Electrostatic Bearings for a Space Elevator? Maybe, maybe not ...
The current space elevator design (2014) suggests an "electrostatic bearing" between photovoltaic panels and the 1 meter diameter tether. Earnshaw's Theorem tells us that arrangements of charges unstable, or at best neutrally stable, and so the restoring forces would be divergent, not convergent.
But perhaps we can use electronics to adjust voltage fields relative to the conductive tether, so let's assume we can adjust our way to stability.
But how strong are these forces, anyway?
Voltages in vacuum are limited. There is always a tiny amount of ionized gas around, even near geosyncronous orbit, and the electrons or ions will get accelerated by an electric field.
What is the attractive force of a parallel-plate capacitor? The energy stored in the electric field is C V^2 / 2 , always positive because of the V^2 whether the voltage difference is positive or negative. You can't make a repulsive force with a capacitor; the foils in an electroscope repel because they are inside a container and are attracted to the walls. The universe can be considered a container, perhaps, but you need an infinite voltage out to infinity ...
Never mind, let's assume we can center with greater and lesser attractive forces. Repulsion will not be larger than the attraction force, so at most we are pessimistic by a factor of 2.
Assume that the bearing surface A is 1 m2, and the gap l is 1 centimeter. The capacitance is C = \epsilon A / l , and \epsilon is 8.85e-12 Farads/meter, so the capacitance is 8.85e-10 meters, 885 picofarads.
I'm not sure of the voltage that will strike and sustain an arc across 1 cm of near-vacuum - since kilovolts can cause destructive arcing across meters of solar cell, let's assume 1 kV. The total energy E stored in our vacuum capacitor is 442 μJoules. Force is the derivative of energy with distance, d E / d l Assuming constant voltage, the energy change is the caused by the capacitance change, so
As the gap narrows, the energy goes up; the force is (as expected) attractive, 44 milliNewtons. The force on a cylinder 1 meter in diameter is 2/π times that ( the integral of the cosine of the force from -π/2 to π/2 ) or 28 mN .
If the collar surface is carbon nanotube and weighs 1.4 grams with a density of 1.4, then it has a volume of 1 cm3, spread out over 2π square meters; 160 nanometers thick. Assume that the solar panel, the structure, the wiring, the one meter high collar, and the control electronics weigh ten grams. That is awfully light ...
The angular velocity ω of the space elevator is 2 π / 86164s or ω = 73 μr/s . The Coriolis acceleration is 4 \pi \omega V If the climber is moving at 100 meters per second, the Coriolis acceleration is 92 mm/s2, so the Coriolis force on 10 grams is 0.9 mN . It looks like we can handle Coriolis force, if the system is extremely light and very well behaved.
If the tether is sloping up from a point 60 kilometers off the equator, and the displacement diminishes with a slope of 60 km/6,000 km near the station, then there is a lateral force on the climber of g / 100 or 98 mm/s2. That is perpendicular to the Coriolis force, so the sum of the forces is 134 mm/s2, or 1.3mN .