- Level: AS and A Level
- Subject: Maths
- Word count: 1357
Case of Doubled screen Maths Investigation
Extracts from this document...
Introduction
This is a portion of the main project taken from between.
Case of Doubled screen(diagram only)
First max.
First Max.
GENERALIZATION FOR N – SCREENS.
First of all let us consider a simple case in which we find the fringe width on nth screen of first maxima.
The diagram may be
S1 S2 S3 S4
N => odd
λ ……………….
……………….
……………….. ………………
……………….. …………………
D D D D
Again if n is odd then we would obtain the fringe width d/2 & thus the separation between two maximas
= d.
Therefore, if n is odd then fringe width of first maxima = d /2.
And if n is even, then we would get fringe width greater then that when n is odd and it would be λD/d.
Note: Here we are considering diagram type A, ie n=>odd suppose we considered B, then also we would get same result.
THIS IS A VERY GENERAL CASE IN WHICH WE CAN CALCULATE “ FRINGE WIDTH”
FOR ANY SCREEN NUMBER .THE MAIN PART OF THIS IS THAT WE HAVE
CONSIDERED THIS FOR FIRST MAXIMA.
A MORE GENERAL CASE IS DISSUSED BELOW:
…………………………….
…………………………….
d/2 X1 X3 X5 X7 X9 Xn-1
d/2 X2 X4 X6 X8 X10 Xn
………………………….
………………………….
The size of First two slits is “d/2” & distance between the slits is “D’. Now a monochromatic source of light is emitted having wavelength λ .
Middle
Since the fringe width is different for different “As” therefore we
Break our series in two parts:
1\2 TYPE “1” 3\4 TYPE “2”
5\6 7\8 have another
typeof fringe width
9\10 have a particular 11\12
type of fringe width
now the general term of TYPE “1” series is {4m-3\4m-2} now if we have a no.
(n)such that m=n+3\4 or m=n+2\4 & both “m” & “n” obtained are integers then
we will say that (n) lies in type 1 now each element of type 1 has certain
common factor in it the uncommon factors are1\2=>1
5\6=>(A1+A2)\(A3+A4)
9\10=>(A1+A2)(A5+A6)\(A3+A4)(A7+A8)
now if we find general term of this series& generalize this situation then we will find general term of type “1” series BY OBSERVATION GENERAL TERM IS
(A1+A2)(A5+A7)(A9+A10)………………………………………………………..[Am+4 + An-4]
(A3+A7)(A7+A8)(A11+A12)…………………..…………………………………..[Am-2 +An-2]
Where m is the maxima no. just prior to n.
NOW FOR TYPE TWO SERIES, the variance is 3\4 =>1\A1+A2
7\8 =>A3+A4\(A1+A2)(A5+A6)
11\12=>(A3+A4)(A7+A8)
(A1+A2)(A5+A6)(A9+A10)
Again by simple observation the general series in this case is:
(A3+A4)(A7+A8)(A11+A12).............................................(Am-4+An-4)
(A1+A2)(A5+A6)(A9+A10)..............................................(Am-2+An-2)
FINALLY WE OBTAIN:
XN {for type 1} =ANλ
Conclusion
=>X{dtanσ−2λ} = λ{D+Y+X1}
> X2= λ {D+Y+X1}
dtanσ−2λ
Consult fig.3 tanσ = λ {D+Y+X1+X2+3X3}
DX3
=>X3{dtanσ-3λ} = λ{D+Y+X1+X2}
THUS X3 = λ {D+Y+X1+X2}
dtanσ-nλ
GENERALISITION:
Xn =λ{D +Y+X1+X2+X3+X4+….+Xn-1}
dtanσ-nλ
We can surely say that f the second max. of screen at an angle will be the second max. of the arbitrary screen because between first & second max., since their is no other max. this light rays cant pass in that region & inferred to produce a maxima. This explanation is also true in case of first-second, second-third and third-forth & ay maxima.
Finally distance of a general max. From central maxima in a screen kept a an angle is:
An= λ{D+L\2(cotσ) +X1+X2+X3+……+nXn}
d sinσ
(A significant result.)
COLOURED FRINGES
If white light of wavelength {400-700} nm is sent through a ydsethen we get colored fringes
For each of wavelength out of 400-700 we will obtain different fringes & hence different
Fringe width. The first maxima will be formed at different positions as shown below.
The central maxima will be white.
RED
ORANGE
YELLOW
GREEN
BLUE
INDIGO
VOILET
WHITE LIGHT
We may obtain some intensity due to λ also. This case may arise a possibility of MAXIMA overlapping between higher order maxima’s.
☺ANALYSIS ENDS ☺
Shubham Verma
E-mail:-shubahmverma@rediffmail.com
78, new civil lines
Behind Gurudev Palace
Kanpur 208024
UP
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