Question

A glass plate 2.50 mm thick, with an index of refraction of 1.40, is placed between a source of light with wavelength 540 nm (in vacuum) and a screen. The source is 1.80 cm from the screen. How many wavelengths are there between the source and the screen?

Answer 1

Thickness of glass plate:

$$ t=2.50 \mathrm{~mm}=2.50 \times 10^{-3} \mathrm{~m} $$

Refractive index of glas s, \(n_{\text {dass }}=1.40\) Wavel ength of light in vaccum, \(\lambda_{v}=540 \mathrm{~nm}=540 \times 10^{-9} \mathrm{~m}\)

Distance between source and screen, \(d=1.80 \mathrm{~cm}=1.80 \times 10^{-2} \mathrm{~m}\)

Solution

Wavel ength of the light in medium:

$$ \begin{aligned} \lambda &=\frac{\lambda_{v}}{n_{\text {ghss }}} \\ &=\frac{540 \times 10^{-9} \mathrm{~m}}{1.40} \\ &=385.71 \times 10^{-9} \mathrm{~m} \end{aligned} $$

Number of wavel ength in glass:

$$ \begin{aligned} N_{1} &=\frac{t}{\lambda} \\ &=\frac{2.50 \times 10^{-3} \mathrm{~m}}{385.71 \times 10^{-9} \mathrm{~m}} \\ &=6.481 \times 10^{3} \end{aligned} $$

Distance traveled by light in air:

$$ D=d-t $$

$$ \begin{array}{l} =1.80 \times 10^{-2} \mathrm{~m}-2.50 \times 10^{-3} \mathrm{~m} \\ =1.55 \times 10^{-2} \mathrm{~m} \end{array} $$

Thus, the number of wavelengths

$$ \begin{aligned} N_{2} &=\frac{D}{\lambda_{v}} \\ &=\frac{1.55 \times 10^{-2} \mathrm{~m}}{540 \times 10^{-9} \mathrm{~m}} \\ &=2.87 \times 10^{4} \end{aligned} $$

Total no. of wave lengths \(N=N_{1}+N_{2}=6.481 \times 10^{3}+2.87 \times 10^{4} \approx 35185.2\)