# Calculating Water Depth Using Snell's Law and Trigonometry

• prehisto
I recommend to use the second form of the Snell's law, n1sinα=n2sinβ, (with the angles) and the above expressions of sinα and sinβ. Try to eliminate x from these equations and you will get a function of α and h only which you can solve numerically.
prehisto

## Homework Statement

The light beam travels from air into water,the angle of incidence is 60 degress. The bent beam creates point F on the bottom of eater tank.If we imagine extension of incident beam inside the water, we obtain point B on the bottom of water tank. The distance between F and B is 1m. Calculate the depth of the water.
http://[url=http://postimg.org/image/rw3c1zipn/][PLAIN]http://s33.postimg.org/rw3c1zipn/bilde.jpg

## The Attempt at a Solution

[/B]
First i used the Snells law to obtain the angle of the bent beam.
Second i thought i use trigonometric functions to obtain the height.
tg(angle1)=x/h and tg(angle2)=(x+1)/h
From which I obtained negative value of x, which confused me.
So guys, do you think this is the way to go or someone has any different ideas?

Last edited by a moderator:
prehisto said:

## Homework Statement

The light beam travels from air into water,the angle of incidence is 60 degress. The bent beam creates point F on the bottom of eater tank.If we imagine extension of incident beam inside the water, we obtain point B on the bottom of water tank. The distance between F and B is 1m. Calculate the depth of the water.
http://[url=http://postimg.org/image/rw3c1zipn/][PLAIN]http://s33.postimg.org/rw3c1zipn/bilde.jpg

## The Attempt at a Solution

[/B]
First i used the Snells law to obtain the angle of the bent beam.
Well, show how you applied Snell's law.
prehisto said:
Second i thought i use trigonometric functions to obtain the height.
tg(angle1)=x/h and tg(angle2)=(x+1)/h
Yes, this is the right way, What values did you substitute for angle1 and angle 2?
prehisto said:
From which I obtained negative value of x, which confused me.
So guys, do you think this is the way to go or someone has any different ideas?
It is impossible to get negative x, Show your work in detail.

Last edited by a moderator:
prehisto
From Snells law I obtained angle value 40,5 degrees (angle1)

ehild said:
t is impossible to get negative x, Show your work in detail.
From trigonometric equations I expressed h, so x/( tg(angle1) )= h and (x+1)/(tg(angle2))=h
Then x/(tg(angle1))=(x+1)/(tg(angle2))
And obtained x=- (tg(angle1)/(tg(angle1)-tg(angle2)

In fact, now I obteined positive value of 0,97. So I think everything is all right :)

From Snell's law n1sinα=n2sinβ where α is the angle of incidence and β is the angle between normal and beam in the water.
Using the given info,
sinα=√3/2=(x+1)\√(x+1)2+h2,
from here you get x= √3h-1 and x=-√3h-1 (however this is negative, so not a physical solution) and sinβ=x\√x2+h2=√3n1/2n2) which gives you an equation with 1 unknown, namely h, after substituting x=√3h-1. If I did not make any mistakes, the last equation should give you h.

prehisto said:
From Snells law I obtained angle value 40,5 degrees (angle1)From trigonometric equations I expressed h, so x/( tg(angle1) )= h and (x+1)/(tg(angle2))=h
Then x/(tg(angle1))=(x+1)/(tg(angle2))
And obtained x=- (tg(angle1)/(tg(angle1)-tg(angle2)

In fact, now I obteined positive value of 0,97. So I think everything is all right :)
Take care at the rounding. It is right otherwise, keep more significant digits during the calculations. You need to determine h from the x value yet.

prehisto

## 1. What is Snell's law?

Snell's law, also known as the law of refraction, is a fundamental principle in optics that describes how light waves change direction when they pass through different mediums with different refractive indices.

## 2. How does Snell's law relate to the depth of water?

Snell's law can be used to calculate the depth of water by measuring the angle of incidence and refraction of light passing from air to water. The depth of water can be determined using the equation: d = (n1/n2) * h, where d is the depth of water, n1 is the refractive index of air (approximately 1), n2 is the refractive index of water (approximately 1.33), and h is the height of the object above the water's surface.

## 3. What factors affect the accuracy of using Snell's law to measure the depth of water?

The accuracy of using Snell's law to measure the depth of water can be affected by several factors, including the clarity of the water, the angle of incidence of light, and the accuracy of measuring the angles of incidence and refraction. Additionally, the temperature and salinity of the water can also affect the refractive index and thus the accuracy of the calculation.

## 4. Can Snell's law be used to measure the depth of any body of water?

No, Snell's law can only be used to accurately measure the depth of transparent, still bodies of water such as pools, lakes, and calm oceans. It cannot be applied to turbid or moving bodies of water, as the refractive index may vary and affect the accuracy of the calculation.

## 5. How is Snell's law used in practical applications?

Snell's law has many practical applications, including in the design of lenses and optical instruments, such as microscopes and telescopes. It is also used in underwater imaging and navigation systems, as well as in the field of geology to study the properties of minerals and rock formations.

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