Question

You are on an engineering team examining the feasibility of a satellite launching system that uses the electric force instead of rocket fuel. The proposed launching system consists of 4 metal spheres, each one 15 m above the ground, arranged at the corners of a square with sides 5 meters long. The satellite is placed on the ground below the center of the square. To launch the satellite, the four spheres are first each given a charge of +Q/4 and the satellite is given a charge –Q. The satellite is then released and is pulled straight upward through the center of the square.

When the satellite reaches 15 m above the ground (the same height as the spheres), the spheres are neutralized. To evaluate this idea, you decide to calculate how big Q would have to be to launch a 100 kg satellite at a speed of 8 km/s. You are also worried about dielectric breakdown of the air, which causes spontaneous sparking in the presence of electric fields larger than 3 x 10⁶ V/m and so decide to check whether that will be a problem. For this purpose, you assume that the spheres have a diameter of 1 m.

(a) Give a purely symbolic answer to the problem in terms of (only) the quantities given in the problem and known constants. Be sure to define each of the symbols you use.
(b) Give a numerical answer to the problem.

Answer 1

(a) we know that the electrostatic potential energy between the charges q ₁ and q ₂ placed at r distance apart is given by

In the above figure, OC = and AC = = 15.411 m (by pythagorean theorem)

For position of satellite just below 15 m from the square and spheres is given by

-----------(1)

and for position of satellite just on the level of square and spheres is given by

-------------(2)

Let ,the velocity of satellite is v;then by conservation of energy,

----------------(3)

The electrostatic field at the center of sphere is not greater than

The electrostatic field at the center of sphere is   

By equations (1),(2) and (3),we get

(b)given v = 8 km/s = 8000 m/s, m = 100 kg

Q = 31978.466 C