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Chapter 33: Q. 2 (page 954)

Two narrow slits $80 μm$ apart are illuminated with light of wavelength localid="1649170764860" $620 nm$ . What is the angle of thelocalid="1649170756737" $m = 3$ bright fringe in radianslocalid="1649170769758" $?$ In degreeslocalid="1649170772775" $?$

Short Answer

Expert verified

The Angle of $m = 3$ bright fringe in radian is $0.0225 rad$ ,in degrees $1.29 °$ .

Step by step solution

01

Formula for angle

For double-slit interference, the $m$ -th order angle is given by

${sinθ}_{m} = \frac{mλ}{d}$

The distance between the slits is $d$ .

And the wavelength is $λ$ .

02

Calculation for angle

Angle is,

$θ = \sin^{- 1} \frac{mλ}{d}$

$= \sin^{- 1} (\frac{3 \cdot 6 \cdot 10^{- 7}}{8 \cdot 10^{- 5}})$

$= \sin (0.0225)$

$= 0.0225 rad$

In degrees,

$θ_{d e g} = \frac{2 π}{360} \cdot θ_{rad}$

$= 1.29 °$

As can be seen, the small angle approximation holds to a high degree of accuracy at this angle.

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Most popular questions from this chapter

A radar for tracking aircraft broadcasts a $12 GHz$ microwave beam from a $2.0 - m$ -diameter circular radar antenna. From a wave perspective, the antenna is a circular aperture through which the microwaves diffract.

a. What is the diameter of the radar beam at a distance of $30 km$ ?

b. If the antenna emits $100 kW$ of power, what is the average microwave intensity at $30 km$ ?

$(a)$ Find an expression for the positions $y_{1}$ of the first-order fringes of a diffraction grating if the line spacing is large enough for the small-angle approximation $\tan θ \approx \sin θ \approx θ$ to be valid. Your expression should be in terms of $d, L$ and $λ$ .
$(b)$ . Use your expression from part a to find an expression for the separation $∆ y$ on the screen of two fringes that differ in wavelength by $∆ λ$ .
$(c)$ Rather than a viewing screen, modern spectrometers use detectors-similar to the one in your digital camera-that are divided into pixels. Consider a spectrometer with a $333 l i n e s / m m$ grating and a detector with $100 p i x e l s / m m$ located $12 c m$ behind the grating. The resolution of a spectrometer is the smallest wavelength separation $∆ λ_{m i n}$ that can be measured reliably. What is the resolution of this spectrometer for wavelengths near localid="1649156925210" $550 n m$ , in the center of the visible spectrum? You can assume that the fringe due to one specific wavelength is narrow enough to illuminate only one column of pixels.

A Michelson interferometer operating at a $600 n m$ wavelength has a $2.00 - c m$ -long glass cell in one arm. To begin, the air is pumped out of the cell and mirror $M 2$ is adjusted to produce a bright spot at the center of the interference pattern. Then a valve is opened and air is slowly admitted into the cell. The index of refraction of air at $1.00 a t m$ pressure is $1.00028$ . How many brightdark-bright fringe shifts are observed as the cell fills with air?

Light from a helium-neon laser $(λ = 633 nm)$ is used to illuminate two narrow slits. The interference pattern is observed on a screen $3.0 m$ behind the slits. Twelve bright fringes are seen, spanning a distance of $52 m m$ . What is the spacing (in mm) between the slits?

FIGURE P33.56 shows the light intensity on a screen behind a circular aperture. The wavelength of the light is $500 nm$ and the screen is $1.0 m$ behind the slit. What is the diameter (in mm) of the aperture?

See all solutions

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