Power from a Tether. A few years ago, the space shuttle Columbia tried an experiment with a tethered satellite. The satellite was released from the shuttle and slowly reeled out on a long conducting cable as shown in the figure (not to scale). For this problem we will make the approximations

- the shuttle is moving at a constant velocity

-the earth's magnetic field is constant and uniform

-the line of the tether, the velocity of the system, and the magnetic field are all perpendicular to each other.

The earth's field produces an electric potential difference from one end of the cable to the other. The idea is to use a system like this to generate electric power in space more efficiently than with solar panels.

a. Explain why a potential difference (voltage) is produced.

b. If the earth's magnetic field has the strength B, the shuttle/satellite system is moving with a velocity y, and the tether has a length L, write an expression to determine the potential difference 1W from one end of the tether to the other. (Hint: Think about all of the forces on a moving charge. While the magnetic field exerts a force to push it one direction, this separates the charge, creating an electric field, which also exerts a force.)

c. At the shuttle's altitude, the earth's field is about 0.3x10^-5 T and the shuttle's speed is about 7.5 km/s. The tether is 20 km long(!). What is the expected potential difference (in Volts)?

d. At the altitude of the shuttle, the thin atmosphere is lightly ionized allowing a current of about 0.5 Amps to flow from the satellite back to the shuttle through the thin air. What is the resistance of the 20 km of ionized air?