Question

Sphere A, has Q units of charge. If we take some little bit of charge q off of sphere A, and put it on sphere B. What should the ratio q/Q be if we want the electrostatic force between the spheres to be a maximum?

Answer 1

`1/2`

Explanation:

The force between the spheres will be proportional to the product of the charges on them. Since total charge is conserved the charges on the two spheres are `Q-q` and `q`, respectively. So the force is proportional to the product `(Q-q)q`. It is easy to see that this product is a maximum when `q=Q/2`. This can be proven in any one of several ways :

1. The arithmetic mean of the two numbers `Q-q` and `q` is fixed (at `Q/2`). The geometric mean `sqrt((Q-q)q)` can never exceed the arithmetic mean, but can at best be equal to it - when the two numbers are equal. Thus, the maximum for `(Q-q)q` occurs for `Q-q=q implies q=Q/2`
2. The product `(Q-q)q` can be interpreted as the area of a rectangle whose sides are `Q-q` and `q`, respectively. The perimeter of such a rectangle must be fixed (at `2Q`). Since among all rectangles with a given perimeter, the square is the largest in area, the maximum occurs when `Q-q=q`.
3. `(Q-q)q = Qq-q^2=Q^2/4-((Q/2)^2-2 Q/2 q+q^2) = Q^2/4-(Q/2-q)^2`. This shows that the largest value of the product will occur for the smallest value of `(Q/2-q)^2` - which of course happens for `Q/2-q=0`.
4. `d/{dq} (Q-q)q = Q-2q` : equating this to zero shows that the extremum must be at `Q-2q=0` (It is easy to check that the second derivative is negative, showing that this, indeed, is a maximum).