Tag: stefan's law

Questions Related to stefan's law

The power radiated by a black body is $P$ and it radiates maximum energy around the wavelength $\lambda  _{o}$ . If the temperature of the black body is now changed so that it radiates maximum energy around a wavelength $3\lambda  _{o}/4$ , the power radiated by it will increase by a factor of :

stefan's law black body radiation heat transfer thermal properties physics

  1. $4/3$

  2. $16/9$

  3. $64/27$

  4. $256/81$

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Correct Option: D
Explanation:
From Wein's displacement law, we know that ${\lambda} _{max}{T}={CONSTANT}$

So, let us assume that initial temperature of the body be ${T} _{1}$
and by changing the temperature from ${T} _{1}   \ to \  {T}$, wavelength will change from $\lambda$ to $\dfrac{3{\lambda}}{4}$,so applying the above relation,

${\lambda} _{0}{T} _{1}=\dfrac{3{\lambda} _{0}}{4}{T}$

Hence, ${T}=\dfrac{4}{3}T _1$

and we know that ${P}={e}{{T}^{4}}$

So, ${P'}=\dfrac{256}{81}P$

The rays of sun are focussed on a piece of ice through a lens of diameter $5$ cm, as a result of which $10$ grams of ice melts in $10$ min. The amount of heat received from Sun is (per unit area per min)

stefan's law black body radiation heat transfer thermal properties physics

  1. 4 $cal/cm^{2} : min$

  2. 40 $cal/cm^{2} : min$

  3. 4 $J/cm^{2} : min$

  4. 400 $J/cm^{2} : min$

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Correct Option: A
Explanation:

The latent heat of fusion is $80cal/g$.

So heat received per unit times is $10g\times 80cal g^{-1}/ 10min=80cal/min$
The are is $\pi r^2=\pi\times(2.5)^2cm^2\approx 20cm^2$
So, amountof heat per unit area per unit time is $80/20=4cal/cm^2\ min$

The emissive power of a sphere of radius $5$cm coated with lamp black is $1500$Wm$^{-2}$. The amount of energy radiated per second is.

stefan's law black body radiation heat transfer thermal properties physics

  1. 15.7 J

  2. 3.14 J

  3. 47.10 J

  4. 4.71 J

Show answer
Correct Option: C
Explanation:

We know that emissive power is given as: ${E}={\sigma}{T}^{4}$
${T}^{4}=\dfrac{E}{\sigma}$
${T}=\dfrac{1500}{5.67\times{10}^{-8}}=403.29 K$
Now, we know that radiation from the surface is given as $E=\sigma \epsilon A{ T }^{ 4 }$
${E}={5.67\times{10}^{-8}\times4\times{\pi}\times{0.05}^{2}\times{403.29}^{4}}=47.15 J $

Match the physical quantities given in Column I with their dimensional formula given in ColumnII

Column-I Column-II
(a) Thermal conductivity (p) is a dimensionless quantity
(b) Stefans constant (q) $ML^{o}T^{o}K$
(c) Wiens constant (r) $ML^2T^{-3}K^{-1}$
(d) Emissivity (s) $ML^{o}T^{-3}K^{-4}$
stefan's law black body radiation heat transfer thermal properties physics

  1. a-s, b-r, c-p, d-q

  2. a-r, b-s, c-q, d-p

  3. a-s, b-q, c-r, d-p

  4. a-p, b-q, c-r, d-s

Show answer
Correct Option: B
Explanation:

Thermal resistance $=\dfrac{kelvin}{watt}$
Unit of power is watt so power = force$\times$ velocity
Force = mass$\times$acceleration = M$\times$$\dfrac{m}{{sec}^{2}}$
Force = $ML{T}^{-2}$
Power = $M{L}^{2}{T}^{-3}$
So, Thermal resistance $={M}^{-1}{L}^{-2}{T}^{3}{\theta}$
and we know that relation between conductivity and resistevity is that they are reciprocal to each other,so thermal conductivity =${M}^{1}{L}^{2}{T}^{-3}{\theta}^{-1}$

Stefan's constant $=5.64\times$${10}^{-8}$$\dfrac { w }{ { m }^{ 2 }{ k }^{ 4 } } $
$\sigma$=${ M{ L }^{ 0 }{ T }^{ -3 }{ \theta  }^{ 4 } }$

Wein's constant: ${\lambda}{\theta}$ = constant = [$L{\theta}$]

Emissivity is define as the ration of two same quantity so it is dimensionless.

