Solve Refraction Problem: Archimedes' Gold Crown

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In summary, Archimedes observed that light appeared to bend as it entered the water in his partially filled bathtub, even though he was not familiar with Snell's Law or the refractive index of dirty bath water. He noticed that the crown appeared to be located one meter away from the edge of the tub when viewed at a 35° angle above the water's edge, with the tub being one meter high. Using Snell's Law, we can determine the depth of the King's bath water by taking into account the angle of incidence and the height of the bathtub. Without a diagram, it is difficult to provide a specific solution.
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Homework Statement



Archimedes left his gold crown lying on the bottom of his partially filled bathtub. Although he did not know Snell's Law in fact he thought it was Smell's Law (they don't bathe very often back then), or even that the refractive index of dirty bath water was 1.35, he did note that light appeared to bend as it entered the water. As he peers into the water at 35° above the water's edge he sees the crown one meter away from the edge of the tub. His tub is one meter high. Given the information provided, determine the depth on the King's bath water.


Homework Equations



Snell's law- n1 sin θ1= n2 sin θ2


The Attempt at a Solution



I found the angle of incidence which was 37.37 but I don't know how to incorporate the apparent location of the crown and the height of the bathtub into this problem. Plz help. ty.
 
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  • #2
We really need a diagram to begin with!
 

1. What is Archimedes' Gold Crown problem?

Archimedes' Gold Crown problem is a famous problem in the field of optics, also known as the "crown jewel" of Archimedes' work. It involves determining whether a given crown is made of pure gold or if it contains a cheaper metal, by using principles of refraction and buoyancy.

2. How did Archimedes solve the problem?

Archimedes used the principle of buoyancy, which states that an object submerged in a fluid will experience an upward force equal to the weight of the fluid it displaces. By measuring the volume and weight of the crown, Archimedes was able to calculate its density and determine if it was made of pure gold or not.

3. What is the role of refraction in this problem?

Refraction is the bending of light as it passes through different mediums, such as air and water. In this problem, Archimedes used the fact that different materials have different refractive indices, or the measure of how much they bend light, to determine the density of the crown and ultimately solve the problem.

4. What are the applications of this problem in modern science?

The principle of refraction and the use of density to determine the purity of materials is still used in various fields such as chemistry, materials science, and geology. The concept of buoyancy is also fundamental in understanding the behavior of fluids and designing objects that can float or sink in them.

5. Why is this problem considered a breakthrough in science?

Archimedes' solution to the Gold Crown problem was a groundbreaking discovery, as it demonstrated the importance of using mathematical principles to solve real-world problems. It also laid the foundation for the development of concepts such as buoyancy and refraction, which are crucial in various scientific fields today.

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