These post series aim to solve examples and problems from Introduction to Optics book which is writen by Frank L. Pedrotti, Leno M. Pedrotti and Leno S. Pedrotti.
Chapter 1: Nature of Lights
Example 1
An electron is accelerated to a kinetic energy Ek of 2.5 MeV. (a) Determine its relativistic momentum, de Broglie wavelength, and speed. (b) Determine the same properties for a photon having the same total energy as the electron.
Solution:
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// //Introduction to Optics //Chapter 1 Example 1 //Cemalettin Yılmaz #include <iostream> #include <math.h> const long double c = 3.0e+8; //speed of light const long double h = 6.626e-34; //Plank's constatn long double m = 9.11e-31; //mass long double mc2 = (m)*pow(c,2.0); //mc^2 const long double eV = 1.6e-19; // 1eV = 1.6*10^-19 J auto calculate_p (auto E, auto mass){ return (sqrt(pow(E,2)-pow((mass)*pow(c,2),2)))/c; }; auto calculate_wl (auto p){ return h/p; }; auto calculate_v (auto E, auto p){ return (p*pow(c,2))/E; }; int main () { long double Ek = 2.5e+6; //2.5MeV std::cout.precision(5); std::cout << "(A) For m = 9.11e-31 "; std::cout << "The electron's total energy E = mc^2 + Ek\n" << std::scientific; std::cout << "mc^2 = "<< mc2 << " J \n"; std::cout << "Ek = "<< Ek << " eV \n"; mc2=mc2/eV; std::cout << "Convert J to eV for mc^2: " << mc2 << "eV \n"; std::cout << "Total energy E = " << mc2 + Ek << " eV\n"; std::cout << "Total energy E = " << (mc2 + Ek)*eV << " J\n"; std::cout << "p (momentum) = " << calculate_p((mc2 + Ek)*eV,m) << " kg.m/s\n"; std::cout << "Wave Lenght(^) = " << calculate_wl(calculate_p((mc2 + Ek)*eV,m)) << " m\n"; std::cout << "V speed = " << calculate_v((mc2 + Ek)*eV,calculate_p((mc2 + Ek)*eV,m)) << " m/s\n"; std::cout << "(B) For m = 0 "; m = 0.0; std::cout << "p (momentum) = " << calculate_p((mc2 + Ek)*eV,m) << " kg.m/s\n"; std::cout << "Wave Lenght(^) = " << calculate_wl(calculate_p((mc2 + Ek)*eV,m)) << " m\n"; std::cout << "V speed = " << calculate_v((mc2 + Ek)*eV,calculate_p((mc2 + Ek)*eV,m)) << " m/s\n"; return 0; } |
Output:
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// (A) For m = 9.11e-31 The electron's total energy E = mc^2 + Ek mc^2 = 8.19900e-014 J Ek = 2.50000e+006 eV Convert J to eV for mc^2: 5.12438e+005eV Total energy E = 3.01244e+006 eV Total energy E = 4.81990e-013 J p (momentum) = 1.58322e-021 kg.m/s Wave Lenght(^) = 4.18515e-013 m V speed = 2.95628e+008 m/s (B) For m = 0 p (momentum) = 1.60663e-021 kg.m/s Wave Lenght(^) = 4.12415e-013 m V speed = 3.00000e+008 m/s |
Example 2
A certain sensitive radar receiver detects an electromagnetic signal of frequency 100 MHz and power (energy/time) 6.63 x 10^-16 J/s
a. What is the wavelength of a photon with this frequency?
b. What is the energy of a photon in this signal? Express this energy in J and in eV.
c. How many photons/s would arrive at the receiver in this signal?
d. What is the energy (in J and in eV) of a visible photon of wavelength 555 nm?
e. How many visible (wavelength 555 nm) photons/s would correspond to a detected power of 6.63 x 10^-16 J/s
f. What is the energy (in J and in eV) of an X-ray of wavelength 0.1 nm?
