# How to get maximum electrostatic force?

• catch22
In summary, the ratio of q/Q should be 1/2 in order to maximize the electrostatic force between two spheres with Q units of charge on one sphere and a small amount of charge q transferred to the other sphere. This can be determined by differentiating the function F = k(Q-q)(q)/r^2 and setting it equal to 0.
catch22

## Homework Statement

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?

F = K Q q / r2

## The Attempt at a Solution

[/B]
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?

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 !

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

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

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

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

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

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 ?

catch22
BvU said:
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 ?
whoops, for a second I thought Q was the charge of the sphere that was taking in q.

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

BvU said:
So what is now the one-variable function you are going to differentiate with respect to ##q## ?
F = k q Q / r^2

catch22 said:
F = k q Q / r^2
whoops, should be F = k (Q-q)(q) / r^2

Agreed. Does it work out OK now ?

BvU said:
Agreed. Does it work out OK now ?
hmm, my q turned into a 1.
dF / dq = K(Q-1) / r^2 = 0

Q = 1

doesn't seem right.

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

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

catch22 said:
F = k (Q-q)(q) / r^2
!

BvU said:
!
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

Bingo ! Well done.

catch22

## 1. How can I increase the electrostatic force between two objects?

To increase the electrostatic force between two objects, the distance between them must be decreased. This can be done by bringing the objects closer together or by decreasing the distance between their charges.

## 2. What factors affect the strength of electrostatic force?

The strength of electrostatic force is affected by the magnitude of the charges, the distance between the charges, and the medium surrounding the charges. The force increases with larger charges, decreases with greater distance, and is influenced by the permittivity of the medium.

## 3. How does the type of material affect electrostatic force?

The type of material can affect electrostatic force in two ways. First, the permittivity of the material can alter the force between two charges. Second, materials with higher conductivity can dissipate or neutralize charges, reducing the overall electrostatic force.

## 4. What is the relationship between electrostatic force and electric fields?

Electric fields are created by electric charges and are responsible for exerting electrostatic force on other charges. The strength of an electric field is directly proportional to the magnitude of the charges and inversely proportional to the square of the distance between them.

## 5. Can I manipulate electrostatic force for practical applications?

Yes, electrostatic force can be manipulated for various practical applications such as in electrostatic precipitators for air pollution control, electrostatic painting, and electrostatic separation of materials. Understanding the principles of electrostatic force can also lead to advancements in technology, such as the development of electrostatic motors and generators.

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