# How to get maximum electrostatic force?

1. Oct 26, 2015

### catch22

1. The problem statement, all variables and given/known data
Let's take two spheres that are somewhat nearby. On one sphere, sphere A, there is Q units of charge. Now, 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?

2. Relevant equations
F = K Q q / r2

3. The attempt at a solution

For this question, I just took plotted numbers and found that the charge q has to be half of Q in order to have the greatest electrostatic force between the spheres. So the ratio of q/Q = 1/2

How would one do this problem without plotting numbers?

2. Oct 26, 2015

### BvU

Hi catch,

Do you know what the criteria are for the maximum of a function of one variable ? If so, all you need to do is write F as a function of one variable !

3. Oct 26, 2015

### catch22

unfortunately, I don't. do you mean like global maximum/minimums of a function?

4. Oct 26, 2015

### BvU

I mean like if f(x) has a maximum in x0, then what about df/dx at x = x0

5. Oct 26, 2015

### catch22

oh, so if I picture a graph, the highest point is where the slope is 0.
and to get the slope, we need the derivative.

6. Oct 26, 2015

### BvU

Yessss! Now, what derivative ? f is easy: F. But what is a candidate for the variable x ?

7. Oct 26, 2015

### catch22

q or Q?

8. Oct 26, 2015

### BvU

Well, that's a no-brainer: Q is a number, so it must be q ! And now it's time to realize that the Q in the problem statement is not the Q in the relevant equation ! But you knew that, right ?

9. Oct 26, 2015

### catch22

whoops, for a second I thought Q was the charge of the sphere that was taking in q.

10. Oct 26, 2015

### BvU

So what is now the one-variable function you are going to differentiate with respect to $q$ ?

11. Oct 26, 2015

### catch22

F = k q Q / r^2

12. Oct 26, 2015

### catch22

whoops, should be F = k (Q-q)(q) / r^2

13. Oct 26, 2015

### BvU

Agreed. Does it work out OK now ?

14. Oct 26, 2015

### catch22

hmm, my q turned into a 1.
dF / dq = K(Q-1) / r^2 = 0

Q = 1

doesn't seem right.

15. Oct 26, 2015

### BvU

Ah ! So not the same as when looking at the plotted figure ?
Or perhaps a second try differentiating F = some constant times ( Qq - q2 ) ?

16. Oct 26, 2015

### catch22

where did ( Qq - q2 ) come from?

17. Oct 26, 2015

### BvU

!

18. Oct 26, 2015

### catch22

oh, we shouldn't have factored out the q?

anyways, k (Q-2q)/ r^2 = 0

Q - 2q = 0

2q = Q

q/Q = 1/2

19. Oct 26, 2015

### BvU

Bingo ! Well done.