Bundle: Physics for Scientists and Engineers, Technology Update, 9th Loose-leaf Version + WebAssign Printed Access Card, Multi-Term
Bundle: Physics for Scientists and Engineers, Technology Update, 9th Loose-leaf Version + WebAssign Printed Access Card, Multi-Term
9th Edition
ISBN: 9781305714892
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 38, Problem 38.31P

A diffraction grating has 4 200 rulings/cm. On a 2.00 m from the grating. it is found that for a particular  order m, the maxima corresponding lo two closely spaced wavelengths of sodium (589.6 nm and 589.6 nm) are separated by 1.54 mm. Determine the value of m.

Expert Solution & Answer
Check Mark
To determine

The value of m .

Answer to Problem 38.31P

The value of m is 2 .

Explanation of Solution

Given info: The number a grating is 4200rulings/cm . The distance between screen and grating is 2.00m . The maxima corresponding to 589.0nm and 589.6nm are separated by 1.54mm .

The first wavelength of sodium is 589.0nm and second wavelength of sodium is 589.6nm .

For the first order the value of m is 1 .

The formula to calculate separation between gratings is,

d=1N

Here,

N is number of rulings on diffraction grating.

Substitute 4200rulings/cm for N in the above expression.

d=14200rulings/cm=14200rulings/cm×(1cm102m)=2380×109m×(109nm1m)=2380nm

It remains same for all the order.

The formula for diffraction grating is,

dsinθ1=mλ1

Here,

θ1 is angle corresponding to first wavelength.

λ1 is first wavelength.

Rearrange the above expression for θ1 ,

θ1=sin1mλ1d

Substitute 589.0nm for λ1 , 2380nm for d and 1 for m in the above expression.

θ1=sin1(1)(589.0nm)2380nm=14.32°

Write the expression for tan of the above angle,

tanθ1

Substitute 14.32° for θ1 in the above expression.

θ2=tan(14.32°)=0.255268582°

The formula for diffraction grating is for second wavelength is,

dsinθ3=mλ2

Here,

θ3 is angle corresponding to second wavelength.

λ2 is second  wavelength.

Rearrange the above expression for θ3 ,

θ3=sin1mλ2d

Substitute 589.6nm for λ3 , 2380nm for d and 1 for m in the above expression.

θ3=sin1(1)(589.6nm)2380nm=14.34°

Write the expression for tan of the above angle,

θ4=tanθ2

Substitute 14.34° for θ3 in the above expression.

θ4=tan(14.34°)=0.255640427°

The difference between the tan of the two angle is,

θ5=θ4θ3

Substitute 0.255640427° for θ4 and 0.255268582° for θ3 in the above expression.

θ5=(0.255640427°)(0.255268582°)=3.71845×104

Write the formula to calculate position of central mth maxima from central maxima is,

y=Ltanθ5

Here,

L is distance between grating and screen.

Substitute 2.00m for L and 3.71845×104 for θ5 in the above expression.

y=(2.00m)(3.71845×104)=7.436×104m×(103mm1m)=0.7436mm

For the second order the value of m is 2 .

The formula for diffraction grating is,

dsinθ6=mλ1

Here,

θ6 is angle corresponding to first wavelength and second order.

λ1 is first wavelength.

Rearrange the above expression for θ1 ,

θ6=sin1mλ1d

Substitute 589.0nm for λ1 , 2380nm for d and 2 for m in the above expression.

θ6=sin1(2)(589.0nm)2380nm=sin1(0.49495°)=29.66°

Write the expression for tan of the above angle,

θ7=tanθ6

Substitute 29.66° for θ6 in the above expression.

θ7=tan(29.66°)=0.569465032°

The formula for diffraction grating is for second wavelength is,

dsinθ8=mλ2

Here,

θ8 is angle corresponding to second wavelength and the order is 2 .

λ2 is second  wavelength.

Rearrange the above expression for θ8 ,

θ8=sin1mλ2d

Substitute 589.6nm for λ2 , 2380nm for d and 2 for m in the above expression.

θ8=sin1(2)(589.6nm)2380nm=sin1(0.49546°)=29.7°

Write the expression for tan of the above angle,

θ9=tanθ8

Substitute 29.7° for θ8 in the above expression.

θ9=tan(29.7°)=0.570389929°

The difference between the tan of the two angle is,

θ10=θ9θ7

Substitute 0.570389929° for θ9 and 0.569465032° for θ7 in the above expression.

θ10=(0.570389929°)(0.569465032°)=9.24897×104

Write the formula to calculate position of central mth maxima from central maxima is,

y=Lθ10

Here,

L is distance between grating and screen.

Substitute 2.00m for L and 9.24897×104 for θ10 in the above expression.

y=(2.00m)(9.24897×104)=1.8497×103m×(103mm1m)=1.8497mm

For the third order the value of m is 3 .

The formula for diffraction grating is,

dsinθ11=mλ1

Here,

θ11 is angle corresponding to first wavelength and second order.

λ1 is first wavelength.

Rearrange the above expression for θ11 ,

θ11=sin1mλ1d

Substitute 589.0nm for λ1 , 2380nm for d and 3 for m in the above expression.

θ11=sin1(3)(589.0nm)2380nm=sin1(0.7424°)=47.936°

Write the expression for tan of the above angle,

θ12=tanθ11

Substitute 47.936° for θ11 in the above expression.

θ12=tan(47.936°)=1.108120806°

The formula for diffraction grating is for second wavelength is,

dsinθ13=mλ2

Here,

θ13 is angle corresponding to second wavelength and the order is 3

λ2 is second  wavelength.

Rearrange the above expression for θ8 ,

θ13=sin1mλ2d

Substitute 589.6nm for λ2 , 2380nm for d and 3 for m in the above expression.

θ13=sin1(3)(589.6nm)2380nm=sin1(0.7431°)=47.99°

Write the expression for tan of the above angle,

θ14=tanθ13

Substitute 29.7° for θ8 in the above expression.

θ14=tan(47.99°)=1.110222778°

The difference between the tan of the two angle is,

θ15=θ14θ12

Substitute 1.110222778° for θ14 and 1.108120806° for θ12 in the above expression.

θ15=(1.110222778°)(1.108120806°)=2.101972×103

Write the formula to calculate position of central mth maxima from central maxima is,

y=Lθ15

Here,

L is distance between grating and screen.

Substitute 2.00m for L and 2.101972×103 for θ15 in the above expression.

y=(2.00m)(2.101972×103)=4.2039×103m×(103mm1m)=4.203mm

From the above calculation it is clear that only for second order the maxima of two closely spaced wavelength of sodium 589.0nm and 589.6nm are separated by 1.54mm .

Conclusion:

Therefore, the value of m is 2 .

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Chapter 38 Solutions

Bundle: Physics for Scientists and Engineers, Technology Update, 9th Loose-leaf Version + WebAssign Printed Access Card, Multi-Term

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