A black body emits maximum radiation of wavelength $\displaystyle \lambda _{1}=2000A $ at a certain temperature $\displaystyle T _{1} $ On increasing the temperature the total energy of radiation emitted is increased $16$ times at temperature $\displaystyle T _{2} $ If $\displaystyle \lambda _{2} $ is the wavelength corresponding to which maximum radiation emitted at temperature  $\displaystyle T _{2} $ Calculate the value of $\displaystyle \left ( \frac{\lambda _{1}}{\lambda _{2}} \right ) $

stefan's law black body radiation heat transfer thermal properties physics

  1. $2:1$

  2. $1:2$

  3. $3:4$

  4. $4:3$

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Correct Option: A
Explanation:

The rate of radiation, emitted by per unit area of a body at temperature $T$ , is given by Stefan-Boltzmann's law as,

   $E\propto T^{4}$ ,
hence , $E _{1}/E _{2}=(T _{1}/T _{2})^{4}$ ,
given , $E _{2}=16E _{1}$ ,
therefore ,
            $E _{1}/16E _{1}=(T _{1}/T _{2})^{4}$ ,
or         $T _{2}/T _{1}=2$ ,
now by Wein's law ,
            $\lambda _{m}\propto 1/T$ ,
hence  $\lambda _{m1}/\lambda _{m2}=T _{2}/T _{1}$ ,
therefore ,  $\lambda _{m1}/\lambda _{m2}=T _{2}/T _{1}=2$ ,
               

All bodies emit heat energy from their surfaces by virtue of their temperature. This heat energy is called radiant energy of thermal radiation. The heat that we receive from the sun is transferred to us by a process which, unlike conduction or convection, does not require the help of a medium in the intervening space which is almost free of particles. Radiant energy travels in space as electromagnetic spectrum. Thermal radiations travel through vacuum with the speed of light. Thermal radiations obey the same laws of reflection and refraction as light does. They exhibit the phenomena of interference, diffraction and polarisation as light does.
The emission of radiation from a hot body is expressed in terms of that emitted from a reference body (called the black body) at the same temperature. A black body absorbs and hence emits radiations of all wavelengths. The total energy $E$ emitted by a unit area of a black body per second is given by $E =\sigma T^{4}$ where $T$ is the absolute temperature of the body and $\sigma $ is a constant known as Stefan's constant. If the body is not a perfect black body, then $E =\varepsilon \sigma  T^{4}$where $\varepsilon $ is the emissivity of the body.

In which region of the electromagnetic spectrum do thermal radiations lie?

stefan's law black body radiation heat transfer thermal properties physics

  1. Visible region

  2. Infrared region

  3. Ultraviolet region

  4. Microwave region

Show answer
Correct Option: B
Explanation:

Stefan's law states:
$E = \sigma T^{4}$ 
Normally temperature of a body never exceeds more than 1000K. 
As the temperature is quite less compared to the sun's temperature. Photons of this radiation have less energy and hence greater wavelength and fall into infrared region.
Alternately by Wein's displacement law
$ \lambda _{max} T = constant$
By applying above law on sun and normal body, wavelength of normal thermal radiation falls in infrared region.

All bodies emit heat energy from their surfaces by virtue of their temperature. This heat energy is called radiant energy of thermal radiation. The heat that we receive from the sun is transferred to us by a process which, unlike conduction or convection, does not require the help of a medium in the intervening space which is almost free of particles. Radiant energy travels in space as electromagnetic spectrum. Thermal radiations travel through vacuum with the speed of light. Thermal radiations obey the same laws of reflection and refraction as light does. They exhibit the phenomena of interference, diffraction and polarization as light does.
The emission of radiation from a hot body is expressed in terms of that emitted from a reference body (called the black body) at the same temperature. A black body absorbs and hence emits radiations of all wavelengths. The total energy E emitted by a unit area of a black body per second is given by $E =\sigma T^{4}$ where T is the absolute temperature of the body and $\sigma $ is a constant known as Stefan's constant. If the body is not a perfect black body, then $E =\varepsilon \sigma  T^{4}$where $\varepsilon $ is the emissivity of the body.

