Electric Field Due to a Line of Charge Problem

[frankfjf]

Homework Statement

Charge is uniformly distributed around a ring of radius R = 2.40cm and the resulting electric field magnitude E is measured along the ring's central axis (perpendicular to the plane of the ring). At what distance from the ring's center is E maximum?

Homework Equations

Therein lies my problem, I'm unsure how to approach this except ink knowing that Calculus can handle it. I know that I need to find the maximum and can obtain it by deriving the equation, finding the value of Z, then plugging it in and deriving a second time. But the only equation I know of here is:

E = kqz/(z^2 + R^2)^3/2

The Attempt at a Solution

I'm not sure if I have to modify the formula before I derive it or if I only worry about one part of it to derive. If I just derive the formula in terms of z I get a huge mess. What am I missing? I'm not sure where to start.

 

[Doc Al]

I'm not sure if I have to modify the formula before I derive it or if I only worry about one part of it to derive. If I just derive the formula in terms of z I get a huge mess. What am I missing? I'm not sure where to start.

By "derive" I assume you mean differentiate. Go ahead--take the derivative! It's a bit messy, but so what? (Use the quotient/product rule.)

 

[frankfjf]

Well after taking the derivative and using the quotient and chain rules I get something like...

kq * (z^2 + R^2)^3/2 - z(3/2(2z^3+2R^3)^1/2) / (z^2 + R^2)^5/2

But I get the feeling I either did something wrong in the differentiation or I can still further simplify.

 

[robphy]

Try factoring.

 

[frankfjf]

I can see that working but I can't figure out how to best factor it out, should I favor the entire top half or just the right side of the top half?

 

[Doc Al]

Well after taking the derivative and using the quotient and chain rules I get something like...

kq * (z^2 + R^2)^3/2 - z(3/2(2z^3+2R^3)^1/2) / (z^2 + R^2)^5/2

But I get the feeling I either did something wrong in the differentiation or I can still further simplify.

Looks to me like you made a few errors. For one, how did you get the term that I highlighted?

I recommend that you use the product & chain rules, writing the field equation as:

E = kq [z] [(z^2 + R^2)^-3/2]

(I put brackets around the two factors of interest, in case we need to refer to them; the kq drops out--since you'll set it to zero.)

 

[frankfjf]

Oh, I had forgotten about that approach. In that case once I've applied both the product and chain rules I end up with..

-3z(z^3 + R^3)^-5/2 + (z^2 + R^2)^-3/2

 

[Doc Al]

Oh, I had forgotten about that approach. In that case once I've applied both the product and chain rules I end up with..

-3z(z^3 + R^3)^-5/2 + (z^2 + R^2)^-3/2

I spoke too soon. What's up with the highlighted term? Where'd you get the cubes?

Let's do this step by step: What's the derivative of the second bracketed factor in post #6?

 

[frankfjf]

Chain rule gives me... -3/2(z^2 + R^2)^-1/2 * (2z + 2R), hence how I get the cubes.

 

[Doc Al]

I'm not sure how to simplify that though.

Forget simplifying until you fix it. Answer my question in my last post.

 

[frankfjf]

Sorry, I replied too soon, I've editted post 9 to answer your question.

 

[Doc Al]

Chain rule gives me... -3/2(z^2 + R^2)^-1/2 * (2z + 2R), hence how I get the cubes.

Several problems here:
(1) The derivative of (z^2 + R^2) is just 2z. (R is a constant!)
(2) Where did you get the power of -1/2? Remember you started with a power of -3/2.
(3) Even with the wrong derivative, how do you get cubes from that?

 

[frankfjf]

Ahh good point, d'oh. Actually, chalk it up to temporary insanity, doing it a second time I get...

(z^2 + R^2)^-3/2 + (2z^2)^-3/2

 

[Doc Al]

Ahh good point, d'oh. Actually, chalk it up to temporary insanity, doing it a second time I get...

(z^2 + R^2)^-3/2 + (2z^2)^-3/2

Not sure what this is. One more time, redo the derivative of the second factor in post #6.

 

[frankfjf]

Trying it again I get -3/2(2z)^-5/2

 

[Doc Al]

Trying it again I get -3/2(2z)^-5/2

You need to review the chain rule. Do it in two steps:
(1) Take care of the exponent part: (something)^-3/2
(2) Take the derivative of the something

(And then multiply them, of course.)

 

[frankfjf]

-3(z^3 + R^2z)^-5/2 ?

 

[Doc Al]

-3(z^3 + R^2z)^-5/2 ?

Nope. Do each step listed in my last post separately. (Don't try to combine anything.)

 

[frankfjf]

I need some clarification then. By "second factor" do you mean what would be considered g'(x) in the product rule?

IE: f'(x)g(x) + g'(x)f(x)?

If so then g'(x), in steps, looks like...

(1) (-3/2)(z^2 + R^2)^-5/2
(2) 2z

When I multiply them, I get -3(z^3 + zR^2)^-5/2, unless I am doing something incorrectly, but I don't think I am, since step 1 would also require the power rule.

But this is identical to my previous answer, so I'm not sure what I'm doing wrong but it seems to be in step 1. But it seems to line up just fine with the power rule. The exponent is -3/2, which when I subtract 1 becomes -5/2, and then the original exponent is multiplied by the inside of the parenthesis.

 

[Doc Al]

If so then g'(x), in steps, looks like...

(1) (-3/2)(z^2 + R^2)^-5/2
(2) 2z

Exactly correct!

When I multiply them, I get -3(z^3 + zR^2)^-5/2, unless I am doing something incorrectly, but I don't think I am, since step 1 would also require the power rule.

