Comprehensive Study Notes: Optics, Diffraction, and Special Relativity

Geometrical Optics: Concave Mirror Calculations## Theoretical Context and Problem ParametersConcave mirrors (konkavno ogledalo) are converging mirrors. The problem explores the relationship between the radius of curvature, focal length, object position, and image position using the mirror equation.## Problem 1: Determining Object Distance and Magnification- Given Data:- Radius of curvature: $R = 50\,\text{cm}$- Image distance from vertex ($T$): $l = 70\,\text{cm}$ - Light source: Point source (tačkasti svjetlosni izvor) located on the main optical axis.- Focal Length Calculation:- The focal length ($f$) of a concave mirror is half the radius of curvature:$f = \frac{R}{2} = \frac{50\,\text{cm}}{2} = 25\,\text{cm} = 0.25\,\text{m}$- Object Distance ($p$) Calculation:- Using the mirror equation:$\frac{1}{f} = \frac{1}{p} + \frac{1}{l}$- Rearranging for $p$:$\frac{1}{p} = \frac{1}{f} - \frac{1}{l}$- Substituting the values in cm:$\frac{1}{25} = 0.04$, $\frac{1}{70} \approx 0.014$- $\frac{1}{p} = 0.04 - 0.014 = 0.026$- Finding the reciprocal:$p = \frac{1}{0.026} = 38.46\,\text{cm}$- Magnification ($u$) Calculation:- Based on the ratio of distances from the vertex:$u = \frac{l}{p} = \frac{70}{38.46} \approx 1.81$# Physical Optics: Interference and Wavelength MeasurementThis section details calculations involving wave phenomena, specifically wavelength determination likely related to Newton's Rings or a similar interference setup.## Problem 2: Wavelength Determination- Parameters:- Curvature radius: $R = 16\,\text{cm} = 16 \times 10^{-2}\,\text{m}$- Ring order: $k = 4$- Ring radius: $r = 0.8\,\text{mm} = 8 \times 10^{-4}\,\text{m}$- Calculations for Wavelength ($\lambda$):- Standard formula for ring radius: $r_k^2 = k \times R \times \lambda$- Rearranging for lambda:$\lambda = \frac{r^2}{k \times R}$- Substituting values:$\lambda = \frac{(8 \times 10^{-4})^2}{4 \times 16 \times 10^{-2}} = \frac{64 \times 10^{-8}}{64 \times 10^{-2}} = 10^{-6}\,\text{m}$- Secondary Data in Records:- Records show alternative calculations or measurements:- $n = \frac{8}{112 \times 5.6 \times 10^{-4}} = 14.4 \times 10^{-4}\,\text{m}$- $R = 144 \times 10^{-6}\,\text{m}$- $R = 0.571 \times 10^{-6}\,\text{m}$- Final result listed for wavelength: $\lambda = 571 \times 10^{-9}\,\text{m} = 571\,\text{nm}$# Physical Optics: Diffraction GratingA diffraction grating (optička rešetka) is used to disperse light into its constituent wavelengths based on the grating constant.## Problem 3: Analysis of Diffraction Grating- Given Data:- Wavelength: $\lambda = 0.540\,\mu\text{m} = 0.54 \times 10^{-6}\,\text{m}$- Grating characteristics: $N = 2000$ lines over a length $l = 1\,\text{cm}$- Grating Constant ($d$) Calculation:- $d = \frac{l}{N} = \frac{0.01\,\text{m}}{2000} = 5 \times 10^{-6}\,\text{m}$ (or $0.0005\,\text{cm}$)- Diffraction Angle for Third Order ($k=3$):- Working from the grating equation: $\sin(\alpha_k) = \frac{k \times \lambda}{d}$- $\sin(\alpha_3) = \frac{3 \times 0.54 \times 10^{-6}}{5 \times 10^{-6}} = \frac{1.62}{5} = 0.324$- Angle calculation: $\alpha_3 = \arcsin(0.324) = 18.91^{\circ}$- Maximum Order of Diffraction ($k_{max}$):- Limit condition: $\sin(\alpha) \le 1 \implies k \le \frac{d}{\lambda}$- $k \le \frac{5 \times 10^{-6}}{0.54 \times 10^{-6}} \approx 9.259$- Conclusion: The maximum observable order is $k = 9$. As noted: "ne postoji k=10" (order 10 does not exist for this setup).# Special Relativity: Time DilationRelativity addresses how measurements of physical quantities change when moving at high speeds relative to an observer.## Problem 4: Relativistic Time Interval- Scenario Description:- A space object moves at a velocity of $v = 0.6c$.- An observer on Earth measures an event in their own system that lasts $\Delta t_0 = 10\,\text{s}$.- Time Dilation Formula:- $\Delta t' = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}$- Calculation performed in notes:- $\Delta t' = \frac{10}{\sqrt{1 - (0.6)^2}} = \frac{10}{\sqrt{1 - 0.36}} = \frac{10}{\sqrt{0.64}}$- $\Delta t' = \frac{10}{0.8} = 12.5\,\text{s}$- Conclusion: In the observer's frame (Earth), the phenomenon duration is calculated as $12.5\,\text{s}$.# Spectrometry and Physical Definitions## Analytical Tools and Concepts- Spectrometer Components:- Instruments utilize either a prism (prizma) or a grating (rešetka).- Spectral Analysis:- Used for determining the composition of stars (sastava zvijezdi) and objects around Earth (Zemlje).- General Constants/Measurements Mentioned:- Speed of sound (ne zvuka)- Stopping/Standing (sto)## Additional Exercises and ReferencesThe notes include references to specific problems likely found in a standardized textbook or workbook, all noted as relating to "kontrakcija" (length contraction):- 115 - 1: Length Contraction Application- 113 - 1: Length Contraction Application- 117 - 1: Length Contraction Application- 112 - 2: Length Contraction Application- 116 - 571: Length Contraction Application