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To verify the Stefan’s law by Electrical method – Physics Practical

Stefan’s law states  that the energy radiated per second by unit area of a black body at thermodynamic  temperature T is directly proportional to T 4 . The constant of proportionality is the Stefan constant, equal to 5.670400 × 10 –8 Wm –2 K –4 . This practical will verify this law using electrical method.

Practical of verification of Stefan’s law

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Experiments on the Stefan-Boltzmann-law

Introduction: Experiments on the Stefan-Boltzmann-law

A body with a certain temperature T emits electromagnetic radiation. This is noticeable in a forge, for example, when the piece of iron is brought to a high temperature. The spectrum of that radiation was already known around 1900, but it was not possible to derive it theoretically satisfactorily.

Two existing theoretical approximations failed to fully describe the intensity curve of this thermal radiation. While the Rayleigh-Jeans law only agreed with the experimental results for large wavelengths (low frequencies) and predicted the so-called ultraviolet catastrophe for increasing frequencies, Wien's radiation law only predicted the correct intensity profile for small wavelengths (high frequencies).

The German physicist Max Planck devoted himself to this theoretical challenge of so-called black body radiation. He postulated that electromagnetic radiation can only be emitted in energy portions E = h · f = h · c / λ. This can be described as the birth of quantum physics. Under this assumption, he received an expected intensity profile which was in excellent agreement with the experimental results.

The formula he derived for the spectral intensity curve is attached.

M_0_λ (λ, T) is the radiation power that is emitted by the surface element dA in the wavelength range between λ and λ + dλ in the entire half-space. In addition to the wavelength λ, the intensity profile depends on the temperature T:

What can you tell from the above spectra? The higher the temperature T, the further the radiation maximum shifts to the left in the direction of the smaller wavelength. At a temperature of 5800 K, for example, the maximum radiation is at a wavelength of approx. 500 nm. This roughly corresponds to the intensity curve of our sun. Their surface temperature must therefore also be in this range. The simple relationship between λ_max and temperature T describes Wien’s law of displacement, which reads:

λ_max [in μm] = 2897.8 / T [in K]

However, it can also be seen that the area under the curve increases sharply with increasing temperature. The area under the intensity curve corresponds to the energy radiated over all wavelengths per second and square meter, i.e. the total radiation intensity.

The two Austrian physicists Josef Stefan and Ludwig Boltzmann found a simple relationship for this, which is:

I_ges = σ · T ^ 4 with σ = Stefan-Boltzmann constant = 5.67 · 10 ^ –8 W / m ^ 2 · K ^ 4

A body with twice the temperature therefore emits a total of 16 times the energy per second and m ^ 2.

Step 1: The Black Body Radiation Source

For the black body radiation source you need a common light bulb. I use a 12V/5W one.

If you know the voltage U and the current I through the light bulb, the resistance R = U / I of the tungsten filament can be calculated very easily. However, this resistance R depends on the temperature T of the filament. The formular for calculating the temperature T with a given resistance R/R_room_temperature is attached. It works for tungsten filaments. You just need to know the resistance R_room_temperature of the filament at room temperature (bulb without a current/voltage just connected to the Ohmmeter) and the resistance R at the calculated temperature T.

Parts you will need:

• a AC/DC power supply. I use a 14V/60W model, but you can take f.e. your old laptop-power-supply: power supply
• a 5A DC/DC step down converter: buck-converter ebay
• a 12V/5W bulb with E10 thread: bulb ebay
• a E10 lamp holder: lamp holder ebay
• a multiturn 50kohm-potentiometer: 50k potentiometer ebay
• a 4 digits digital panelmeter for voltage and current: panelmeter ebay

Step 2: The DIY Thermopile

The radiation-power emitted by the light bulb at temperature T is measured with a so-called thermopile. You can simply build your own thermopile using a peltier-element and some other parts like plastic-tubes, a heatsink and some resistors for calibrating your apparatus.

The power (energy per second) or intensity (energy per second and per m²) of thermal radiation can be determined with a thermopile. The heart of a thermopile is a so-called Peltier element. This is based on the so-called thermo-electric voltage. If you combine two pieces of wire from different metals and bring the two contact points to different temperatures, you measure a so-called thermal voltage. In the Peltier element, many such “pieces of wire” are arranged one behind the other and alternately in order to intensify the effect. If, for example, a laser is shone on a side of the Peltier element blackened with soot, its radiation is absorbed and this side heats up slightly. As a result, a likewise very small thermal voltage U can be measured on the two cables of the Peltier element. This is a measure of the absorbed radiant power P.

