Summer 2016 - Q.1 - Grad Plus

# Summer 2016 – Q.1

1. Attempt any five: [15M]
(a) Why does an excessively thin film appear to be perfectly dark when illuminated by white light?

The expression for path difference in case of thin film is given by

$latex \triangle=2\mu t\cos\theta\pm\frac\lambda2$

If the film is excessively thin i.e. t << λ and the path difference Δ = λ/2. In this case condition for destructive interference is satisfied and therefore the film appears to be perfectly dark when illuminated by white light.

(b) A grating has 620 rulings/mm & is 5.05 mm wide. What is the smallest wavelength-interval that can be resolved in the third order at λ = 481 nm?

Since grating has 620 rulings/mm, number of rulings (N) in 5.05 mm = 620 x 5.05 = 3131

We have,

Resolving Power$latex =\frac\lambda{d\lambda}=Nn$ or

$latex \begin{array}{l}d\lambda=\frac\lambda{Nn}\\\\\;\;\;\;\;=\frac{481\times10^{-9}}{3131\times3}\\\\\;\;\;\;\;=0.512\times10^{-10}\\\\\;\;\;\;\;=0.512\;\overset\circ A\\\end{array}$

Which is the smallest wavelength interval that can be resolved.

(c) Why would you recommend use of optical fibre in communication system?

The reasons are as follows. Optical fibre are made from silica which is one of the most abundant material on the earth. The overall cost of optics communication is lower than that of an equivalent cable communication system. The cross section of an optical fibre is about few microns. Hence, the fibres are less bulky. Optical fibre is quite flexible and strong.

In optical fibres, information is carried by photons which are electrically neutral and cannot be disturbed by high voltage fields, lightening, etc. Therefore, fibres are immune to externally caused background noise generated through EMI and RFI.

The light waves propagation along the optical fibre are completely trapped within the fibre and cannot leak out further light cannot couple into the fibre from sides. Therefore, the possibility of cross talk is minimised when optical fibre is used. Thus, transmission is more secure and private.

Optical fibres have ability to carry large amounts of information. A 1mm fibre can transmit 50000 calls. The transmission loss per unit length of an optical fibre is about 1Db/km. Therefore, longer cables run between repeaters and are feasible.

(d) An electron is bound in a one dimensional potential well of width 2A0 but of infinite height. Find its energy values to the ground state and first excited state?

We have,

$latex E=\frac{n^2h^2}{8mL^2}$

For ground state, n = 1,

$latex \begin{array}{l}E=\frac{\left(6.63\times10^{-34}\right)^2}{8\times9.1\times10^{-31}\times\left(2\times10^{-10}\right)^2}\\\\\;\;\;=0.3015\times10^{-17}\;J\\\\\;\;\;=18.8\;eV\end{array}$

For first excited state, n = 2,

$latex \begin{array}{l}E=\frac{25\times\left(6.63\times10^{-34}\right)^2}{8\times9.1\times10^{-31}\times\left(2\times10^{-10}\right)^2}\\\\\;\;\;=0.5375\times10^{-17}\;J\\\\\;\;\;=471\;eV\end{array}$

(e) Explain measurement of frequency of AC signal using Cathode Ray Oscilloscope?

An AC signal whose frequency is to be measured is given to Y-input of CRO. The time-base sensitivity (time/div) is adjusted so as to obtain two or three cycles of AC on the CRO screen. The horizontal spread of one cycle is noted in terms of number of division. By multiplying it with time/div, the time for one cycle i.e. time period is obtained. The reciprocal of time period gives the frequency of the AC signal.

(f) Explain the term Stimulated emission & Population inversion?

Stimulated emission:
A photon of energy hυ = E2 – E1 can induce an excited atom in higher energy state E2 to make downward transition to ground state E1, releasing excess energy in the form of photon. This phenomenon of forced emission of photon is called as induced emission or stimulated emission.

The no. of stimulated transitions occurring in the material at any instant depends upon the no. of atoms at energy level E2 and spectral energy density. Thus, the number of atoms Nst that undergoes downward transition during the time Δt is

$latex N_{st}\;/\Delta t\;\propto\;N_{2\;}Q$

Hence, rate of spontaneous emission is,

$latex \frac{dN_{st}}{dt}=B_{21}N_2Q$

Where, B12 is the Einstein B- coefficient, N2 is the no. of atoms in state E2 and Q is the spectral energy density.

Population Inversion:
The no. of active atoms occupying an energy state is called as population of that state. The
population N of an energy state depends upon absolute temperature T and energy E.

In the state of thermal equilibrium there are more atoms in ground state than excited state.
However, in order to achieve stimulated emission, more no. of atoms should be in excited state than
ground state. Such a non-equilibrium condition in which number of atoms in excited state is greater than number of atoms in ground state is called as Population Inversion.

If N1 and N2 are the population of ground and excited state having energy E1 and E2 respectively
then population ratio is given by,

$latex \frac{N_2}{N_1}=exp\left[\frac{\displaystyle-\left(E_2-E_1\right)}{kT}\right]$

(g) Define superconductivity, critical temperature & critical magnetic field.

Superconductivity: The sudden disappearance of electrical resistance in materials below a certain
temperature is known as superconductivity. The materials that exhibit superconductivity and which are in the superconducting state are called superconductors.

Critical Field: The temperature at which a normal material turns into a superconductor is called as
critical temperature, Tc. Every superconductor has it’s own critical temperature at which it passes over into superconducting state. The superconducting transition is sharp for chemically pure and structurally pure specimen but broad for impure specimens and with structural defects.

Critical Magnetic Field: It is observed that superconductivity vanishes if a sufficiently strong magnetic field is applied. The minimum magnetic field which is necessary to regain the normal resistivity is called as critical magnetic field, Hc.

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