$$\lambda_1$$ and $$\lambda_2$$ are the de-Broglie wavelengths of the particle, when $$0 \le x \le 1$$ and $$x > 1$$ respectively. The total energy of the particle is $$42E_0$$. If the ratio $$\dfrac{\lambda_1}{\lambda_2}$$ is $$\sqrt{x} $$, find $$x$$.

A proton is fired from very far away towards a nucleus with charge $$Q=120e$$, where $$e$$ is the electronic charge. It makes a closest approach of $$10 fm$$ to the nucleus. The de-Broglie wavelength (in unit of $$fm$$) of the proton at its start is:

(take the proton mass, $$m_p=(5/3)\times 10^{-27} kg; h/e=4.2\times 10^{-15}Js/C; $$

$$\dfrac {1}{4\pi \epsilon_0}=9\times 10^9m/F; 1fm=10^{-15} m)$$

mind that the orbital angular momentum of the valence electron in the silver atom is zero.

**What is the momentum of a photon if the
wavelength of X-rays is 1 angstrom?**

The number of photons absorbed per second by the surface is $$k\times 10^{19}$$. what is a value of k?

(a) Quarks inside protons and neutrons are thought to carry fractional charges $$[(+2/3)e ; (1/3)e]$$. Why do they not show up in Millikan's oil-drop experiment ?

(b) What is so special about the combination e/m? Why do we not simply talk of e and m separately?

(c) Why should gases be insulators at ordinary pressures and start conducting at very low pressures?

(d) Every metal has a definite work function. Why do all photoelectrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?

(e) The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations: $$E = h \nu$$, p = $$\displaystyle\frac{h}{\lambda }$$.

But while the value of $$\displaystyle\lambda$$ is physically significant, the value of $$\nu$$ (and therefore, the value of the phase speed n $$\displaystyle\lambda)$$ has no physical significance. Why?(b) From your answer to (a), guess what order of accelerating voltage(for electrons) is required in such a tube?

The energy of a photon is given by the formula _____________

At what rate are photons emitted from the lamp?

At what distance from the lamp will the average flux of photons be 1.00 photons /$$(cm^2s)$$ ?

A. Microwaves of wavelength $$1.5\ cm$$

B. Red light of wavelength $$660\ nm$$

C. Radiowaves of frequency $$96\ MHz$$

D. $$X-$$rays of wavelength $$0.17\ nm$$

$$E = h v, p = \dfrac{h}{\lambda}$$

But while the value of $$\lambda$$ is physically significant, the value of $$v$$ (and therefore, the value of the phase speed $$v$$ $$\lambda$$) has no physical significations. Why?

$$V(x)=E_{o} ; 0\leq x \leq 1$$

$$=0; x >1$$

For $$0\leq x \leq 1$$, de Broglie wavelength is $$\gamma_{1}$$ and for $$x >1$$ the de Broglie wavelength is $$\gamma_{2}$$.

Total energy of the particle is $$2E_{o}$$. If $$\gamma_{1}/\gamma_{2}=\sqrt x$$. Find $$x$$

Laser is used in cutting metals.

i) Kinetic energy of photo electrons.

ii) Number of photo electrons.

Ordinary key chain laser light should not be directly viewed.

$$\displaystyle { \lambda }_{ 1 }$$ and $$\displaystyle { \lambda }_{ 2 }$$ are the de-Broglie wavelengths of the particle, when $$\displaystyle 0\le x\le 1$$ and $$\displaystyle x>1$$ respectively.

If the total energy of particle is $$\displaystyle { 2E }_{ 0 }$$, find $$\displaystyle \left( { { \lambda }_{ 1 } }/{ { \lambda }_{ 2 } } \right) $$.

(b) Write the basic features of photon picture of electromagnetic radiation on which Einstein's photoelectric equation is based.

and $$\omega_{o}=3.60\times 10^{15}\ rad/s$$.

(i) What is the energy of the photon in the light beam?

(ii) How many photons per second, on an average are emitted by the source ?

The wavelength $$'\lambda'$$ of a photon and the de Broglie wavelength of an electron have the same value.Show that the energy of photon is $$\dfrac{2\lambda mc}{h}$$ times the kinetic energy of electron, where m,c,h have their usual meanings.

(b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature ($$27^o C$$). Hence, explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.

Exercise 11.31:

Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wave length of the probe equal to 1 A∘A∘ which is of the order of inter-atomic spacing in the lattice) ($$m_e=9.11 \times 10^{-31} kg$$)

Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about photons. The second number tells you why our eye can never count photons, even in barely detectable light.

(a) The number of photons emitted per second by a medium wave transmitter of 10 kW power, emitting radio waves of wave length 500 m.(b) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive $$(~\,10^{10}\, W\, m^{2})$$. Take the area of the pupil to be about $$0.4\, cm^2$$, and the average frequency of white light to be about $$6\, \times\, 10^{14}\, Hz$$.

(a) the energy received by the surface per second,

(b) the number of photons hitting the surface per second,

(c) if surface is tilted such that plane of the surface makes an angle $${30}^{o}$$ with light beam, find the number of photons hitting the surface per second.

what will be the minimum energy of a photon which can be absorbed by hydrogen atom at ordinary temperature ?

$$\dfrac{\Delta E}{E}=\dfrac{h f^{\prime}}{m c^{2}}(1-\cos \phi)\\$$

where $$E$$ is the energy of the incident photon, $$f^{\prime}$$ is the frequency of the scattered photon, and $$\phi$$ is defined as in Fig. $$38-5$$

(a) Number of photons per second arriving at $$1$$ squared metres at earth.

(b) Number of photons emitted from the sun per second assuming that the average radius of earth's orbit is $$1.49 m$$.

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