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51.

In YDSE, the two slits are separated by 0.1 mm and they are 0.5 m from the screen. The wavelength of light used is 5000 Å. Find the distance between 7th maxima and 11 th mimima on the upper side of screen.

Answer» Given, `d = 0.1 mm =10^-4m, D = 0.5 m and lambda = 5000 Å = 5.0 xx (10^-7)m.`
`Delta y = y_11 -y_7 = ((2xx 11xx -1)lambda D)/(2d) -(7lambdaD)/(d)`
` or Delta y = (7lambdaD)/(2d) = ( 7xx 5.0 xx 10^(-7) xx 0.5)/(2 x 10^(-4))`
` = 8.75 xx (10^-3) `
` = 8.75 mm.`
Discussion
52.

In interference, `I_(max)/I_(min) = alpha` , find (a) ` A_(max)/A_(min)` (b)` A_1/A_2` (c)`I_1/I_2` .

Answer» Correct Answer - A::B::C
(a) `A_(max)/A_(min) = sqrt (I_(max)/I_(min)) = sqrt alpha`
(b) `A_(max)/A_(min) = sqrt alpha = (A_1 +A_2)/(A_1-A_2) = (A_1//A_2 +1)/(A_1//A_2-1)`
Solving this equation, we get ` A_1/A_2 = sqrt(alpha+1)/(sqrt alpha-1)`
` (c) I_1/I_2 = (A_1/A_2)^2 = (((sqrt(alpha))+1)/((sqrt alpha)-1)) ^2 ` .
Discussion
53.

Two waves of equal frequencies have their amplitudes in the ratio of 3:5. They are superimposed on each other. Calculate the ratio of maximum and minimum intensities of the resultant wave.

Answer» Correct Answer - A
Given, `A_1/A_2 = 3/5 `
` :. sqrt(I_1/I_2) = 3/5` (as `I prop A^2`)
Maximum intensity is obtained, where
` cos phi = 1`
and `I_(max) = (sqrtI_1 + sqrtI_2)^2`

Minimum intensity is found, where
` cos phi = -1`
and `I_(min) = (sqrtI_1 - sqrtI_2)^2`
Hence, `I_(max)/I_(min) = ((sqrt I_1+sqrtI_2)/(sqrt I_1 -sqrt I_2))^2 = ((sqrt (I_1/I_2)+1)/(sqrt((I_1/I_2))-1))`
` = ((3/5 +1)/(3/5-1))^2 = 64/4 = 16/1`.
Discussion
54.

In interference, two individual amplitudes are `A_0` each and the intensity is `I_0` each. Find resultant amplitude and intensity at a point, where: (a) phase difference between two waves is `60^@`. (b) path difference between two waves is `lambda/3`.

Answer» (a) Substituting `phi= 60^@` in the equations,
`A = 2A_0 "cos" phi/2` and `I = 4I_0 "cos"^2 phi/2`
We get, `A= sqrt (3)A_0` and `I = 3I_0`
(b) Given , `Delta x = lambda/3`
` :. Phi or Delta phi = (2pi)/lambda.Delta x `
` = (2pi)/lambda(lambda/3)= (2pi)/3` or `120^@`
Now, substituting `phi=120^@` in the above two equations, we get
` A = A_0` and `I = I_0 ` .
Discussion
55.

In interference, `I_(max)/I_(min) = alpha` , find (a) ` A_(max)/A_(min)` (b)` A_1/A_2` (c)`I_1/I_2` .

Answer» Correct Answer - A::B::C
(a) `A_(max)/A_(min) = sqrt (I_(max)/I_(min)) = sqrt alpha`
(b) `A_(max)/A_(min) = sqrt alpha = (A_1 +A_2)/(A_1-A_2) = (A_1//A_2 +1)/(A_1//A_2-1)`
Solving this equation, we get ` A_1/A_2 = sqrt(alpha+1)/(sqrt alpha-1)`
` (c) I_1/I_2 = (A_1/A_2)^2 = (((sqrt(alpha))+1)/((sqrt alpha)-1)) ^2 ` .
Discussion
56.

In interference, two individual amplitudes are 5 units and 3 units. Find (a)`A_(max)/A_(min)` (b) ` I_(max)/I_(min)` .

Answer» (a) `A_(max)/A_(min) = (A_1+A_2)/(A_1-A_2)`
` (b) I_(max)/I_(min) = (A_(max)/A_(min))^2`
Discussion
57.

Two coherent sources of intensity ratio ` beta^2` interfere. Then, the value of `(I_(max)- I_(min))//(I_(max)+I_(min))` isA. `(1+beta)/(sqrt beta)`B. `(sqrt(1+beta)/beta)`C. `(1+beta)/beta`D. None of these

Answer» Correct Answer - D
`I_1/I_2 =beta^2`
So let, `I_2 = 1unit, then I_1=beta`
`I_(max) = (sqrtI_1+sqrtI_2)^2 = (I+beta)^2`
`I_(min) = (sqrtI_1+sqrtI_2)^2 = (I+beta)^2`
`= I_(max) - I_(min) =4beta`
` I_(max) + I_(min) = 2(1+beta^2)`
:. The asked ratio is `(2beta)/(1+beta^2).`
Discussion
58.

Two coherent sources of intensity ratio `alpha` interfere in interference pattern `(I_(max)-I_(min))/(I_(max)+I_(min))` is equal toA. `(2alpha)/(1+alpha)`B. `(2sqrt(alpha))/(a+alpha)`C. `(2alpha)/(1sqrt(alpha))`D. `(1+alpha)/(2alpha)`

Answer» Correct Answer - B
`(I_("max")-I_("min"))/(I_("max")+I_("min"))=((a_(1)+a_(2))^(2))/((a_(1)+a_(2))^(2))`
`[because I_("max")=(a_(1)+a_(2))^(2), I_("min")=(a_(1)-a_(2))^(2), "where a = amplitude"]`
`=(4a_(1)a_(2))/(2(a_(1)^(2)+a_(2)^(2)))=(2a_(1)a_(2))/(a_(1)^(2)+a_(2)^(2))`
Now, dividing the numerator and denominator by `a_(1)a_(2)`, we get
`(I_("max")-I_("min"))/(I_("max")+I_("min"))=(2)/([(a_(1))/(a_(2))+(a_(2))/(a_(1))])`
`=(2)/([sqrt(alpha)+(1)/(sqrt(alpha))])=(2sqrt(alpha))/((alpha+1))`
Discussion