# Experiments on the Stefan-Boltzmann-law

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## 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:

## 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).

## 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.

Heureka...

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