Coulomb's Law of a sphere

[Lavid2002]

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 don't even know where to start. This is my very very first HW question in physics 2 and its blowing my mind.

 

[tiny-tim]

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?

 

[Lavid2002]

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]

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

surely you can do that? ​

 

[Lavid2002]

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 don't 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]

Hi Lavid2002!

(just got up :zzz: …)

But I don't 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

 

[Lavid2002]

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 don't have values for Q I don't know what to do for Q-q2.

 

[tiny-tim]

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 don't have values for Q I don't 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"

 

[Lavid2002]

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]

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

 

[Lavid2002]

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 don't think that's 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 don't 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]

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. ​

 

[Lavid2002]

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]

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!

 

[Lavid2002]

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]

Usually, yes. The hard part can be in figuring out how to set up an equation for the function that is to be maximized. Or figuring out which quantities are variable, and which are constants.

 

[Lavid2002]

I see... I will go to my teacher during her office hours for help and stay on top of this stuff!


A in physics this semester.