## Have Formed a Layer Around the Moon and Swallowed the Apollo Lander?

Before the first moon landing, several NASA scientists were concerned that about the possibility that the lunar lander might be engulfed by a layer of dust hovering just above the moon’s surface, keep there by a balance between gravity and electrostatic repulsion.
a. What must the net charge on the moon be if it can support a 1g dust particle that has charge of 1mC upon it at a distance of 10km above its surface? (Assume the moon acts like a uniformly charged sphere)

b. This grain is in an equilibrium position – what sort of equilibrium is this? (Hint, what direction is the net force on the grain in if you move it slightly closer to or further from the moon?) Explain.

Solution:

The weight of the grain of dust, mass m (the force on it due to the gravity of the moon, mass M) must be equal to the electrical repulsion between the moon, charge Q, and the dust, charge q.

Fg=Fe

GMm/r2 = kqQ/r2

The magnitude of the charge of a single electron is typically denoted by e.

GMm = keQ

Q = GMm/(ke)

Q = 6.67300 × 10-11 m3 kg-1 s-2 * 7.36 * 1022kg * 1.0 * 10-3/(9 * 109 N m2 C-2 * 0.001

Q = 545.7 C

As can be seen above in the force balance equation, the distance between the dust and the moon cancels out from both sides – there is no particular distance that the dust will hover above the moon: if the forces cancel at any one distance, then they cancel at every distance. Thus it is a Metastable equilibrium.

This explains why the dust layer was not seen - without a single place where the layer of dust would want to form, it is unable to accumulate to any degree that is a danger to moon landings.