What is the capacitance?

The capacitance of a conducting sphere is C. Two such identical spheres touch each other externally. What is t?

• The capacitance of a conducting sphere is C. Two such identical spheres touch each other externally. What is the resulting capacitance of this system?

• Answer:

  Maxwell solved this problem in his treatise. The answer is not C/2. _________________ Using the method of images, place a unit positive charge at the point of contact (origin) of the spheres and two parallel planes each distant 2a from said origin. We then inquire as to what series of charges placed along the line connecting the centers of said spheres results in UNIT potential. Let the origin charge via induction on planes create a series of positive unit charges with locations qp = s*2/a from the origin, and a series of negative unit charges at qn = 1/a + s*2/a. Where a=sphere radius. The infinite sum of these positive and negative unit charges with reciprocal distance from the origin is then proportional to total charge and thus total capacitance: Q ≈ [s: 0,∞]∑ 1/qn - [S: 1,∞]∑ 1/qp Since the combined series converges we find: Q ≈ lim ∑ 1/(2*s(2s-1)) [1,∞] Q = 4πεo*a lim ∑ 1/(2*s(2s-1)) [1,∞] Q = 4πεo*a*ln(2) Given two spheres: C = 2*Q / V More simply: Let v=1 unit C = Q*v C = Q The total negative charge required to meet the constraint of v=1 equates to the total reciprocal distance from the contact point: -q = [1,∞] a ∑- 1 / (S * (1 / a + 1 / a)) And similarly for positive charge we need: +q = [0,∞] a ∑+ 1 / (1 / a + S * (1 / a + 1 / a)) Since C = Q, C = a*(-q+q) = a*ln(2) Two spheres = 2*C rohan:re:email: There is no link for my solution. There is now.