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Chapter 37, Problem 45AP

(a)

To determine

To show: The dispersion is given by dλdy=L2dm(L2+y2)32 .

(a)

Expert Solution
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Answer to Problem 45AP

The dispersion is, dλdy=L2dm(L2+y2)32 .

Explanation of Solution

Formula to calculate the angles of bright beams diffracted from the grafting is,

dsinθ=mλ (1)

Here,

d is the spacing between adjacent slits.

m is order number of intensity maximum.

λ is wavelength of light.

θ is the angle by which ray is diffracted.

Write the expression for sine of angle θ .

sinθ=yL2+y2

Substitute yL2+y2 for sinθ in equation (1)

dyL2+y2=mλ

Differentiate the above equation with respect to y .

ddy(dyL2+y2)=ddy(mλ)

Apply product rule of differentiation to differentiate above equation.

d(L2+y2)12+(d)y(12)(L2+y2)32(0+2y)=mdλdyd(L2+y2)12(d)y2(L2+y2)32=mdλdy(d)(L2+y2)(d)y2(L2+y2)32=mdλdydλdy=L2dm(L2+y2)32

Conclusion:

Therefore, the dispersion is dλdy=L2dm(L2+y2)32 .

(b)

To determine

The dispersion in first order.

(b)

Expert Solution
Check Mark

Answer to Problem 45AP

The dispersion in first order is 3.77nm/cm .

Explanation of Solution

Given info: The mean wavelength of light is 550nm , grating is 8000ruling/cm , and screen is placed at a distance of 2.40m .

Formula to calculate the angles of bright beams diffracted from the grafting is,

dsinθ=mλ (2)

Here,

d is the spacing between adjacent slits.

m is order number of intensity maximum.

λ is wavelength of light.

θ is the angle by which ray is diffracted.

The spacing between adjacent slit is inverse of number of rulings per centimeter is,

d=18000cm=1.25×104cm

Substitute 1.25×104cm for d , 1 for m and 550nm for λ in equation (2).

1.25×104cm×102m1cm(sinθ)=1550nm×109m1nm1.25×106msinθ=550×109mθ=sin1(550×109m1.25×106m)=26.10°

For the value of y ,

tanθ=yLy=Ltanθ

Substitute 2.40m for L , and 26.1° for θ in above equation.

y=2.40tan(26.1°)=2.40m(0.489)=1.18m

Formula to calculate the dispersion is,

dλdy=L2dm(L2+y2)32

Here,

L is the distance between slits and screen.

m is order number of intensity maximum.

y is position relative to the center of a diffraction pattern.

Substitute 2.40m for L , 1 for m , 1.18m for y and 1.25×106m for d to calculate dλdy .

dλdy=(2.4m)2(1.25×106m)1((2.4m)2+(1.18m)2)32=7.2×106m(7.1524)32m=3.77107m1m×102cm1m=3.77109mcm

Further solve the above equation.

dλdy=3.77×109m×(109nm1m)cm=3.77nm/cm

Conclusion:

Therefore, the dispersion is 3.77nm/cm .

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Chapter 37 Solutions

Bundle: Physics For Scientists And Engineers With Modern Physics, 10th + Webassign Printed Access Card For Serway/jewett's Physics For Scientists And Engineers, 10th, Multi-term

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