Fortune teller's 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?
Note that the question mark in the image below is the unknown you'll need to figure out, but the requested answer is the distance from P.
To determine the distance from point P to the image of the jewel on your side, we can use the principles of geometric optics. Here's how you can solve this problem:
1. Begin by drawing a diagram to represent the situation. Draw a circle to represent the crystal ball, labeling the center as O and the equatorial plane passing through point P as a line XY.
2. Place the small red jewel on the fortune teller's side of the ball, precisely in the equatorial plane (line XY). Label the position of the jewel as J.
3. Since the fortune teller is holding the jewel in the equatorial plane, the light rays from the jewel will pass through the center of the ball, O, without any deviation. Therefore, the light rays will continue straight across to point P on your side of the ball.
4. The distance from the jewel (J) to the nearest point on the surface of the ball (P) is given as the diameter of the ball, which is 0.2 meters.
5. Now, we need to find the distance from point P to the image of the jewel on your side of the ball. To do this, we can use the concept of refraction.
6. The refractive index of the crystal ball is given as 1.5. When light passes from a medium with a higher refractive index to a medium with a lower refractive index, as is the case here, the light rays bend away from the normal line.
7. In this case, the normal line is perpendicular to the surface of the ball at point P. Draw the normal line passing through point P and label it as N.
8. Using the laws of refraction, we can determine the angle of incidence (i) and the angle of refraction (r) at the interface between the crystal ball and air.
9. Since the normal line N is perpendicular to the surface, the angle of incidence (i) is the angle between the incoming light ray and the normal line. This angle is determined by drawing a line from point J to point P and measuring the angle formed between that line and the normal line N.
10. Knowing the angle of incidence (i) and the refractive index (n1) of the crystal ball, we can use Snell's law to find the angle of refraction (r) at the air-crystal ball interface. Snell's law states that n1 * sin(i) = n2 * sin(r), where n1 is the refractive index of the first medium (crystal ball) and n2 is the refractive index of the second medium (air).
11. Once we have the angle of refraction (r), we can extend the refracted light ray from point P to meet the surface of the ball. This point of intersection is the image of the jewel on your side.
12. Measure the distance from point P to the image of the jewel to find the final answer.
By following these steps and utilizing the principles of geometric optics and refraction, you can determine the distance from point P to the image of the jewel on your side of the crystal ball.