g. How many X-ray photons/s would correspond to a detected power of 6.63 x 10^-16 J/s
Solution:
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// //Introduction to Optics //Chapter 1 Example 2 //Cemalettin Yılmaz #include <iostream> #include <math.h> const long double c = 3.0e+8; //speed of light const long double eV = 1.6e-19; // 1eV = 1.6*10^-19 J const long double h = 6.626e-34; //Plank's constatn long double wave = 0.0e+1; auto calculate_wl (auto v){ return c/v; }; auto calculate_energy (auto v,bool Joule, bool wl,auto Wavelength){ if (wl){ if (Joule) return h*c/Wavelength; else return (h*c/Wavelength)/eV; } else{ if (Joule) return h*v; else return (h*v)/eV; } }; auto calculate_number_photon (auto E,auto P){ return P/E; }; int main () { long double f = 100e+6; // 100MHz long double P = 6.63e-16; //Power J/s std::cout.precision(3); std::cout << "a) Wavelength of a photon with this frequency " << f << " is = "<< std::scientific; std::cout << calculate_wl(f) << "\n"; std::cout << "b) The energy of a photon in this signal is = "<< calculate_energy(f,false,false,wave) << " eV or " << calculate_energy(f,true,false,wave) << " J\n"; std::cout << "c) Number of photons is = " << calculate_number_photon(calculate_energy(f,true,false,wave),P) << "\n"; wave = 555e-9; std::cout << "d) The energy of a photon with 555nm Wavelength = " << calculate_energy(f,false,true,wave) << " eV or " << calculate_energy(f,true,true,wave) << " J\n"; std::cout << "e) Number of photons is (555nm) = " << calculate_number_photon(calculate_energy(f,true,true,wave),P) << "\n"; wave = 1e-10; std::cout << "f) The energy of a photon with 0.1nm Wavelength = " << calculate_energy(f,false,true,wave) << " eV or " << calculate_energy(f,true,true,wave) << " J\n"; std::cout << "g) Number of photons is (0.1nm) = " << calculate_number_photon(calculate_energy(f,true,true,wave),P) << "\n"; return 0; } |
Output:
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// a) Wavelength of a photon with this frequency 1e+008 is = 3.000e+000 b) The energy of a photon in this signal is = 4.141e-007 eV or 6.626e-026 J c) Number of photons is = 1.001e+010 d) The energy of a photon with 555nm Wavelength = 2.239e+000 eV or 3.582e-019 J e) Number of photons is (555nm) = 1.851e+003 f) The energy of a photon with 0.1nm Wavelength = 1.242e+004 eV or 1.988e-015 J g) Number of photons is (0.1nm) = 3.335e-001 |
Example 3
Consider a 5 milliwatt Helium-Neon laser emitting a “pencil-like” beam with a divergence angle a of 1.3 milliradians. The laser cavity is designed so that the beam emerges from a surface area 2.5*10^-3 cm^2 at the output mirror.
a. Determine the solid angle in terms of R and a
b. Determine the radiance of this laser light source in units of W/cm2*sr.
