Winter 2016 - Q.4 - Grad Plus

# Winter 2016 – Q.4

4. (a) What is grating clement? Derive condition for absent spectra in plane transmission grating and explain with example. [5M]

The grating is a plane glass plate on which large number of lines is ruled on it with the help of diamond point. The distance between the centers of adjacent clear spaces is called as grating element or grating spacing. If a is the width of a clear space and b is the width of the ruled line, then the grating element or grating spacing is (a+b). The points in two adjacent clear spaces separated by distance (a+b) are called as corresponding points.

Absent Spectra
The condition for the formation of principle maxima corresponding to wavelength λ in the spectrum of order n is,

$latex \begin{array}{l}\left(a+b\right)\sin\theta=n\lambda\\\\or\\\\\sin\theta=n\lambda\left(a+b\right)—–\left(i\right)\end{array}$

where, n = 0,1,2,3,…..
The condition for destructive interference of all secondary wavelets from a single clear space in the direction of ‘α’is

$latex \begin{array}{l}a\sin\alpha=m\lambda\;\\\\or\\\\\sin\alpha=m\lambda/a—–\left(ii\right)\end{array}$

This condition for destructive interference is true for all clear spaces on grating surface. Hence equation (ii) is condition for zero intensity in the direction of ‘α’.
When both the conditions for wavelength λ are simultaneously satisfied i.e. when

$latex \begin{array}{l}\sin\theta=\sin\alpha\\\\\frac{n\lambda}{a+b}=\frac{m\lambda}a\\\\n=\left(\frac{a+b}a\right)m\end{array}$

The spectra of order n for given values of a, b and m will be absent. Since n and m are both integers $latex \left(\frac{a+b}a\right)$ must be in the ratio of two integers for absent orders.

Examples:
1. When b=a, the order of spectra must be absent given by

$latex \begin{array}{l}n=\left(\frac{a+b}a\right)m\\\\=\left(\frac{a+a}a\right)m\\\\=2m\\\\=2,\;2,\;6,….\;\;\;\left(As,\;m=1,\;2,\;3,…0.\right)\end{array}$

Thus in this case all even spectra will be absent.

2. when b=2a, the order of spectra must be absent are given by

$latex \begin{array}{l}n=\left(\frac{a+b}a\right)m\\\\=\left(\frac{a+2a}a\right)m\\\\=3m\\\\=3,\;6,\;9,….\;\;\;\left(As,\;m=1,\;2,\;3,…0.\right)\end{array}$

(b) What is Heisenbregs’ uncertainty principle? An electron has a speed of 300 m/sec. With an accuracy of 0.001%. Calculate the certainty with which we can locate the position of an electron. [5M]

Statement: “It is impossible to determine exact position and exact momentum both of a particle simultaneously with unlimited accuracy.”

It means that, if the position of the particle is determined accurately then there is uncertainty in the determination of momentum and if the momentum of the particle is determined accurately then there is uncertainty in the determination of its position.

Mathematically, “The product of uncertainties in simultaneous measurement of position and momentum of a microscopic particle is always of the order of Planck’s constant or greater than or equal to h/2π”.
Thus mathematically,

Δx Δpx ≥ ђ [ ђ = h/2 π ]

Or

Δx Δpx ≥ h/2π

This is called as Heisenberg’s uncertainty relation. Where, Δx = Uncertainty in measurement of position, ΔpX = uncertainty in measurement of momentum.

Given that, $latex m_e=9.1\times10^{-31}kg,$

Accuracy$latex =\frac{\triangle V}V$

$latex \begin{array}{l}=0.001\%\\\\=10^{-5}\end{array}$

$latex \begin{array}{l}\triangle V=300\times10^{-5}\\\\\;\;\;\;\;\;=3\times10^{-7}\end{array}$

The uncertainty in location of electron,

$latex \begin{array}{l}\begin{array}{l}\triangle x=\frac{\hslash}{\triangle p}\\\\\;\;\;\;\;\;=\frac{\hslash}{m\triangle v}\end{array}\\\\\;\;\;\;\;\;=\frac{1.05\times10^{-34}}{9.1\times10^{-31}\times3\times10^{-7}}\\\\\;\;\;\;\;\;=384.6m\end{array}$

(c) What do you mean by critical magnetic field and critical temperature ? A lead superconductor with Tc=7.2k has critical magnetic field of 6.5×103 A/m at absolute zero. What would he the magnitude of critical magnetic field at 5k temperature. [5M]

Critical Temperature: 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.

We have,

$latex \begin{array}{l}H_c\left(T\right)=H_c\left(0\right)\left[1-\left(\frac T{T_c}\right)\right]^2\\\\H_c\left(T\right)=6.5\times10^3\left[1-\left(\frac5{7.2}\right)\right]^2\\\\=3.39\times10^3\;A/m\end{array}$

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