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In a physics lab, light with a wavelength of 530 nm travels in air from a laser to a photocell in a time of 16.7 ns . When a slab of glass with a thickness of 0.870 m is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light a time of 21.3 ns to travel from the laser to the photocell.

What is the wavelength of the light in the glass?

Use 3.00×108 m/s for the speed of light in a vacuum. Express your answer using two significant figures.

Respuesta :

Answer:

[tex]\lambda'=78.086\ nm[/tex]

Explanation:

Given:

  • wavelength of light in the air, [tex]\lambda=530\times 10^{-9}\ m[/tex]
  • time taken to travel from the source to the photocell via air, [tex]t=16.7\ s[/tex]
  • time taken to reach the photocell via air and glass slab, [tex]t'=21.3\times 10^{-9}\ s[/tex]
  • thickness of the glass slab, [tex]x=0.87\ m[/tex]

Now we have the relation for time:

[tex]\rm time=\frac{distance}{speed}[/tex]

hence,

[tex]t=\frac{d}{c}[/tex]

c= speed of light in air

[tex]16.7\times 10^{-9}=\frac{d}{3\times 10^8}[/tex]

[tex]d=16.7\times 10^{-9}\times 3\times 10^8[/tex]

[tex]d=5.01\ m[/tex]

For the case when glass slab is inserted between the path of light:

[tex]\frac{(d-x)}{c} +\frac{x}{v} =t'[/tex] (since light travel with the speed c only in the air)

here:

v = speed of light in the glass

[tex]\frac{(5.01-0.87)}{3\times 10^8} +\frac{0.87}{v} =21.3\times 10^{-9}[/tex]

[tex]v=4.42\times 10^7\ m.s^{-1}[/tex]

Using Snell's law:

[tex]\frac{\lambda}{\lambda'} =\frac{c}{v}[/tex]

[tex]\frac{530}{\lambda'} =\frac{3\times 10^8}{4.42\times 10^7}[/tex]

[tex]\lambda'=78.086\ nm[/tex]

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