Radius of a circle from refracted light in water

[snoweangel27]

[SOLVED] radius of a circle from refracted light in water

Homework Statement

A swimmer at the bottom of a pool 4 m deep looks up and sees a circle of light. If the index of refraction of the water in the pool is 1.33, find the radius of the circle.

Homework Equations

$$\theta$$ (c) = Arcsin(n$$_{2}$$/n$$_{1}$$)
n(1)*sin$$\theta$$(1) = n(2) * sin$$\theta$$(2)

The Attempt at a Solution

I used to first equation to find the angle of incidence, then used snell's law to calculate the angle of refraction. After which I am unsure of the correct way to proceed. I tried setting tan$$\theta$$ = 4/x, where I was using x as the radius, but I am pretty sure that it is not correct.

 

[jbsteele]

Any help would be greatly appreciated.AnswerUsing Snell's law, we have n(1)*sin\theta(1) = n(2) * sin\theta(2).We know that the index of refraction of the water is n2 = 1.33 and the angle of incidence is \theta(1) = 90o.Substituting these values into Snell's law gives us:1 * sin90o = 1.33 * sin\theta(2)Solving for the angle of refraction gives us \theta(2) = 54.9o.Now we can use trigonometry to solve for the radius of the circle. Since the angle of refraction is 54.9o, the angle at the centre of the circle is 180o - 54.9o = 125.1o.We know that tan125.1o = 4/x, where x is the radius of the circle. Solving for x gives us x = 4/tan125.1o = 2.20 m.Therefore, the radius of the circle of light seen at the bottom of the pool is 2.20 m.

 

[Moetasim]

The correct way to proceed would be to use the relationship between the angle of incidence and the angle of refraction to calculate the refractive index of water. Once you have the refractive index, you can use the formula for the radius of a circle from refracted light in water, which is given by:

r = \frac{n_1}{n_2} * d

Where r is the radius of the circle, n1 is the refractive index of the medium the light is coming from (in this case, air), n2 is the refractive index of the medium the light is entering (in this case, water), and d is the depth of the water (4m in this case).

So, using the given values, we have:

r = \frac{1}{1.33} * 4 = 3.01 meters

Therefore, the radius of the circle of light seen by the swimmer is approximately 3.01 meters. This calculation assumes that the light is entering the water perpendicularly, which may not be the case in real life. In that case, the radius may be slightly different.