(a) How long does it take light to travel through a 3.0 mm thick piece of window glass
(nglass = 1.5)?
(b) Through what thickness of water (nwater = 1.33) could light travel in the same amount of
time?
(c) In which case is the number of wavelengths larger?
(a) To calculate the time it takes for light to travel through a given thickness of a material, we can use the equation:
time = thickness / speed
The speed of light in a vacuum is approximately 3.00 x 10^8 m/s, but in a material like glass, it slows down. The speed of light in glass is given by:
speed in glass = speed in vacuum / refractive index
The refractive index of window glass is given as nglass = 1.5. Therefore, the speed of light in glass is:
speed in glass = (3.00 x 10^8 m/s) / 1.5 = 2.00 x 10^8 m/s
Plugging this speed into the equation, we find:
time = thickness / speed in glass
time = 3.0 mm / (2.00 x 10^8 m/s)
To ensure units are consistent, we convert 3.0 mm to meters:
3.0 mm = 3.0 x 10^-3 m
Plugging in the values:
time ≈ (3.0 x 10^-3 m) / (2.00 x 10^8 m/s) ≈ 1.5 x 10^-11 s
Therefore, it takes light approximately 1.5 x 10^-11 seconds to travel through a 3.0 mm thick piece of window glass.
(b) To calculate the thickness of water through which light can travel in the same time, we use the same equation as before:
time = thickness / speed
We need to find the thickness of the water, so we rearrange the equation:
thickness = time x speed
The speed of light in water is given by:
speed in water = speed in vacuum / refractive index
The refractive index of water is given as nwater = 1.33. Therefore, the speed of light in water is:
speed in water = (3.00 x 10^8 m/s) / 1.33 ≈ 2.256 x 10^8 m/s
Plugging this speed into the equation, we find:
thickness = time x speed in water
thickness = (1.5 x 10^-11 s) x (2.256 x 10^8 m/s)
Calculating the product:
thickness ≈ 3.384 x 10^-3 m
Therefore, light can travel through approximately 3.384 mm of water in the same amount of time it takes to travel through 3.0 mm of window glass.
(c) The number of wavelengths is determined by the distance traveled divided by the wavelength:
number of wavelengths = distance / wavelength
Since we're comparing the same distance (thickness) for both cases, the one with the smaller speed will have a larger number of wavelengths.
In this case, since the speed of light is slower in glass compared to water, the number of wavelengths will be larger in glass than in water for the same thickness.