Two small metallic balls of radii R 1 and R 2 are in vacuum at a distance considerably exceeding their dimensions and have a certain total charge. Find the ratio q 1 /q 2 between the charges of the balls at which the energy of the system is minimal. What is the potential difference between the balls in this case?
Text Solution
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Sol. The electric energy of this system is
W = W 1 + W 2 + W 12 = 
,
where W 1 and W 2 are the intrinsic energies of the balls (q ϕ /2), W 12 is the energy of their interaction (q 1 ϕ 2 or q 2 ϕ 1 ), and l is the distance between the balls. Since q 2 = q – q 1 , where q is the total charge of the system, we have
W = 
.
The energy W is minimal when δ W/ δ q 1 = 0. Hence
q 1 ≈ q 
and q 2 ≈ q 
,
where we took into account that R 1 and R 2 are considerably smaller than l and
q 1 /q 2 = R 1 /R 2 .
The potential of each ball (they can be considered isolated) is ϕ ∝ q/R. Hence it follows from the above relation that ϕ 1 = ϕ 2 , i.e. the potential difference is equal to zero for such a distribution.
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