A satellite dish is designed to pick up radio waves from geostationary satellites about 36000 km about the Earth's surface.
If the dish's surface is a part of a sphere of radius R, where should the receiver be placed compared to the center of the dish?
1. At a distance R/2 from the center of the dish, behind the dish (on the convex side).
2. At the center of the sphere, a distance R from the dish.
3. At a distance R/2 from the center of the dish, in front of the dish (on the concave side).
4. At a distance R behind the dish.
The satellte dish reflects the signal from the satellite to a detector in front of the concave side, at the focal point. It acts just like a mirror for light.
You should know the forcal point of a mirror is at R/2.
DirecTV satellite dishes are off-axis paraboloids, not part of a sphere. The focal point and detectors are thus kept out of the way of the incoming beam.
To determine where the receiver should be placed compared to the center of the dish, we need to consider the geometry of the situation. The question states that the satellite dish's surface is part of a sphere with radius R.
Since the satellite dish is designed to pick up radio waves from geostationary satellites, which are about 36,000 km above the Earth's surface, we can assume that the radio waves from the satellite will be parallel to each other when they reach the dish.
To receive these parallel radio waves as a focused beam, the receiver should be placed at the focal point of the dish. The focal point of a spherical dish is located at a distance equal to half of the dish's radius (R/2) on the concave side, which is on the side facing the dish.
Therefore, the correct answer is option 3: At a distance R/2 from the center of the dish, in front of the dish (on the concave side).