Problems on Properties of Light & Modern Physics
Speed of a wave
Speed (c) of any wave is given in terms of its frequency (n) and wavelength (l)
Energy (E) of a particle is given by the frequency of the wave that can be imagined to be associated with it; as is the momentum (p) given in terms of the wavelength. The constant of proportionality h ~ 6.634 x 10-34 Joule-sec is called as Planck’s constant.
Frequency / wavelength of light changes when the source is in motion w.r.t. the observer. The shift (z) is given as the change in the quantity, per unit original quantity. The shift is thus related to the relative speed of the object (as seen by the observer) as a fraction of light-speed.
The Bohr Atom
Bohr’s concept of atomic structure was based on Planck’s hypothesis that energy is given and taken in packets (called quanta) and not continuously as was previously believed. Bohr assumes that the electrons perform uniform circular motion around the positive nucleus, where the centripetal force needed is provided by Coulomb’s force, i.e. electrostatic interaction, which depends upon the charges of the nuclei (Q) and electron (q) and the radius of orbit (R). Next Bohr assumes that the electrons cannot remain in any orbit but only those in which the angular momentum (L) is an integral multiple of h/2p. Finally he also assumes that the electrons will radiate (i.e. emit a photon) only when jumping from an outer orbit to the inner one and that the energy of the photon would be the energy difference between these orbits. This leads us to the following formulae.
Energy of Orbit
Using Bohr’s first postulate, we acquire the velocity of orbit (v) and also the kinetic energy (K). As the electrostatic force of attraction is a central force, the electrostatic potential energy (U) is given as ~ F.R. Hence the total energy of an electron in an orbit around the nucleus is proportional to 1/R. Also, the energy is negative as the electron is bound to the nucleus. U is also called as the binding energy of the electron and if the electron is the outermost in the atom, supplying positive energy equal to the binding energy will release the electron from the atom, reducing the atom to an ion. This positive energy is called as the ionization potential.
Quantization of Space & Energy
The second postulate restricts the electrons to have orbits with only certain radii, as can be seen above. The term inside the bracket in the last equation above is a constant giving R µ n2. Using this formula for R in the equation for total energy of orbit we get;
Thus energy is quantized, i.e. restricted to be in certain packets only depending upon the value of n. Hence n is called the Principal Quantum Number and it determines largely the energy, momentum and hence the behavior of an electron in any orbit around the nucleus. The constant in the bracket is called as the Rydberg Number. Thus the energy of the photon emitted when an electron jumps from an outer to inner orbit or the photon absorbed in jumping to the outer orbit in the first place, is given as DE.
Q1) A photon (particle of light) has wavelength 7000 A0. What is its frequency?
Q2) If the frequency of the photons in a beam of light coming to us from a star is 5 GHz (Giga = billion), then in what wavelength would we see the light.
Q3) A star is moving towards us at a speed of 100 km/s. What would be the shift in the frequency of the Sodium spectral lines of wavelength 5890 A0? (Hint: First find the frequency using the wavelength and then compute )
Q4) A star has recently exploded as a supernova, and the gaseous material emitted from it is coming towards us at a certain speed. If the frequency of the Hydrogen light coming from the nebula (gaseous shell), with l = 21 cm, is shifted by 100 A0, then what is the speed of the nebula? (Hint: Use Doppler’s law and also take care of conversions)
Q5) When a photon of light with wavelength 5000 A0, (green) hits a electronic counter, it imparts some energy and momentum to it. What is that energy and momentum? If the counter is made of silicon chips such that they emit electrons when energy of 10 Joules hits the counter, how many photons will have to hit before the counter starts electric current?
Q6) 1 Joule of energy is required to heat 1 gram of water. How many photons of normal sunlight (yellow color ~ 6000 A0) would be required to heat the water? (Calculate the energy of 1 photon of yellow light and hence work it out)
Q7) If the spectral line of frequency 2000 A0 as seen in a star’s spectra, redshifts by 2 A0, what is the velocity of the star with respect to us?
Q8) Calculate the Coulomb force between the electron and proton (i.e. nucleus) in a Hydrogen atom, if the electron is orbiting at a distance of 0.5 A0 from the proton. Compare this force with the force of gravity between the same electron-proton pair.
Q9) What would be the electrostatic force of repulsion between two Chlorine ions (Cl-), if they are separated by 8A0?
Q10) Consider an electron in the outermost orbit of Mg. How much energy (ionization potential) will be required to evict the electron from its orbit?
Q11) What would be the total energy of an electron in orbit around a gold nucleus, at a distance of ~ 2A0 from it?
Q12) What is the period of revolution of an electron orbiting a Sodium nucleus at a distance of ~ 3A0?
Q13) What is the radius of the third orbit of Hydrogen? Hence what is the energy of an electron in the third orbit?
Q14) What is the radius of the third orbit of Calcium? Hence what is the energy of an electron in this orbit? What is the difference between the radii of orbit and energies for Calcium & Hydrogen as compared for the third orbit?
Q15) What would be the period of revolution of an electron in the third shell (n = 3) of the Carbon atom? What would be its frequency? (i.e. no. of times it goes around in one second)
Q16) An electron jumps from the first orbit (n = 1) of Hydrogen to its second orbit (n = 2), because it has absorbed an incoming photon. What was the energy of the photon? What was its frequency & hence to which part of the spectrum did it belong?
Q17) What is the energy of the photon which is emitted when an electron jumps from the L-shell to the K-shell in Hydrogen? What is its frequency?
Q18) What is the energy of the photon emitted when an electron jumps from the N-shell of Potassium to its L-shell? What is its frequency?
Q19) There is Hydrogen in the photosphere (upper atmosphere) of the Sun. If the temperature of the photosphere is ~ 5000 0K, in what state will be the hydrogen – atomic or ionized? Also what will be the state of Tungsten? (Hint: If the surroundings have enough temperature, i.e. heat, to knock out the outermost electron, then the atom is ionized.)