What is the SI unit of Stefan's constant?

stefan's law black body radiation heat transfer thermal properties physics

  1. $Js^{-1}K^{-4}$

  2. $Wm^{-1}K^{-4}$

  3. $Wm^{-2}K^{-4}$

  4. $Jm^{-2}K^{-4}$

Show answer
Correct Option: C
Explanation:

Stefan's law:
$E = \sigma T^{4}$ 
$\sigma = E/ T^{4}$
$E = energy/(area*time) = W m^{-2}$
$\sigma =  W m^{-2} K^{-4}$

Match the physical quantities given in Column I with their SI units given in Cloumn II :

Column-I Column-II
(a) Thermal conductivity (p) Wm$^{-2}$K$^{-4}$
(b) Stefans constant (q) m-K
(c) Wiens constant (r) J kg$^{-1}$K$^{-1}$
(d) Specific heat (s)Wm$^{-1}$K$^{-1}$
stefan's law black body radiation heat transfer thermal properties physics

  1. a-s, b-p, c-q, d-r

  2. a-s, b-p, c-r, d-q

  3. a-s, b-r, c-p, d-q

  4. a-r, b-s, c-p, d-q

Show answer
Correct Option: A
Explanation:

Thermal conductivity =$\dfrac{Watt}{Metre.Kelvin}$

Stefan's constant $=5.64\times$ ${10}^{-8}$$\dfrac { w }{ { m }^{ 2 }{ K }^{4}}$

Wien's constant $={\lambda}{\theta}={constant}={Metre}  {Kelvin}$

Specific heat $=\dfrac{Joule} {{Kg}  {Kelvin}}$

Which of the following statements is true/correct?

stefan's law black body radiation heat transfer thermal properties physics

  1. During clear nights, the temperature rises steadily upward near the ground level

  2. Newton's law of cooling, and approximate form of Stefan's law, is valid only for natural convection

  3. The total energy emitted by a black body per unit time per unit area is proportional to the square of its temperature in the Kelvin scale

  4. Two spheres of the same material have radii $1 m$ and $4 m$ and temperatures $4000 K$ and $2000 K$ respectively. The energy radiated per second by the first sphere is greater than that radiated per second by the second sphere

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Correct Option: B
Explanation:

During clear nights object on surface of earth radiate out heat and temperature falls. Hence option (a) is wrong.
The total energy radiated by a body per unit time per unit area $E \propto {T}^{4}$. Hence option (c) is wrong.
Energy radiated per second is given by
$\dfrac { Q }{ t } =PA\varepsilon \sigma { T }^{ 4 }$
$\Rightarrow \dfrac { { P } _{ 1 } }{ { P } _{ 2 } } =\dfrac { { A } _{ 1 } }{ { A } _{ 2 } } { \left( \dfrac { { T } _{ 1 } }{ { T } _{ 2 } }  \right)  }^{ 4 }={ \left( \dfrac { { r } _{ 1 } }{ { r } _{ 2 } }  \right)  }^{ 2 }\cdot { \left( \dfrac { { T } _{ 1 } }{ { T } _{ 2 } }  \right)  }^{ 2 }$
$={ \left( \dfrac { 1 }{ 4 }  \right)  }^{ 2 }\left( \dfrac { 4000 }{ 200 }  \right) =\dfrac { 1 }{ 1 } $
$\because    {P} _{1} = {P} _{2}$ hence option (d) is wrong.
Newton's law is an approximate from of Stefan's law of radiation and works well for natural convection. Hence option (b) is correct.

STATEMENT-1 : Animals curl into a ball, when they feel very cold.
STATEMENT-2 : Animals by curling their body reduces the surface area.

stefan's law black body radiation heat transfer thermal properties physics

  1. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

  2. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

  3. STATEMENT-1 is True, STATEMENT-2 is False

  4. STATEMENT-1 is False, STATEMENT-2 is True

Show answer
Correct Option: A
Explanation:

Both statements are true as the animals curl their body in very cold environment so they can reduce their surface area , and reduce the heat leaving their body in form of radiation as heat emitted is directly proportional to surface area.