C.6.2 Motion across the magnetic field

C.6.2.1. Induced potential

A difference of potential appears along conductors moving across a magnetic field. For a conductor of length L moving with a velocity vector v in a magnetic field B , the difference of potential V is:

For instance a 5 kV potential difference is expected along a 20 km tether in low Earth orbit (which can have applications for power generation).

This potential difference can power a current. The value of both depends on the ability of surfaces at each end to collect ions and electrons. Hence the plasma density and the conductive surface area of the satellites are important. Currents flowing through the plasma along field lines connected to the ionosphere complete the circuit.

C.6.2.2. Current through a long conductor

The current passing through a conductor depends on the geometry of the anode and the cathode and whether they are passive or active. For a spherical probe, electron collection at the positive (anode) end can be approximated using the Alpert Law (see [12]).

if and

Here j₀ is the random current of the ambient plasma. Generally, current collection at the anode is not a problem. For the negative (cathode) end, the slower velocity of the ions means that an equivalent collecting area results in a much lower current which does not balance the anode current. Hence, unless a far larger collection area is available, it is common practice to use electron emitters on the cathode.

C.6.2.3. Force on conductors

When a conductor of length dl having a current passing through it i(l), is moving across a magnetic field B it experiences a force acting perpendicular to the direction of flow of the current tether and to the magnetic field

Therefore, the total force acting on a tether of length L is given by the equation:

While the electro-dynamic forces act perpendicular to the tether, gravitational and centrifugal forces act along its length and provide a restorative force in the frame of motion of the tether’s centre of mass. If the electro-dynamic force varies (e.g. due to varying magnetic field or current collection along the orbit) oscillations can develop. These cause undesired accelerations, stresses on long tether material and the possibility of being caught up on the deployment mechanism or another part of the spacecraft.

When a closed loop is moving across a magnetic field an electromotive potential V appears along the loop.

where F is the flux of magnetic field intercepted by the loop.

A loop having a current passing through has a magnetic moment M. The ambient magnetic field exerts a force on M which is

and a torque T,