1. Problem5. Mirage

2. Mirage Create a mirage like a road or desert mirage in the laboratory and study its parameters.

3. Introduction • What is mirage? • How is it formed? • Which are parameters of a mirage?

4. Experiment Effect: heated surface observedobject

5. Formation of mirage Snell’slaw: φ1 n1 – index of refraction of first medium n2 – index of refraction of second medium φ1 – ray angle in first medium φ2 – ray angle in second medium n1 φ2 n2 n1 > n2 → ray is refracting away from vertical

6. Formation of mirage - applying Snell’s law on sequence of parallel planes of infinitesimal thickness whose index of refraction is changing monotonously: Why would ray return ???

7. Formation of mirage In one moment (sooner or later), if sequence of ni isn’t convergent, will be: Total reflection occurs and ray is returning.

8. Mirage parameters • index of refraction dependence of temperature • temperature dependence of shift in direction parallel with temperature gradient n1 n2 n3 … nN index of refraction temperature

9. Experiment horizontal line vesselwith flourescein termometer

10. Experiment • temperature distribution was measured with thermometer put in the water • rays are photographed and deviation from horizontal line is measured

11. Theoretical model It is sufficient to derive ray equation. • assumptions: • linear dependence of temperature of depth i.e.: • linear dependence of index of refraction of temperature i.e.:

12. x y Theoretical model We will use following coordinate system and following equations: -dy φ dx Snell’s law in differential form: …….

13. Theoretical model Ray equation: y0 – initial depth n0 – initial index of refraction φ0 – initial ray angle

14. Results

15. Results kT1=-0.78Kmm-1 kT3=-0.40Kmm-1 kT2=-0.32Kmm-1 kT4=-0.27Kmm-1

16. Results

17. Results

18. Results kn1=-4.86*10-4K-1 kn3=-4.58*10-4K-1 kn2=-4.68*10-4K-1 kn4=-4.75*10-4K-1

19. Results For coefficient of dependence of index of refraction of water on temperature we got (in interval ≈ 318K – 358K): According to literature:

20. Conclusion • we created mirage in a laboratory • we constructed theoretical model whose results are good compared to literature • mirage parameters are dependence of temperature on depth and dependence of index of refraction on temperature • for water: • for practical reasons we photographed and measured ray in water but we have also created mirage in air