase and so consumed power will be less, so it will emit less light. Temperature distribution will be non uniform. At the position where temperature is maximum, filament will break. Black body radiation curve will become flat so the filament consumes less electrical power towards the end of the life of the bulb.

Explanation:

1. Energy Density (u): The formula ( \(u = \sigma T^4 \)) represents the Stefan Boltzmann law, where ( u ) is the energy density, ( \(\sigma\) ) is the Stefan Boltzmann constant, and ( T ) is the temperature. When the temperature is doubled ( \(T \rightarrow 2T \)), the new energy density ( u' ) becomes 16 times the original value, since (\( u' = \sigma (2T)^4 = 16 \sigma T^4\) ).

2. Wien’s Displacement Law (\(λ_m\)): The formula ( \(\lambda_m T = b\) ) represents Wien's displacement law, where ( \(\lambda_m\) ) is the wavelength at the maximum radiation, ( T ) is the temperature, and ( b ) is Wien’s constant. When the temperature doubles ( \(T \rightarrow 2T\) ), the peak wavelength decreases by half ( \(\lambda_m' = \frac{\lambda_m}{2}\) ).

\( \ u = \sigma T^4 \quad \text{as} \quad T \rightarrow 2T \\ u' = \sigma 16T^4 \implies \frac{u'}{u} = 16 \\ \lambda_m T = b \quad \text{as} \quad \lambda_m' 2T = b \\\implies \frac{\lambda_m' 2}{\lambda_m} = 1 \implies \lambda_m' = \frac{\lambda_m}{2} \)

The correct options are (1) and(3) .

Concept:

From Wein's displacement law 

\({\lambda _{max}}T = 2898\;\mu m - k\left( {constant} \right)\)

Thus, \({\lambda _{peak}}T = \lambda _{peak}'T'\)

Calculation:

Given, 

Black body λpeak = 1.45 μm at 2000 K.

Now, \(\lambda _{peak}' = 2.90\;\mu m\)

1.45 × 2000 = 2.90 × T'

T' = 1000 K

Explanation:  

Stephen's Boltzmann law:

According to Stefan’s law, the radiant energy emitted by a black body is directly proportional to the fourth power of its absolute temperature.

Q̇ = єσAT⁴

Where Q̇ is Radiate energy, σ is the Stefan-Boltzmann Constant, T is the absolute temperature in Kelvin, є is Emissivity of the material, and A is the area of the emitting body.

• Stefan's Law is used to accurately find the temperature Sun, Stars, and the earth.
• A black body is an ideal body that absorbs or emits all types of electromagnetic radiation.

From above it clear that the amount of radiation emitted by a perfectly black body is proportional to the fourth power of temperature on an ideal gas scale. Thus, option 3 is correct.

Concept:

• Thermal radiation is the transmission of heat in the form of radiant energy from one body to another body by the thermal motion of charged particles of matter.
• The mechanism of radiation is divided into 3 phases
  • Conversion of thermal energy of the hot source into electromagnetic waves
  • Passage of wave motion through intervening medium
  • Transformation of waves into heat
• Thermal radiation is limited to wavelengths ranging from 0.1 to 100 µm of the electromagnetic spectrum.

• It includes some portion of UV radiation (0.1 to 0.4 µm), entire visible radiation (0.4 to 0.7 µm), and entire infrared radiation (0.7 to 100 µm)

Explanation:  

Stefen's Boltzmann law:

According to Stefan’s law, the radiant energy emitted by a black body is directly proportional to the fourth power of its absolute temperature.

P = σAT4

Where P is Radiate energy, σ is the Stefan-Boltzmann Constant, T is the absolute temperature in Kelvin, є is Emissivity of the material, and A is  Area of the emitting body.

• Stefan's Law is used to accurately find the temperature Sun, Stars, and the earth.
• A black body is an ideal body that absorbs or emits all types of electromagnetic radiation.

Additional Information

If anybody radiates equally in all directions, then it is called a diffuse body.

For diathermonous bodies also known as transparent bodies, the transmissivity (τ) = 1

⇒ absorptivity (α) = 0; reflectivity (ρ) = 0;

For perfect black body,

• Absorptivity = 1; Emissivity = 1;
• Transmissivity = 0; Reflectivity = 0;

Gray body

• If a body absorbs or emits radiation in constant proportions irrespective of wavelength then it is called a Gray body. 
• Absorptivity = Transmissivity = Reflectivity = constant

Opaque body

• For opaque bodies, Transmissivity = 0 ⇒ absorptivity + reflectivity = 1;

White body

• For white bodies, reflectivity = 1 ⇒ transmissivity = 0; absorptivity = 0;

Explanation:

White Body:

• A body that is called a white body is one that reflects almost all radiations incident upon it and does not absorb or transmit any part of it.
• For a white body, α = τ = 0, ρ = 1

Additional Information

Black body

• A black body is an object that absorbs all the radiant energy reaching its surface.
• No actual body is perfectly black; the concept of a black body is an idealization with which the radiation characteristics of real bodies can be compared.
   

