A film of magnesium fluoride (n=1.38), 1.25x10^-5 cm thick, is used to coat a camera lens (n=1.55). Are any wavelengths in the visible spectrum intensified in the reflected light?
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Bats use the reflections from ultra-high frequency sound to locate their prey. Estimate the typical frequency of a bat's sonar. Take the speed of sound to be 3.40x10^2 m/s, and a small moth 3.00mm across, to be a typical target.
answers in the back of the book are
6.9 x 10^-5 cm
and
1.13 x 10^5 Hz
Δ=2bn-λ/2 = 2k(λ/2)
2bn =(2k+1)λ/2
λ=4bn/(2k+1)
For k=0,
λ=4bn=4•1.25•10⁻⁷•1.38=6.9•10⁻⁷m=
=6.9•10⁻⁵cm
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λ=3mm=3•10⁻³m
v=340 m/s
f=v/ λ=340/3•10⁻³=1.13 x 10⁵Hz
Thanks so much Elena !
why did you calculatebit as destructive interference? wouldnt we want it to be constructive if we want to intensify the light
To determine if any wavelengths in the visible spectrum are intensified in the reflected light from the magnesium fluoride film, we need to consider the phenomenon of interference.
Interference occurs when two or more waves interact with each other. If the waves are in phase (i.e., their crests and troughs align), they can reinforce each other and produce constructive interference. If the waves are out of phase (i.e., their crests and troughs do not align), they can cancel each other out and produce destructive interference.
In this case, the magnesium fluoride film acts as a thin film, which can cause interference between the incident light and the reflected light. The thickness of the film, in combination with the refractive indices of the film and the camera lens, determines the behavior of the interference.
To determine if any wavelengths in the visible spectrum are intensified, we can use the concept of phase shift. When light reflects off a medium with a higher refractive index (like the camera lens in this case), there is a phase shift of 180 degrees. If the phase shift between the incident and reflected light is an odd multiple of half-wavelength (λ/2), we observe constructive interference. On the other hand, if the phase shift is an even multiple of half-wavelength (λ/2), we observe destructive interference.
Let's assume the incident light is white light, which consists of a range of wavelengths in the visible spectrum. To determine if any wavelengths are intensified in the reflected light, we can use the formula for the phase shift:
Phase shift = 2π * (film thickness) * (refractive index difference) / wavelength
In this case, the refractive index difference would be (n_camera lens - n_magnesium fluoride). Plug in the values:
Phase shift = 2π * (1.25x10^-5 cm) * (1.55 - 1.38) / wavelength
For each wavelength in the visible spectrum, calculate the corresponding phase shift using the above formula. If the phase shift is an odd multiple of half-wavelength (λ/2), then that particular wavelength will be intensified in the reflected light.
It is important to note that this calculation assumes the magnesium fluoride film is truly thin, meaning its thickness is much smaller than the wavelengths of visible light. If the film thickness is comparable to or larger than the wavelengths of visible light, more complex calculations and interference phenomena need to be considered.
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To estimate the typical frequency of a bat's sonar, we can use the given information about the speed of sound and the size of a typical target (a small moth).
Sonar is a technique that uses sound waves to navigate or detect objects. Bats emit high-frequency sound waves and listen for the echoes reflected back by objects in their surroundings. The frequency of bats' sonar can vary, but they typically use frequencies in the ultrasonic range, beyond the frequency range of human hearing.
To estimate the typical frequency, we can use the equation:
Frequency = Speed of Sound / Wavelength
Given the speed of sound as 3.40x10^2 m/s, we need to find the wavelength of the sound waves. To do this, we can use the size of the small moth (3.00 mm) as the target.
Imagine the sound wave emitted by the bat reaches the moth and reflects back as an echo. The wavelength of the sound wave can be estimated as twice the size of the moth. Therefore:
Wavelength = 2 * 3.00 mm
Convert the wavelength to meters:
Wavelength = 2 * 3.00 mm = 6.00 mm = 6.00x10^-3 m
Now, plug in the values into the frequency equation:
Frequency = 3.40x10^2 m/s / (6.00x10^-3 m)
Calculate the result to estimate the typical frequency of a bat's sonar.