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.
To find the distance between the point P on the crystal ball and the image of the jewel on your side, we can use the concept of refraction and apply Snell's law.
Snell's law states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the speeds of light in the two media. In this case, the incident medium is air, and the refracting medium is the crystal ball.
Let's set up the coordinate system with the center of the crystal ball as the origin. The point P can be represented as (0, y) where y represents the distance from P to the center along the y-axis.
Given that the diameter of the crystal ball is 0.2 m, we can conclude that the radius is 0.1 m.
According to the problem statement, the fortune teller holds up the jewel on her side of the crystal ball in the equatorial plane. Therefore, the jewel is also at a distance of 0.1 m from the origin along the x-axis.
Now, let's assume that the light ray from the jewel enters the crystal ball and refracts towards point P, creating an image on your side. Let the angle of incidence (angle between the incident ray and the normal to the surface) be i and the angle of refraction (angle between the refracted ray and the normal) be r.
By Snell's law, we have:
sin(i) / sin(r) = n2 / n1
Here, n2 is the index of refraction of the crystal ball, given as 1.5, and n1 is the index of refraction of air, which is approximately 1.
Since we are interested in finding the distance y, we need to analyze the path of the refracted ray. Let's assume that the refracted ray passes through a point Q at a distance x from the origin along the x-axis.
For the right-angled triangle formed by the points Q, P, and the origin (O), we have:
OQ = x,
OP = y,
and the distance OQ is equal to the radius of the crystal ball.
Now, using trigonometry, we can express sin(i) and sin(r) in terms of x and y:
sin(i) = OP / OQ = y / x
sin(r) = OQ / OP = x / y
By substituting these expressions into Snell's law, we get:
y / x = n2 / n1
y / x = 1.5 / 1
y = 1.5x
So, the distance y is 1.5 times the distance x.
Now, let's find the value of x. We can use the Pythagorean theorem for the right-angled triangle formed by the points P, Q, and the center of the crystal ball (C):
OQ^2 + CQ^2 = OQ^2
x^2 + (0.1)^2 = (0.1)^2
x^2 = (0.1)^2 - (0.1)^2
x^2 = 0
x = 0
Therefore, the distance x is 0.
Substituting the value of x back into our previous equation, we get:
y = 1.5(0) = 0
This means that the image of the jewel is on the same side of the crystal ball as the jewel itself, and it's located at the point P. So, the distance from P to the image of the jewel on your side is 0 meters.
Therefore, the final answer is 0 + 0.2 m = 0.2 m. The image of the jewel is located 0.2 meters away from point P on your side of the crystal ball.