Properties of the black body:

• It absorbs all the incident radiation falling on it and does not transmit or reflect regardless of wavelength and direction
• It emits the maximum amount of thermal radiation at all wavelengths at any specified temperature
• It is a diffuser emitter (i.e. the radiation emitted by a black body is independent of direction)
• Transparent body: τ = 1, α = ρ = 0
• Opaque body: τ = 0

Explanation:

According to Planck's law, the wavelength corresponding to the maximum energy is inversely proportional to Temperature. It is obtained from Wien's displacement law. The Wien’s displacement law can be obtained by determining the maxima of Planck’s law.

Wein’s displacement law:

• According to this law, the radiation curve for a black body is different at different temperatures and peak wavelengths are inversely proportional to the temperature of the body.
• Maximum wavelength = \(\lambda ~=~\frac{b}{T}\)

Where b is Wein's constant = 3 × 10-3 m.K and T is the temperature of the body (In kelvin unit).

Concept:

Wein’s displacement law gives the relationship between wavelength and temperature.

λ_peak ⋅ T = 2.898 × 10^-3 m ⋅ K

λ_peak ⋅ T = 2898 micrometre ⋅ K

Calculation:

Given,T = 17°C = 290 K 

λ_peak × 290 = 2898

λ_peak = 10 micrometres

Explanation:

Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T.

Planck’s law for the energy Eλ radiated per unit volume by a cavity of a blackbody in the wavelength interval λ to λ + Δλ can be written in terms of Planck’s constant (h), the speed of light (c = λ × v), the Boltzmann constant (k), and the absolute temperature (T):

Energy per unit volume per unit wavelength:

\({E_\lambda } = \frac{{8\pi hc}}{{{\lambda ^5}}} \times \frac{1}{{{e^{\frac{{hc}}{{kT\lambda }} - 1}}}}\)

Energy per unit volume per unit frequency:

\({E_\nu } = \frac{{8\pi h}}{{{c^3}}} \times \frac{{{\nu ^3}}}{{{e^{\frac{{hv}}{{kT}} - 1}}}}\)

So Planck’s distribution function:

\(E\left( {\omega ,T} \right) = \frac{1}{{{e^{\frac{{h\omega }}{\tau }}} - 1}}\)

Using planck’s law, when we plot Ebλ with λ, we get the curve as shown below.

As temperature increases, the peak of the curve shift towards a lower wavelength.

Explanation:  

Stephen's Boltzmann law:

According to Stefan’s law, the radiant energy emitted by a black body is directly proportional to the fourth power of its absolute temperature.

P = σAT4

Where P is Radiate energy, σ is the Stefan-Boltzmann Constant, T is the absolute temperature in Kelvin, є is Emissivity of the material, and A is  Area of the emitting body.

• Stefan's Law is used to accurately find the temperature Sun, Stars, and the earth.
• A black body is an ideal body that absorbs or emits all types of electromagnetic radiation.

Additional Information

Newton's law of cooling

According to Newton’s law of cooling, the rate of loss of heat from a body is directly proportional to the difference in the temperature of the body, and its surroundings.

Rate of cooling ∝ ΔT

\(\frac{{dT}}{{dt}} = - k\left( {T - {T_\infty }} \right)\)

Fourier law

According to Fourier law "rate of heat flow by conduction is proportional to the area measured normal to the direction of heat flow, and to the temperature gradient in that direction". 

 \(Q = -kA\frac{{dT}}{{dx}}\)

where Q = rate of heat flow, k = proportionality constant which is the thermal conductivity of the material, A = cross-sectional area perpendicular to the direction of heat flow, \(\frac{dT}{dx}\) = temperature gradient in the direction of heat flow

Here negative sign shows that heat always flows from higher temperature to lower temperature, means the temperature gradient will always be negative in the direction of heat flow.

CONCEPT:

Wien's Displacement Law:        

• According to Wien's law the product of wavelength corresponding to the maximum intensity of radiation and temperature of the body (in Kelvin) is constant, i.e.

λmT = b = constant

Where b = Wien's constant and has value 2.89 10-3 m-K.

EXPLANATION:

• As the temperature of the body increases, the wavelength at which the spectral intensity (E_λ) is maximum shifts towards left. Therefore option 2 is correct.
• Therefore it is also called Wien's displacement law.