Final

How are the final potentials V1 and V2 of the spheres related?

Consider two widely separated conducting spheres, 1 and 2, the second having six times the diameter of the first. The smaller sphere initially has a positive charge q = 1.00×10-5 C, and the larger one is initially uncharged. You now connect the spheres with a long thin wire. How are the final potentials V1 and V2 of the spheres related? Find the final charges q1 and q2. What is the ratio of the final surface charge density of sphere 1 to that of sphere 2?

V1_final and V2_final will be the same. This is a necessary condition for electrostatic equilibrium; if it were not so, the potential difference would cause charge to continue to flow. The potential at the surface of the first sphere is given by Q1/(4πε0r1) = Q1/(4πε0r), and the potential at the surface of the second sphere is Q2/(4πε0r2) = Q2/(4πε0(6r)) = Q2/(24πε0r).

Equating these gives:
Q1/(4πε0r) = Q2/(24πε0r). Most of this cancels, leaving
6Q1 = Q2.

Conservation of charge assures us that the initial and final charge is the same: hence,

Q1 + Q2 = 10 μC

Solving these equations simultaneously gives:

Q1 + 6Q1 = 10
7Q1 = 10
Q1 = (10/7) μC
Q2 = 6(10/7) = 60/7 μC.

Since the diameter (and thus radius) of sphere 2 is 6 times the radius of sphere 1, its surface area is 6² = 36 times that of sphere 1. You can solve for the charge density explicitly if you like, but even without doing so you can see that sphere 2 has six times as much charge, spread out over 36 times the surface area, so its final surface charge density is one-sixth that of sphere 1.

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