Solution:
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// //Introduction to Optics //Chapter 1 Example 3 //Cemalettin Yılmaz #include <iostream> #include <math.h> long double calculate_r (long double a, long double R){ return R*tan(a/2); }; long double calculate_delta_At (long double a, long double R){ return M_PI*(pow(calculate_r(a,R),2)); }; long double calculate_solid_angle (long double a, long double R=1){ return calculate_delta_At(a,R)/pow(R,2); }; auto calculate_Ie (auto P, auto angle){ return (P/angle); }; auto calculate_radiance (auto Ie, auto area, auto angle){ return Ie/(area*cos(angle*M_PI/180)); }; int main () { long double total_radiant_power = 5e-3; // 5 miliWatt long double divergence_angle = 1.3e-3; // 1.3 miliradians long double surface_area = 2.5e-3; //2.5 cm^2 long double angle = 0.0; std::cout.precision(3); std::cout << "a)Solid Angle (w) in terms of R's and a's " << std::scientific; std::cout << calculate_solid_angle(divergence_angle) << " sr \n"; std::cout << "b)Radiance of laser light source"; std::cout << calculate_radiance(calculate_Ie(total_radiant_power,calculate_solid_angle(divergence_angle)),surface_area,angle) << " W/(cm^2*sr) \n"; return 0; } |
Output:
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// a)Solid Angle (w) in terms of R's and a's 1.327e-006 sr b)Radiance of laser light source1.507e+006 W/(cm^2*sr) |
Problems
1-6 , 9-10, 12-17 Problems
Solution:
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// //Introduction to Optics //Chapter 1 Problems //Cemalettin Yılmaz #include <iostream> #include <math.h> const long double h = 6.626e-34; //Plank's constatn long double m = 9.11e-31; //mass long double m_p = 1.673e-27; //mass const long double c = 3.0e+8; //speed of light long double mc2 = (m)*pow(c,2.0); //mc^2 const long double eV = 1.6e-19; // 1eV = 1.6*10^-19 J auto calculate_p (auto E, auto mass,bool Joule){ //Relativistic momentum if (Joule) return (sqrt(pow(E,2)-pow((mass)*pow(c,2),2)))/c; else return (sqrt(pow(E,2)-pow((mass)*pow(c,2)/eV,2)))/c; }; auto calculate_p_mass_speed (auto speed, auto mass){ return mass*speed; }; auto calculate_wl_mass_speed (auto speed, auto mass){ return h/calculate_p_mass_speed(speed,mass); }; auto calculate_wl (auto p){ return h/p; }; auto calculate_power (auto e, auto t){ return e/t; }; auto energy_of_light (auto n, auto v){ return n*v*h; }; auto convert_WLtoV_photons (auto wl){ return c/wl; }; auto convert_VtoWL_photons (auto v){ return c/v; }; auto calculate_energy (auto v,bool Joule, bool wl,auto Wavelength){ if (wl){ if (Joule) return h*c/Wavelength; else return (h*c/Wavelength)/eV; } else{ if (Joule) return h*v; else return (h*v)/eV; } }; auto calculate_mc2 (auto m){ return ((m)*pow(c,2.0)); }; auto calculate_number_of_photon (auto E,auto v){ return (E/(h*v)); }; auto calculate_time_with_distance (auto d, long double speed=c){ return (d/speed); }; auto calculate_radiant_intensity (auto p){ return (p/(4*M_PI)); }; auto calculate_radiant_excitance (auto p,auto area){ return (p/area); }; auto calculate_radiant_irradiance (auto p,auto r){ return (p/(4*M_PI*pow(r,2))); }; auto calculate_radiant_flux_hole (auto E,auto r){ return (E*M_PI*pow(r,2)); }; auto divergence_angle (auto r,auto L){ return tan(r/L); }; long double calculate_delta_At (long double r){ return M_PI*(pow(r,2)); }; long double calculate_solid_angle (long double r,long double L){ return calculate_delta_At(r)/pow(L,2); }; auto calculate_Ie (auto P, auto angle){ return (P/angle); }; auto calculate_radiance (auto Ie, auto area, auto angle){ return Ie/(area*cos(angle*M_PI/180)); }; int main () { std::cout.precision(3); std::cout << "1.Problem \na)de Broglie wavelength of a golf ball of mass 50 g moving at 20m/s is " << std::scientific; std::cout << calculate_wl_mass_speed(20,50e-3) << " m \n"; std::cout << "b)de Broglie wavelength of an electron with kinetic energy of 10eV " << calculate_wl(calculate_p((mc2/eV + 10)*eV,m,true)) << " m \n"; std::cout << "2.Problem \nHumans eyes threshold sensivity is 100 photons per second in this case threshold in watts of power for 550nm wavelength is "; std::cout << calculate_power(energy_of_light(100,convert_WLtoV_photons(550e-9)),1) << " W \n"; std::cout << "3.Problem \nEnergy of light photons at the ends of visible spectrum that is, at wavelengths of 380 and 770nm\na)380nm: "; std::cout << calculate_energy(1,false,true,380e-9) << " eV \nb)770nm: "; std::cout << calculate_energy(1,false,true,770e-9) << " eV \n"; std::cout << "4.Problem \nWavelength and momentum of an electron whose energy equals the rest mass energy of electron is:\nMomentum: "; std::cout << mc2/c << " J \nb)Wavelenght: "; std::cout << calculate_wl(mc2/c) << " m \n"; std::cout << "5.Problem \nRest mass energy of electron is: "; std::cout << (mc2)/eV << " eV\n"; std::cout << "6.Problem \nRelativistic momentum for 1 million volts potential difference is: "; std::cout << calculate_p(((mc2/eV)+1e+6),m,false)*c << " eV/c\n"; std::cout << "9.Problem \nA proton is accelerated to kinetic energy of 2 billion electron volts (2GeV) so that\na)Momentum is : "; std::cout << calculate_p((calculate_mc2(m_p)+(2e+9*eV)),m_p,true) << " kg.m/s\nb)de Broglie wavelength is: "; std::cout << calculate_wl(calculate_p((calculate_mc2(m_p)+(2e+9*eV)),m_p,true)) << " kg.m/s\nc)Wavelength of a photon with the same total energy : "; std::cout << calculate_wl(calculate_p((calculate_mc2(m_p)+(2e+9*eV)),0,true)) << " m\n"; std::cout << "10.Problem \nSolar radiation is incident at the Earth's surface at an average of 1000W/m^2 on a surface normal to the rays. For a mean wavelength of 550 nm so that,\nNumber of photons falling on 1cm^2 of surface each second : "; std::cout << calculate_number_of_photon((1000*1e-4),convert_WLtoV_photons(550e-9)) << "\n"; std::cout << "12.Problem \nBand of frequencies of electromagnetic radiation capable of producing a visiual sensation in the normal eye (380-770nm) is : "; std::cout << "Band of frequencies " << convert_WLtoV_photons(380e-9) << " to " << convert_WLtoV_photons(770e-9) << " Hz\n"; std::cout << "13.Problem \nLength of half-wave dipole antenna designed to brodcast FM radia waves at 100MHz is : "; std::cout << convert_VtoWL_photons(100e+6)/2 << " m\n"; std::cout << "14.Problem \nLength of quater-wave dipole antenna designed to brodcast FM radia waves at 90MHz is : "; std::cout << convert_VtoWL_photons(90e+6)/4 << " m\n"; std::cout << "15.Problem \nA soprano's voice is sent by radio waves to a listener in a 90 km away. \na)Voice reach to the listener : "; std::cout << calculate_time_with_distance(90e+3) << " second\nb)If sound speed is 340m/s with same time distance will be "; std::cout << 340*calculate_time_with_distance(90e+3) << " m\n"; std::cout << "16.Problem \nA small, monochromatic light source, radiating at 500 nm , is rated at 500W\na)If the source radiates uniformly in all directions, radiant intensity is :"; std::cout << calculate_radiant_intensity(500) << " W/sr\nb)If the surface area of the source is 5cm^2, radiant excitance is: "; std::cout << calculate_radiant_excitance(500,5e-4) << " W/m^2\nc)Irradiance on a screen situated 2 m from the source, with its surface normal to the radiant flux so that irradiance is : "; std::cout << calculate_radiant_irradiance(500,2) << " W/m^2\nd)If the receiving screen contains a hole with diameter 5 cm, number of radiant flux gets trough is: "; std::cout << calculate_radiant_flux_hole(calculate_radiant_irradiance(500,2),(5e-2/2)) << " W\n"; std::cout << "17.Problem \n15 mW helium-neon laser beam delivers a spot of light 5mm in diameter