Fortune tellers use crystal balls to see the future. A fortune teller has a crystal ball with an index of refraction of 1.5 and diameter of 0.2 m. You sit on one side of the ball and the fortune teller sits on the other. She holds up a small red jewel on her side in the equatorial plane of the ball. Let P be the point on the ball nearest to the jewel. How far away in m from P is the image of the jewel on your side?
Hint: the question mark in the image below is an unknown you'll need to figure out as an intermediate step. However, the requested answer is the distance from P, i.e. ? + 0.2 m.
Details and assumptions
The jewel is at a distance of 0.8 m from P.
To solve this problem, we can use the principles of geometric optics and the concept of refraction.
First, let's draw a diagram to visualize the situation. The diagram will consist of the crystal ball, the jewel, and the points P and Q.
```
_
| |
| |
|Q|
| |
----------------------P----------------------------
```
In this diagram, P represents the point on the ball nearest to the jewel, and Q represents the location of the image of the jewel on your side. The distance between P and Q is what we need to find.
Now, let's consider the path of light rays through the crystal ball. When light travels from one medium to another with a different refractive index, it undergoes refraction. In this case, the light travels from air (refractive index = 1) to the crystal ball (refractive index = 1.5).
The key concept here is that light rays incident on the crystal ball will bend towards the normal. The normal is the line perpendicular to the surface of the crystal ball at the point of incidence.
We can use Snell's law to calculate the angle of refraction:
n₁sinθ₁ = n₂sinθ₂
Where:
n₁ = refractive index of the initial medium (air)
θ₁ = angle of incidence
n₂ = refractive index of the final medium (crystal ball)
θ₂ = angle of refraction
In this case, the light travels from air (n₁ = 1) to the crystal ball (n₂ = 1.5). The angle of incidence (θ₁) can be calculated using basic trigonometry:
θ₁ = tan^(-1)(0.8 / 0.2)
Now, we can use Snell's law to find the angle of refraction (θ₂):
1 * sin(θ₁) = 1.5 * sin(θ₂)
Solving this equation for θ₂, we get:
θ₂ = sin^(-1)(1 * sin(θ₁) / 1.5)
Once we have the angle of refraction (θ₂), we can use basic trigonometry to find the distance between P and Q. Since P is the point on the ball nearest to the jewel, and the jewel is at a distance of 0.8 m from P, we can use the tangent function to find the unknown distance:
tan(θ₂) = ? / 0.8
Solving this equation for ?, we get:
? = 0.8 * tan(θ₂)
Finally, we need to add the distance between P and the surface of the ball (0.2 m) to find the total distance between P and Q:
Distance P-Q = ? + 0.2 m
By plugging in the values we have calculated, you can get the distance from P to the image of the jewel on your side.