To determine the relationship U = U (P), several SMD resistors are glued in series on one side of the Peltier element. I used 10 pieces of 1 kohm SMD-resistors connected in parallel to get a total resistance of 100 ohms.

Then you apply a certain voltage to this series of resistors, calculate the electrically supplied power P and measure the thermal voltage U. In my case, I got the linear relationship U = (1 / 11.26) * P. A power of e.g. 20 mW generated a thermal voltage of 1,776 mV. Conversely, a voltage of 1 mV corresponds to a radiation power of 11.26 mW. This relationship is required to determine the radiated power P in the Arduino program.

The Peltier element has an area of 40 × 40 mm² = 1600 mm². In order to be able to deduce the radiation intensity (power per m²), the power applied to the Peltier element must simply be multiplied by the factor (1000000/1600) = 625. If, for example, the power impinging on the Peltier element is 5 mW, then exactly 5 * 625 = 3125 mW = 3,125 W would impinge on 1 m², which then corresponds to an intensity of 3,125 W / m².

Step 3: The Arduino-part

Since the output voltages of the thermopile are very low (in the µV-mV range), they are first amplified 10 times with an operational amplifier of the AD8551 type. You need an operational amplifier with a very low input offset voltage!

Then the amplified voltage goes to the ADS1115 AD converter module. Finally, the 16x2 display shows the thermal voltage (in mV), the total radiation power (in mW) and the radiation intensity per m² (= radiation power * 625; in W / m²).

The displayed values can be set to 0 with a button if a voltage/power/intensity WITHOUT radiation source is displayed (= offset).

Attachments

Step 4: the experiment.

In this experiment to verify the Stefan-Boltzmann-law, the radiation-power emitted by the light bulb at temperature T is measured with the thermopile. To do this, the light bulb is positioned a certain distance in front of the thermopile and then the light bulb voltage U is slowly increased. The radiant power P recorded by the thermopile is then measured as a function of the bulb voltage U. With the values for the voltage U and current I you can simply determine the resistance R of the tungsten filament. With the formular shown in step 1 you can calculate the temperature T.

Finally you draw a graph with the radiant-power P as a function of T^4 - T_room^4. If everything works perfect, you should get a straight line. This means, that the radiation-power emitted by a black body increases with the fourth power of the temperature as the Stefan-Boltzmann-law predicts.

Step 5: The Simpler Variant

If you don't have a thermopile for direct measurement of the radiant power emitted by the light bulb, you can instead simply plot the electrical power supplied to the light bulb P = U · I as a function of T^4 - T_room^4. Because in the case of equilibrium, the electrical power supplied to the light bulb must correspond to the emitted radiation power!

You should also get a straight line for the graph P = P(T^4 - T_room^4) as a proof of the Stefan-Boltzmann-law.

If you are interested in more physics experiments, take a look at my youtube-channel or my homepage:

my youtube-channel

my homepage

Stefan's Law ( AQA A Level Physics )

Revision note.

Stefan's Law

• It's absolute temperature
• It's surface area
• The relationship between these is known as Stefan's Law or the   Stefan-Boltzmann Law , which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

• The Stefan-Boltzmann Law can be calculated using:
• P   = total power emitted across all wavelengths  (W)
• σ = the -8 W m -2 K -4 " data-title="Stefan-Boltzmann Constant" data-toggle="popover">Stefan-Boltzmann constant
• A   = surface area of the body (m)
• T   = absolute temperature of the body (K)
• The Stefan-Boltzmann law is often used to calculate the luminosity of celestial objects, such as stars
• Where r = radius of the star
• The Stefan-Boltzmann equation then becomes:
• L   = luminosity of the star (W)
• r   = radius of the star (m)
• T   = surface temperature of the star (K)

Worked example

The surface temperature of Proxima Centauri, the nearest star to Earth, is 3000 K and its luminosity is 6.506 × 10 23 W.

Calculate the radius of Proxima Centauri in solar radii and show your working clearly. 