across a room 15 m wide. The beam radiates from a small circular area of diameter 0.5 mm at the output mirror of the laser. Assume that the beam irradiance is constant across the diverging beam. So that;\na)Beam divergence angle of this laser is :"; std::cout << divergence_angle((5e-3)/2,15)*180/M_PI << " degree\nb)What solid angle is the laser sending its beam : "; std::cout << calculate_solid_angle((5e-3)/2,15) << " sr\nc)What is the radiance at the spot on the wall 15 m from the laser: "; std::cout << calculate_radiant_irradiance(1.5e-3,(5e-3)/2)*4 << " W/m^2\nd)What is the radiance of the laser"; std::cout << calculate_radiance(calculate_Ie(1.5e-3,calculate_solid_angle((5e-3)/2,15)),M_PI*pow(2.5e-4,2),0) << " W\n"; return 0; } |
Output:
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// 1.Problem a)de Broglie wavelength of a golf ball of mass 50 g moving at 20m/s is 6.626e-034 m b)de Broglie wavelength of an electron with kinetic energy of 10eV 3.881e-010 m 2.Problem Humans eyes threshold sensivity is 100 photons per second in this case threshold in watts of power for 550nm wavelength is 3.614e-017 W 3.Problem Energy of light photons at the ends of visible spectrum that is, at wavelengths of 380 and 770nm a)380nm: 3.269e+000 eV b)770nm: 1.613e+000 eV 4.Problem Wavelength and momentum of an electron whose energy equals the rest mass energy of electron is: Momentum: 2.733e-022 J b)Wavelenght: 2.424e-012 m 5.Problem Rest mass energy of electron is: 5.124e+005 eV 6.Problem Relativistic momentum for 1 million volts potential difference is: 1.423e+006 eV/c 9.Problem A proton is accelerated to kinetic energy of 2 billion electron volts (2GeV) so that a)Momentum is : 1.486e-018 kg.m/s b)de Broglie wavelength is: 4.459e-016 kg.m/s c)Wavelength of a photon with the same total energy : 4.224e-016 m 10.Problem Solar radiation is incident at the Earth's surface at an average of 1000W/m^2 on a surface normal to the rays. For a mean wavelength of 550 nm so that, Number of photons falling on 1cm^2 of surface each second : 2.767e+017 12.Problem Band of frequencies of electromagnetic radiation capable of producing a visiual sensation in the normal eye (380-770nm) is : Band of frequencies 7.895e+014 to 3.896e+014 Hz 13.Problem Length of half-wave dipole antenna designed to brodcast FM radia waves at 100MHz is : 1.500e+000 m 14.Problem Length of quater-wave dipole antenna designed to brodcast FM radia waves at 90MHz is : 8.333e-001 m 15.Problem A soprano's voice is sent by radio waves to a listener in a 90 km away. a)Voice reach to the listener : 3.000e-004 second b)If sound speed is 340m/s with same time distance will be 1.020e-001 m 16.Problem A small, monochromatic light source, radiating at 500 nm , is rated at 500W a)If the source radiates uniformly in all directions, radiant intensity is :3.979e+001 W/sr b)If the surface area of the source is 5cm^2, radiant excitance is: 1.000e+006 W/m^2 c)Irradiance on a screen situated 2 m from the source, with its surface normal to the radiant flux so that irradiance is : 9.947e+000 W/m^2 d)If the receiving screen contains a hole with diameter 5 cm, number of radiant flux gets trough is: 1.953e-002 W 17.Problem 15 mW helium-neon laser beam delivers a spot of light 5mm in diameter across a room 15 m wide. The beam radiates from a small circular area of diameter 0.5 mm at the output mirror of the laser. Assume that the beam irradiance is constant across the diverging beam. So that; a)Beam divergence angle of this laser is :9.549e-003 degree b)What solid angle is the laser sending its beam : 8.727e-008 sr c)What is the radiance at the spot on the wall 15 m from the laser: 7.639e+001 W/m^2 d)What is the radiance of the laser8.754e+010 W |
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