proving maximum by differentiation

Question

Assume I have a large amount of charge, +Q, that I can distribute between 2 point particles such that one particle gets +fQ of charge, and the other gets +(1-f)Q, where 0≤f≤1. (a) Use the first and second derivative from calculus to prove that the electrostatic force between the particles (when they are a distance r apart) is maximum when f=0.5. (b) Why is this result not valid if Q = 1.602 × 10^-19 coulombs?

Answer 1

a) If you have 2 point charges, the force between them is

F = k*q_1*q_2/r^2, where k is a constant, r is the distance between the charges, and q_1 and 1_2 are the charge of each of the points.

If you distribute the total charge Q between the two charges, then q_1 = f*Q and q_2 = (1-f)*Q, where f is a number between 0 and 1 representing the fraction of the total charge on q_1

Substituting these values into the force equation gives F = k*f*Q*(1-f)*Q/r^2 = (k*Q^2/r^2)*f*(1-f).

(b)

To find the value of f that maximizes the force, you want to find the critical values for f, that is, the values of f where if you plot Force as a function of f, you have a slope of 0. Only these points are possible minima or maxima.

set F(f) = (k*Q^2/r^2)*f*(1-f)

Differentiate with respect to f

F'(f) = (dF)/(df) = (k*Q^2/r^2)*(d/df)[f*(1-f)] = (k*Q^2/r^2)*(d/df)[f-f^2] = (k*Q^2/r^2)*(1-2f)

To find the critical values, set this derivative equal to 0:

(k*Q^2/r^2)*(1-2f) = 0 => (1-2f) = 0 [because (k*Q^2/r^2) is a constant multiple that cannot be 0]

1-2f = 0 => 2f=1 => f=0.5

Thus the critical value of f is f=0.5. To determine if this is a maximum or a minimum, you need to check the second derivative

F''(f) = (d/df)[(k*Q^2/r^2)*(1-2f)] = (k*Q^2/r^2)*(d/df)[(1-2f)] = (k*Q^2/r^2)*(-2) = -2k*Q^2/r^2

This number is a negative constant, meaning that at f=0, the second derivative is negative. This means that the point behaves like a downward facing quadratic, and has a maximum at this point.

b) In part a, we assumes that the charge could be distributed in any way we wanted. However, charges are quantized, meaning that all charge amounts are whole-number multiples of the smallest possible unit of charge, which is 1.6*10^{-19} C. If the total charge Q is equal to this number, that would require putting a charge less than the minimum charge on one of the point charges, which is not possible.