Problem 17.6
A helium-neon laser beam has a wavelength in air of 633nm . It takes 1.48 ns for the light to travel through 31.0 cm of an unknown liquid.
Part A -
What is the wavelength of the laser beam in the liquid?
Express your answer with the appropriate units.
To find the wavelength of the laser beam in the liquid, we can use Snell's Law, which relates the angle of incidence and refraction of light passing through different mediums.
The formula for Snell's Law is:
n1 sin(theta1) = n2 sin(theta2)
In this case, we don't have the angle of incidence and refraction, but we can assume that the laser beam is incident perpendicularly, so the incident angle can be considered as 90 degrees. In this case, sin(theta1) = 1.
The refractive index (n) is given as the ratio of the speed of light in a vacuum to the speed of light in the medium.
Using the formula for the refractive index:
n = c/v
where c is the speed of light in a vacuum (approximately 3.00 x 10^8 m/s) and v is the speed of light in the medium.
Since the refractive index of air is approximately 1 (to three significant figures), we can consider it as 1 in our calculation.
To find the speed of light in the unknown liquid, we can use the equation:
v = d/t
where v is the velocity, d is the distance traveled, and t is the time taken.
Given that the distance traveled is 31.0 cm (or 0.31 m) and the time taken is 1.48 ns (or 1.48 x 10^-9 s), we can substitute these values into the equation to find the speed of light in the liquid.
v = (0.31 m) / (1.48 x 10^-9 s)
Once we have the speed of light in the liquid, we can find the refractive index by dividing the speed of light in vacuum by the speed of light in the liquid.
n2 = (3.00 x 10^8 m/s) / v
Finally, we can rearrange Snell's Law to solve for sin(theta2) and then use this to find theta2.
sin(theta2) = (n1 sin(theta1)) / n2
Since sin(theta1) is 1 and n1 is 1, we can simplify this to:
sin(theta2) = 1 / n2
Using the value of n2 calculated earlier, we can substitute it into the equation to find sin(theta2).
With sin(theta2), we can find theta2 by taking the inverse sine (sin^(-1)).
Now, since the angle of incidence is assumed to be 90 degrees (perpendicular), the angle of refraction (theta2) will also be 90 degrees. This means that the light does not change direction when it passes from air to the liquid.
Therefore, the wavelength of the laser beam in the liquid is the same as the wavelength in air, which is 633 nm (or 6.33 x 10^-7 m).