Page 217 - Physics Coursebook 2015 (A level)

Chapter 14: Superposition of waves
WORKED EXAMPLE
2 Monochromatic light is incident normally on a diffraction grating having 3000 lines per centimetre. The angular separation of the zeroth- and first-order maxima is found to be 10°. Calculate the wavelength of the incident light.
Step1 Calculatetheslitseparation(grating spacing) d. Since there are 3000 slits per centimetre, their separation must be:
1 cm
d = 3000 = 3.33 × 10−4 cm = 3.33 × 10−6 m
Step2 Rearrangetheequationdsinθ=nλand substitute values:
θ =10.0°,n=1
λ = d sin θ = 3.36 × 10−6 × sin 10°
n1 λ=5.8×10−7m=580nm
QUESTIONS
QUESTION
14
A student is trying to make an accurate measurement of the wavelength of green light from a mercury lamp (λ = 546 nm). Using a double slit of separation 0.50 mm, he finds he can see ten clear fringes on a screen at a distance of 0.80 m fromtheslits.Thestudentcanmeasuretheir overall width to within ±1 mm. He then tries an alternative experiment using a diffraction grating that has 3000 lines per centimetre. The angle between the two second-order maxima can be measured to within ±0.1°.
a What will be the width of the ten fringes that he can measure in the first experiment?
b What will be the angle of the second-order maximum in the second experiment?
c Suggest which experiment you think will give the more accurate measurement of λ.
12 a
For the case described in Worked example 2, at what angle would you expect to find the second-order maximum (n = 2)?
Diffracting white light
A diffraction grating can be used to split white light up into its constituent colours (wavelengths). This splitting of light is known as dispersion, shown in Figure 14.27. A beam of white light is shone onto the grating. A zeroth- order, white maximum is observed at θ = 0°, because all waves of each wavelength are in phase in this direction.
On either side, a series of spectra appear, with violet closest to the centre and red furthest away. We can see why different wavelengths have their maxima at different angles if we rearrange the equation d sin θ = nλ to give:
sinθ = nλ d
■■ With a diffraction grating, there are many slits per centimetre, so d can be measured precisely. Because the maxima are widely separated, the angle θ can also be measured to a high degree of precision. So an experiment with a diffraction grating can be expected to give measurements of wavelength to a much higher degree of precision than a simple double-slit arrangement.
b Repeat the calculation of θ for n = 3, 4, etc. What is the limit to this calculation? How many maxima will there be altogether in this interference pattern?
13 Consider the equation d sin θ = nλ. How will the diffraction pattern change if:
a the wavelength of the light is increased?
b the diffraction grating is changed for one with more lines per centimetre (slits that are more closely spaced)?
BOX 14.5: Diffraction gratings versus double slits
It is worth comparing the use of a diffraction grating to determine wavelength with the Young two-slit experiment.
■■ With a diffraction grating, the maxima are very sharp.
■■ With a diffraction grating, the maxima are also very
bright. This is because rather than contributions from only two slits, there are contributions from a thousand or more slits.
■■ With two slits, there may be a large inaccuracy in the measurement of the slit separation a. The fringes are close together, so their separation may also be measured imprecisely.
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