Charge Distribution on Connected Spheres

Question:

There are two charges: 1 positively charged sphere (holding + 10nC) with redius 1 cm & 1 negatively charged sphere (holding - 2 nC) with radius 2 cm. Their centres are seperated by 1 m. The two spheres are now connected by a conducting wire, what is the amount of charge finally stayed on each sphere?

Answer:

The charge distribution after the connection is made will be such that the electric potential on both spheres is the same. Otherwise charge would transfer along the wire to balance the potentials at each end of the apparatus. Since the spheres are far apart compared to their radii, the the charge distribution on each sphere may be taken as uniform.

You may use Gauss's law to find that the electric field at the surface of each sphere is E=k*q/r², where k is the coulomb constant, q is the charge on the sphere and r is its radius. The electric potential on the surface of each sphere is found by integrating the work done in moving a unit charge in from infinity. This gives us V=k*q/r for each sphere.

The ratio of the charges then will be q1/q2=r2/r1=2/1. The total charge is 10-2 or 8 Coulombs. q1=2q2 and q1+q2=8. Solve for q1 and q2.

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