Experiment No.2Inverse Square Law for Heat and Stefan-Boltzmann Law

I. Objectives

1. To show that the intensity of radiation on the surface is inversely proportional to the square of the distance of the surface from the radiation source.

2. To show that the intensity of radiation varies as the fourth power of the source temperature.

II. Materials/Equipments Needed

1. Thermal Radiation Unit

III. Equipment Set Up

IV. Theory

Inverse Square Law for Heat

The total energy dQ from an element dA can be imagined to flow through a hemisphere of radius r. a surface element on this hemisphere dA1 lies on a line making an angle with the normal and the solid angle subtended by dA1 at dA is dω1 = dA1/r2. (Note: Solid angle which is by definition the intercepted area on a sphere divided by r2).

If the rate of flow through dA1 is dQ1 then dQ1 = iφω1 dA where iφis the intensity of radiation in the φ direction i.e., dQ1 α 1/ r2.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that:

qb=δ (T S4−T A

4 )

Where:

qb = energy emitted by unit area of a black body surface (Wm-2)

δ = Stefan-Boltzmann constants (5.67 x 10-8 W m-2K4)

Ts = Source temperature (K)

TA = Temperature of radiometer and surroundings (K)

Incident Radiation and Emitted Radiation

The digital meter indicates the intensity of the radiation received by the radiometer (in W/m2) and not the radiation emitted by the heated surface at which it is pointed.

Though beyond the scope of this manual it can be shown that the relationship between radiation received by the sensor and radiation emitted by the heated source is as follows:

Hence as the sensor is removed from the heated surface and L increased the angle 9 decreases.

The model is exact for a black circular emitter and receiver. As it is not possible to utilize circular places due to the shape of the heater available an approximation is made to the “effective diameter” of a circular plate that would be equivalent to the rectangular plates supplied. This diameter is 126mm and hence r = 63mm.

For the diagram sin2 θ=( v2

v2+L2 )Hence q incident=qemitted∗( v2

v2+L2 )q incident=qemitted∗( 0.0632

0.0632+L2 )Or

Radiometer Reading (Wm2 )=qemitted∗( 0.0632

0.0632+L2 ) ¿

Or

qemitted=Radiometer Reading( Wm2 )∗( 0.0632+L2

0.0632 )Note that the sensor surface is 65mm from the centre line of the radiometer mounting rod. Hence for the position of the radiometer sensor 65mm must be subtracted from the marked centre of the detector stand.

V. Procedure

A. Inverse Square Law for Heat

1. Set power control to wide position and follow approximately 15 minutes for the heater to reach a stable temperature before beginning the experiment.

2. Record the radiometer reading (R) and the distance from the heat source (X) for a number of positions of the radiometer along horizontal track. It will take approximately 2 minutes for the radiometer to stabilize after being moved to each new position.

Initial Values of Variables to be Used

Distance from the heat source (X) = 800 mm. note that radiometer sensor surface is 65mm from the center line of detector carriage and therefore center line position will be 865 mm.

B. Stefan-Boltzmann Law

1. Set power control to maximum on the instrument console.2. Record the radiometer reading (R) and the temperature (T) at ambient

conditions then for selected increments of increasing temperature up to maximum within a practical range. Both readings should be noted simultaneously at any given point. It is recommended that while waiting for the black plate temperature to stabilize between each increase of the heater power control the reflective disc is placed in the radiometer aperture to prevent heating effects and zero drifts.

Initial Values of Variables to be Used

Distance from radiometer to black plate (X) = 200 mmDistance from black plate to heat source (Y) = 50 mm

VI. Results and Discussion

A. Inverse Square Law of Heat

B. Stefan-Boltzmann Law

Readings CalculationsTemp

Reading(T)

RadiometerReading

(R)Ts TA Qb = 11.07*R Qb = σ(Ts

4 - TA4)

℃ W/m2 K K W/m2 W/m2

56 20 329.15 300.15 221.4 205.3458 21 331.15 300.15 232.47 221.6768 27 341.15 300.15 298.89 307.8484 41 357.15 300.15 453.87 462.3889 46 362.15 300.15 509.22 515.14103 57 376.15 300.15 630.99 674.94133 90 406.15 300.15 996.3 1082.76

VII. Conclusion

The inverse square law is important as it gives a measure of how the intensity of radiation falls off with distance from a source. This has implications for the storage and use of radioactive sources. A point source of gamma rays emits in all directions about the source. It follows that the intensity of the gamma rays decreases with distance from the source because the rays are spread over greater areas as the distance increases.Any object at elevated temperature gives off light known as thermal radiation. The hotter an object gets the more light it emits. As the temperature of the object increase, it emits most of its light at higher and higher energies. As one moves further from the source, the emitted particles are dispersed and are therefore less likely to strike the radiation measurement device. Since the area over which the emissions are dispersed is that of an expanding sphere about the source, the radiation intensity follows the inverse square law as one move away from the source

VIII. References

http://www.s-cool.co.uk/a-level/physics/radioactive-decay-equations/revise-it/inverse-square-law-and-radiation

Appendices

Appendix A: Experimental Data

A. Inverse Square Law of Heat

Distance, X (mm) 100Radiometer Reading, R (W/m2)

B. Stefan-Boltzmann Law

Temperature Reading (°C)Radiometer Reading, R (W/m2)TA (K)

Appendix B: Sample Computation

A. Inverse Square Law for Heat

- Log-Log Plot

Log (Distance, X (mm))Log (Radiometer Reading, R (W/m2))

B. Stefan-Boltzmann Law

Appendix C: Attendance Sheet