Solar radius R ☉ = 6.96 × 10 8 m

Step 1: List the known quantities: 

• Surface temperature,  T = 3000 K
• Luminosity,  L  = 6.506 × 10 23 W
• Stefan's constant, σ = 5.67 × 10 −8 W m −2 K −4
• Radius of the Sun, R ☉ = 6.96 × 10 8 m

Step 2: Write down the Stefan-Boltzmann equation and rearrange for radius r

Step 3: Substitute the values into the equation 

Radius of Proxima Centauri:  R = 1.061 × 10 8 m

Step 4: Find the ratio of the radii of Proxima Centauri and the Sun

• Proxima Centauri has a radius which is about 0.152 times smaller than the Sun

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Verification of Stefan’s law

For black bodies, Stefan’s law is E = σ (T 4 -T 0 4 )     (1) Where E is the net amount of radiation emitted per second per unit area by a body at temperature T and surrounded by another body at temperature T 0 . σ is called Stefan’s constant. A similar relation can also hold for bodies that are not black. In such case, we can write P = C (T α -T 0 α )      (2) Where, P is the total power emitted by a body at temperature T surrounded by another at temperature T 0 , α is a power quite closed to 4 and C is some constant depending on the material and area of such a body. Further the relation can be put as P = C T α (1-T 0 α / T α )      (3) If T>> T 0 (e.g., T = 1500K, and T 0 ≈ 300K), we can write P = C T α     (4) Or Log 10 P = αLog 10 T + Log 10 C      (5) The graph between Log 10 P and Log 10 T should be a straight line whose slope gives α. When electrical current flows through filament of an electrical bulb, filament gets heated up. There are two modes through which the filament loses heat:

• Conduction and
• Electromagnetic radiation The heat conducted from the filament increases linearly with temperature. There is very little loss of heat due to convection. The filament resistance is directly proportional to the filament temperature and follows the relation R t = R o [ 1+α(T-T o ) ]     (6) Where R 0 is the resistance of the filament at 0 K. R t is the resistance of the filament at T (=t 0 C+273) K α is the temperature coefficient of the resistance of the filament T is the temperature of the filament in K and T 0 is the temperature of the filament at 0 K

Stefan’s Law

Table of Contents

Josef Stefan was an Austrian physicist who made significant contributions to understanding blackbody radiation. In 1879, he formulated Stefan’s Law, which states that the radiant energy of a blackbody is proportional to the fourth power of its temperature. This means that a perfect blackbody, an object that absorbs all radiation falling on it, emits more energy as it gets hotter. Stefan’s Law was a crucial step in studying blackbody radiation and paved the way for the quantum theory of radiation.

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Stefan began his career as a lecturer in mathematical physics in 1858 and became a professor of physics in 1863. By 1866, he was the director of the Physical Institute at the University of Vienna. Five years later, he empirically derived Stefan’s Law , which was later theoretically confirmed by Ludwig Boltzmann. This combined work led to the law being named the Stefan-Boltzmann Law. Understanding “ What is Stefan’s Law ” helps us appreciate how energy emission changes with temperature.

Also Check: Wein’s Displacement Law

Stefan’s Law, also known as the Stefan-Boltzmann Law, explains the power radiated by a black body based on its temperature and surface area. Widely used in thermodynamics and astrophysics, Stefan’s Law helps us understand how objects like stars and planets emit radiation. Scientists use Stefan’s Law to study these celestial bodies and their behaviors. Additionally, Stefan’s Law has practical applications, such as in designing solar panels and other energy conversion devices. By applying Stefan’s Law, we can improve the efficiency of these technologies and gain deeper insights into the natural world.

What is Stefan’s Law?

Stefan’s Law, also known as the Stefan-Boltzmann Law , states that the total energy emitted per unit surface area of a black body across all wavelengths per unit of time is directly proportional to the fourth power of the black body’s thermodynamic temperature and its emissivity. In simpler terms, Stefan’s Law explains that the power emitted by a black body increases rapidly with its temperature. This law highlights that as the temperature of a black body rises, the amount of energy it emits also increases significantly. Stefan’s Law is crucial for understanding the relationship between temperature and radiation in black bodies.

