# Coulomb's Law of a sphere

## Homework Statement

Of the charge Q initially on a tiny sphere a portion q is transferred to a second nearby sphere. Both spheres can be treated as particles. For what value of q/Q will the electrostatic force between the two spheres be maximized?

## Homework Equations

Coulomb's Law
We want to maximize the force on F(q) = k [(Q-q)q/r^2]
Find dF/dq=o
Solve for q in terms of Q

## The Attempt at a Solution

I dont even know where to start. This is my very very first HW question in physics 2 and its blowing my mind.

tiny-tim
Homework Helper
Hi Lavid2002! (try using the X2 icon just above the Reply box )

I don't get it We want to maximize the force on F(q) = k [(Q-q)q/r^2]
Find dF/dq=o
Solve for q in terms of Q

… what can you not do? Hello!

I'm stuck in a rut. Right now I'm blowing through my calc 2 HW no problem. But this basic differentiation and solving for a variable is blowing my mind.

I would love some help : /

tiny-tim
Homework Helper
Well, you need d/dq of (Qq - q2)/r2, where Q and r are constants …

surely you can do that? Basically I'm asking for a lesson on differentiation...

I know with X I subtract one from the exponent and multiply by the old exponent

x3 turns into 3x2

But I dont know what to do when I am differentiating other letters.... How in the world do I differentiate r2 with respect to q?

If I knew how to do this it would be much easier than typing all this online. So clearly I am not just asking to get someone else to do the work for me. I want to learn this. I need to learn this for this semester and my major.

Edit: The one part I can figure out is the q^2 It turns into 2q, and with the Qq the small q dissapears into a 1 onle leaving big Q.

If I hat to take a guess at the bottom, the r^2 turns into 2r, this leaves us with Q-2q/2r?

tiny-tim
Homework Helper
Hi Lavid2002! (just got up :zzz: …)
But I dont know what to do when I am differentiating other letters.... How in the world do I differentiate r2 with respect to q?

Ah! Now I see! ok, the simple rule for other letters is …

ignore them! … they're just constants, so far as the original variable is concerned.

Even if they're obviously variables themselves (for example, a function of x y and z which you want to differentiate wrt x), you still treat them as constants.
Edit: The one part I can figure out is the q^2 It turns into 2q, and with the Qq the small q dissapears into a 1 onle leaving big Q.

Yup! And since you've done the hard bit yourself, I'll just confirm for your peace of mind that the full answer is d/dq ((Qq - q2)/r2) = (Q - 2q)/r2. btw, there's another way of maximising Qq - q2

you can complete the square … Qq - q2 = Q2/4 - ((Q/2 - q)2

that's a maximum when the square is zero, ie when q = Q/2 Cool that was easier than I thought : )

So now we have Q-q2/r2 ... But how do I know how to "Maximize" This equation? I know if I want to make this as big as possible I want my denominator to be as small as possible. So we make r=1? 12=1
And since I dont have values for Q I dont know what to do for Q-q2.

tiny-tim
Homework Helper
Hi Lavid2002! So now we have Q-q2/r2 ... But how do I know how to "Maximize" This equation? I know if I want to make this as big as possible I want my denominator to be as small as possible. So we make r=1? 12=1
And since I dont have values for Q I dont know what to do for Q-q2.

Let's see, the original question was …
Of the charge Q initially on a tiny sphere a portion q is transferred to a second nearby sphere. Both spheres can be treated as particles. For what value of q/Q will the electrostatic force between the two spheres be maximized?

ok, when it says "a second nearby sphere", it means in a particular place nearby …

in other words, although it hasn't told you what r is, you can assume that there is an r, and it's fixed (same with Q) …

so your answer could have both Q and r in (though in fact it'll only have Q) …

express your answer as "q = a function of Q" q= A function of Q

I worked out the q and got

q = Square root of r2-Q

But I looked in the book. the final answer is .5 or 1/2

I don't know how they got that numerical answer from these variables.

hmmmm...

tiny-tim
Homework Helper
q= A function of Q

I worked out the q and got

q = Square root of r2-Q

how did you get that? you were maximising (Qq - q2)/r2, with r a constant,

so the solution is … ? But I looked in the book. the final answer is .5 or 1/2

I don't know how they got that numerical answer from these variables.

D'oh read the question!
For what value of q/Q will the electrostatic force between the two spheres be maximized?

q/Q I'm sorry, I know how frustrating it can as a teacher when the student doesn't understand.

Something very simple isn't clicking though.

Is it safe to make the assumption that q is one half of Q so that makes q/Q 1/2?

I dont think thats how these variables are being used.

How did you get that
I solved for q and pulled it to the other side of the equation. I dont think that was the right method.

you were maximising (Qq - q2)/r2, with r a constant,

so the solution is … ?

So I want to make F as big as possible

F=(Qq-q2)/r2

If I want to make F as big as possible I want to make r very small. The smaller the denominator, the bigger the number.

Similarly, I want to make the numerator as big as possible to maximize the force.

I don't know how to do this tiny-tim
Homework Helper
You're maximising (Qq-q2)/r2

r is fixed, you can't change it, you don't have to maximise it or even take it into account.

So all you need to do is maximise Qq-q2.

To do that, you can either differentiate and equate to zero, or you can complete the square …

either way, you should get q = Q/2, ie q/Q = 1/2.

Try it again. Qq-q2 When differentiated with respect to q is equal to
Q-2q

So Q-2q=0
+2q
Q=2q
/2

Q/2=q

OHHHH So were saying if we multiply Q by .5 (1/2) this maximizes our equation.

This is the part I needed
Differentiate and set = to zero

Did you just know to do that? How do I know how to "Maximize" another equation when I get one?

Your a big help : )

Redbelly98
Staff Emeritus
Homework Helper
Did you just know to do that? How do I know how to "Maximize" another equation when I get one?
The question explicitly said to maximize the force:
"For what value of q/Q will the electrostatic force between the two spheres be maximized?"​
Whenever a problem statement says to maximize or minimize something, then think about setting a derivative equal to zero. It's a standard technique.

Your a big help : )
Yup, tiny-tim rocks!

So just like subtraction means take away x numbers from the original number maximizing means find the derivative and set it equal to zero in these physics problems?

Redbelly98
Staff Emeritus
A in physics this semester. 