3rd of September, 2008
(Note: The curl of any radially symmetric vector field is 0.)
The curl of an electrostatic field is 0, so if and are not parallel, then only the fields in points (b) and (c) may be electrostatic fields.
Let us calculate the two terms of this sum separately:
where = ∕r, and
Thus the complete charge density is
Thus as r →∞, the total charge enclosed by the surface goes to 0.
The sphere has a homogeneous charge distribution, so q = r3ρ and dq = 4πρr2dr.
Thus the energy needed to assemble a charged sphere of radius R and charge Q is
The electric field is
and the potential is
Creating a gap by separating the halves is equivalent to removing (“mining out”) all the matter from the gap, and distributing it on the surface (i.e. it causes the same change ΔW in energy). Provided that the gap between the halves is very narrow compared to the planet’s radius (x ≪ R), the change in the gravitational field is negligible.
Using Gauß’s law, it is easily shown that the gravitational potential below the surface of the spherical planet is
where γ is the gravitational constant, M is the mass of the planet, and r is the distance from the centre.
When moving a piece of mass dm up to the surface, the increase in energy is V (R) - V (r)dm. To get the total change in energy, we must integrate over the volume of the gap:
The force keeping the halves together is
(I.e. the same as the weight of an object of mass M on the surface of the planetoid.)
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