Stefan-Boltzmann Constant

The Stefan-Boltzmann Constant, named after physicists Josef Stefan and Ludwig Boltzmann, is a key component in Stefan’s Law. This constant, symbolized by the Greek letter σ, serves as the proportionality factor in Stefan’s Law.

Also Check: Ohm’s Law

Value of the Stefan-Boltzmann Constant

In the SI unit system, the value of the Stefan-Boltzmann Constant is approximately 5.67 × 10 -8 watts per square meter per Kelvin to the fourth power (W/(m ² K ⁴ )). Here are its values in different unit systems:

• SI Units: 5.670367 × 10 -8 W/(m ² K ⁴ )
• CGS Units: 5.6704 × 10 5 erg/(cm ² s K ⁴ )
• Thermochemistry Units: 11.7 × 10 8 cal/(cm ² day K ⁴ )
• U.S. Customary Units: 1.714 × 10 9 BTU/(ft ² hr °R ⁴ )

The dimension of the Stefan-Boltzmann Constant is [M] 1 [L] 0 [T] -3 [K] -4 , representing mass (M), length (L), time (T), and temperature (K).

Understanding the Stefan-Boltzmann Constant is crucial for applying Stefan’s Law, which helps determine the radiant energy emitted by a blackbody based on its temperature.

Examples of Stefan’s Law

Welding is a process that joins two pieces of metal by heating them until they fuse together. During welding, sparks can be seen because energy is radiated into the surroundings. This demonstrates Stefan’s Law, which explains how energy is emitted as heat.

Calculating the Radius of Stars

The radius of a star can be calculated based on its luminosity, which is the total power radiated by the star into space. This power depends on the star’s surface area and temperature. Stefan’s Law shows the relationship between an object’s surface area, temperature, and the rate of radiation it emits, helping astronomers determine the size of stars.

Aluminium Foil

Aluminium foil is another example of Stefan’s Law in action. This law explains that objects with lower emissivity radiate less energy. Since aluminium foil has a low emissivity of about 0.1 units, it effectively keeps food warmer for longer periods by minimizing radiation loss.

These examples illustrate how Stefan’s Law applies to various real-world scenarios, from welding and astronomy to everyday uses like aluminium foil.

Also Check: Kirchoff’s Law

Understanding Stefan’s Law in Physics

Stefan’s Law , also known as the Stefan-Boltzmann Law, explains that the total radiant heat power emitted by a surface is proportional to the fourth power of its absolute temperature. The formula for Stefan’s Law is:

In this formula:

• E represents the radiant heat energy emitted per unit area per second.
• σ is the Stefan-Boltzmann constant.
• T is the absolute temperature.

Using Stefan’s Law, we can estimate the energy emitted by the Sun. Given the Sun’s photosphere temperature is around 6000 K, the energy emitted can be calculated as:

E sun ​=ϵσT 4

E sun ​=1×5.67×10 −8 ×(6000) 4

This results in an approximate energy emission of:

E sun ​≈25.12×10 9 J m −2 s −1

Stefan’s Law helps us understand the relationship between temperature and emitted radiation, providing insights into various physical phenomena.

Applications of Stefan-Boltzmann Law

Stefan’s Law, also known as the Stefan-Boltzmann Law , has many real-world applications. Here are a few important uses:

• Stefan’s Law helps scientists calculate the luminosity of celestial bodies such as stars, planets, and galaxies.
• The law is also used to understand how greenhouse gases affect our atmosphere. By using Stefan’s Law, scientists can calculate the amount of energy absorbed by the atmosphere and predict the impact of rising temperatures.
• Engineers apply Stefan’s Law to compare surface temperatures of different materials. This helps them design more power-efficient systems that do not require active cooling.

In these ways, Stefan’s Law plays a crucial role in astronomy, environmental science, and engineering.

Also Check: Charles Law

FAQs on Stefan’s Law

Who proposed stefan's law and when.

The Austrian physicist Josef Stefan proposed Stefan's Law in 1879.

Is the Stefan-Boltzmann law applicable to all bodies?

No, the Stefan-Boltzmann law is only applicable to black bodies, which are surfaces that can absorb all incident heat radiation.

What is the value of Stefan-Boltzmann’s constant?

In physics, the value of Stefan-Boltzmann’s constant is approximately 5.670374419 × 10−8 watts per meter² per Kelvin⁴.

