Question

Tethers are long dangling wires that use the velocity of a spacecraft through the Earth's magnetic field to generate electrical power. Alternatively, electrical currents pushed through the tether might be used to change the spacecraft velocity for either propulsion or for braking and reentry. Several 'space tether experiments have been flown to investigate the possible use if Faraday's law for power generation and/or propulsion. Examples are the 1996 joint US/Italy effort to deploy a 20 km tether from one of the space shuttles to generate power, or the European Space Agency's Young Engineers Satellites (YES2) experiment that deployed a 31 km tether as a means of reentry for a small payload dropped from a Russian Foton spacecraft. (The radius of the Earth's surface is about 6400 km. Random side note: originally the meter was defined such that the Earth's circumference along the prime meridian was exactly = 4x10' m.) (a) (Mechanics review question.) What is a typical low-Earth orbital velocity? To be definite, assume an orbit 300 km above the surface of the Earth. (This is the distance from the surface of Earth, not from the Earth's center.) Solve this two ways: once by balancing forces in the rotating reference frame and again by using the Virial Theorem. (b) The Earth has a magnetic field that is approximately described by an ideal magnetic dipole. This field apparently arises from circulating charge currents within the convective parts of the Earth's mantle. These flows are directly related to the flows causing plate tectonics. Fascinating stuff... OK, if you look up the typical size of the Earth's magnetic field on the Earth's surface at the equator you get approximately B = 5 x 10$T. Use this value to estimate the magnitude of the Earth's magnetic dipole moment, m. (See Griffiths Equation 5.88, and be careful with angles: What is at the equator?) (c) Now combine the typical orbital velocity and the typical magnetic field magnitude and direction to calculate the motional EMF that could be generated along a 20 km conductive cable extended down from the satellite (d) Lovely, so now you have a cable stretched out and it has some EMF generated along its length. However, to generate useful POWER, the product of current and voltage, you need to have a closed circuit to allow current to flow. Perhaps you could stretch a second cable out from your spacecraft to touch the st cable and to complete the circuit? Explain why that's not going to work (e) In fact, the circuit is completed by the ionized plasma that surrounds the Earth, the so-called ionosphere. In the shuttle tether, there was an insulating, roughly 1cm diameter nylon cable surrounded by conductive copper braid of 3 mm thickness, all wrapped in an insulating Kevlar layer. Data taken during the experiment suggested that the ionosphere was 'somewhat less conductive than the copper braid. Make life simple by assuming that the ionosphere conducted in a cylindrical pipe 20 km long and with cross sectional area equal to the areal size of the space shuttle (roughly 200 square meters) estimate the plasma conductivity. How much power runs through the tether-plasma circuit? What effect will this power have on the spacecraft?

Answer 1

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We have £g* Ex 20. m 20m - GMm 20 Roth o alguna V Rth = 76-67x10" x 6 x1024 v (6400+300 ) *103 = 40.02 X1013 V 67x105 V0.597 x108 : 0.772 2104 U 1.72 x 10 visms! m Bdipele = Mom (2 Core it sino ê ) 40 g 3 l na magnetic dipole moment - At equator, or 7 Beanaker . tom se o 40 g New, 8= Rth , a ma B har² = B x 44 x (R+ h)" le Scanned with CamScanner

>> m ) 5x10-5x4ū *(6700 x1000) - Mo. Now, Mo - m = 5x10.5 x 3 x 102° 107 2 15 X 100 o m : 1.5*109 A m2 CC) we know that , Motional emf E = (u x B le where , cú= velocity of the conductor Å External magnetic field la length of the conductor {z Ubl sino New, at the equatar , I in perpendicular to B, T9 To a z Ubl 7.72 X10 x 5 x10X 20 X 10 = 7.72 x 103 v € 2 772 V Scanned with CamScanner

(a) orbital velocity =

7.72 x 10'ms-1

(b) magnetic dipole moment of earth =

1.5 x 10 Am

(C) Motional emf =

(d) If we use a second wire to complete the circuit, the second wire itself will develope motional emf identical to the 1st wire due to motion through earth's magnetic field. As a result no current would flow the circuit. The situation is identical to two batteries with same potential difference, connected with reverse polarity, i. e, positive of one battery is connected to the positive of another battery and the negatives are connected together.