How was the Stefan-Boltzmann constant discovered?

The Stefan-Boltzmann constant is named after Josef Stefan and Ludwig Boltzmann. Josef Stefan discovered this constant based on John Tyndall's 1864 measurements of infrared emissions from a platinum filament. In 1879, Stefan deduced the proportionality to the fourth power of absolute temperature from Tyndall’s work. Ludwig Boltzmann then derived the constant from theoretical principles in 1884, using the work of Adolfo Bartoli on radiation pressure and applying thermodynamics principles.

What is Stefan's Law?

Stefan's Law, or the Stefan-Boltzmann Law, states that the total energy emitted per unit time and per unit area of a blackbody is proportional to the fourth power of the blackbody's temperature.

How is the Stefan-Boltzmann constant derived?

One way to derive the Stefan-Boltzmann constant is by integrating Planck’s Radiation Formula.

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Stefan Boltzmann Law - Formula, Derivation

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What Is Stefan Boltzmann Law?

According to Stefan Boltzmann law , the amount of radiation emitted per unit time from area A of a black body at absolute temperature T is directly proportional to the fourth power of the temperature.

u/A = σT 4  . . . . . . (1)

Where σ is Stefan’s constant = 5.67 × 10 -8 W/m 2 k 4

A body that is not a black body absorbs, and hence emits less radiation, given by equation (1)

For such a body, u = e σ AT 4 . . . . . . . (2)

Where e = emissivity (which is equal to absorptive power), which lies between 0 to 1.

With the surroundings of temperature T 0 , net energy radiated by an area A per unit time.

Δu = u – u o  = eσA [T 4  – T 0 4 ] . . . . . . (3)

⇒ Also Read

• Newton’s Law of Cooling

Stefan Boltzmann Law relates the temperature of the blackbody to the amount of power it emits per unit area. The law states that,

“The total energy emitted/radiated per unit surface area of a blackbody across all wavelengths per unit time is directly proportional to the fourth power of the black body’s thermodynamic temperature. ”

Derivation of Stefan Boltzmann Law

The total power radiated per unit area over all wavelengths of a black body can be obtained by integrating Plank’s radiation formula. Thus, the radiated power per unit area as a function of wavelength is:

• P is the power radiated
• A is the surface area of a blackbody
• λ is the wavelength of the emitted radiation
• h is Planck’s constant
• c is the velocity of light
• k is Boltzmann’s constant
• T is temperature.

On simplifying Stefan Boltzmann equation, we get:

On integrating both sides with respect to λ and applying the limits, we get;

The integrated power after separating the constants is:

This can be solved analytically by substituting:

As a result of substituting them in equation (1)

The above equation can be comparable to the standard form of integral:

Thus, substituting the above result, we get,

On further simplifying, we get,

⇒ P/A = σ T 4

Thus, we arrive at a mathematical form of Stephen Boltzmann law:

This quantum mechanical result could efficiently express the behaviour of gases at low temperature , which classical mechanics could not predict!

Problems with Stefan Boltzmann Law

Example: A body of emissivity, e = 0.75, the surface area = 300 cm 2 and temperature = 227 ºC is kept in a room at a temperature of 27 ºC. Using the Stephens-Boltzmann law, calculate the initial value of net power emitted by the body.

Using equation (3);

P = eσA (T 4  – T 0 4 )

= 69.4 Watts.

Example 2:  A hot black body emits energy at the rate of 16 J m -2 s -1, and its most intense radiation corresponds to 20,000 Å. When the temperature of this body is further increased, and its most intense radiation corresponds to 10,000 Å, then find the value of energy radiated in Jm -2 s -1 .

Wein’s displacement law is, λ m .T = b

Here, λ m becomes half, and the temperature doubles.

Now, from Stefan Boltzmann Law, e = sT 4

e 1 /e 2 = (T 1 /T 2 ) 4

⇒ e 2  = (T2/T1) 4  . e 1 = (2) 4 . 16

= 16.16 = 256 J m -2 s -1

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What are some common mistakes students make with Stefan's Law?

VTU Engineering Physics Practical(Lab)

Monday, august 11, 2008.

• VERIFICATION OF STEFAN’S LAW

5 comments:

its helped me. thanx

How can we calculate T temperature

Would you mind to help me with the values please

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