You are doing something wrong when you multiply:
z*(something)^-5/2 does not equal (z*something)^-5/2 !

(Don't waste time multiplying this out correctly--that will only make it harder to simplify the final expression.)

 

[frankfjf]

So I should leave the final expression for g'(x) as 2z(-3/2)(z^2 + R^2)^-5/2

and continue from there?

 

[Doc Al]

So I should leave the final expression for g'(x) as 2z(-3/2)(z^2 + R^2)^-5/2

and continue from there?

Yes. (But please cancel the 2! )

 

[frankfjf]

So -3z(z^2 + R^2)+^-5/2?

 

[Doc Al]

So -3z(z^2 + R^2)+^-5/2?

Yes. (Except for that + sign.) Now put it all together.

 

[frankfjf]

Oh, I apologize, that was a typo.

Since we've got f'(x)g(x) + f(x)g'(x), wouldn't that be...

(z^2 + R^2)^-3/2 - 3z^2(z^2 + R^2)^-5/2?

 

[Doc Al]

Looks good to me. Now simplify. (Eliminate the common factor in both terms)

 

[frankfjf]

That's where it gets tricky. I'm not sure if I can eliminate z^2 + R^2 since that leaves 1^-3/2 and 1^-5/2 and mathematics life isn't usually that easy, so... I guess the only common term I can think of is z^2 itself since it appears 3 times? That'd give...

z^2[(1 + R^2)^-3/2 - 3(1 + R^2)^-5/2?

 

[Doc Al]

That's where it gets tricky. I'm not sure if I can eliminate z^2 + R^2 since that leaves 1^-3/2 and 1^-5/2 and mathematics life isn't usually that easy, so... I guess the only common term I can think of is z^2 itself since it appears 3 times? That'd give...

z^2[(1 + R^2)^-3/2 - 3(1 + R^2)^-5/2?

Yikes no! You cannot just pull things out from under a power!

(z^2 + R^2)^-3/2 is not equal to z^2*(1 + R^2)^-3/2

(z^2 + R^2)^-3/2 is not equal to (z^2 + R^2)*1^-3/2

You need to review your understanding of exponents and powers.

Hint: (z^2 + R^2)^-3/2 itself is a common factor--divide it out from all terms.

 

[frankfjf]

Ahhhh okay. Sorry, been a while since I dealt with messy expressions. However, for me the tricky part of factoring (z^2 + R^2)^-3/2 out of the second term. I'll try though... I get...

(z^2 + R^2)^-3/2 [1 - 3z^2(z^2 + R^2)^-1] I'm not sure if this is correct.

 

[Doc Al]

Looks good--you corrected your typo.

Keep going.

 

[frankfjf]

Hmm well to be honest I'm kind of stumped again, I'm not sure if there's anything left to factor out... couldn't I just set equal to zero and attempt to find the value for z that would cause that to happen, or is there further simplification to be done?

 

[frankfjf]

Well, I think I can re-arrange the equation if I can't simplify further to be...

(1-3z^2) / (z^2 + R^2)^3/2 in which case z must equal the square root of 1/3?

 

[Doc Al]

Hmm well to be honest I'm kind of stumped again, I'm not sure if there's anything left to factor out... couldn't I just set equal to zero and attempt to find the value for z that would cause that to happen, or is there further simplification to be done?

Just set it equal to zero and then simplify.

Well, I think I can re-arrange the equation if I can't simplify further to be...

(1-3z^2) / (z^2 + R^2)^3/2 in which case z must equal the square root of 1/3?

Not sure how you got this.

 

[frankfjf]

I fear I might need a hint again. If I put an "= 0" at the end, I'm not sure how I'd get something on the other side. Only thing I can think of is to use the binomial theorem on the expression in the brackets, but I'm not sure if such a thing is fair play or helps to simplify it.

 

[Doc Al]

Take the derivative (your expression in post #29) and set it equal to zero. Realize that each factor can be set equal to zero, if possible. (Don't dare use the binomial theorem.)

 

[frankfjf]

That's just it though, I'm not sure what to set z to that would make any of the factors zero, due to z being squared on both factors, I can't reliably use negative numbers...

 

[frankfjf]

I think that Doc Al's gone offline, could someone help me finish this problem? Doc Al's patience with me has been legendary and I hope he helps me finish it, but this problem has really been plaguing me.

 

[robphy]

Well, I think I can re-arrange the equation if I can't simplify further to be...

(1-3z^2) / (z^2 + R^2)^3/2 in which case z must equal the square root of 1/3?

dimensionally, this is incorrect... since in 1-3z^2, you'd be adding a pure number and something with m^2.

Go back to

Oh, I apologize, that was a typo.

Since we've got f'(x)g(x) + f(x)g'(x), wouldn't that be...

(z^2 + R^2)^-3/2 - 3z^2(z^2 + R^2)^-5/2?

Factor out (z^2 + R^2)^-5/2 from each term.

 

[frankfjf]

So should I abandon the equation I got in post #29?

 

[frankfjf]

Regardless, I'll give it a shot. If I factor that out, my guess is I get something like...

(z^2 + R^2)^-5/2[(z^2 + R^2) - 3z^2]

is that correct?

Wait, ah-ha! Now I've got..

(R^2 - 2z^2) / (z^2 + R^2)^-5/2

Set equal to zero, z would have to be R/2^1/2. Since I know R, I can just plug it in now. Rmax = 1.70cm which lines up with the answer given in the book.

Thanks Doc Al and robphy. I apologize for trying your patience and am thankful for your help.

 

[robphy]

#29 looked okay... but I think my suggestion is neater.

#40 looks correct.
Now, the bracketed term can be simplified.
Note that this entire expression must be zero at an extremum.
Note that (z^2 + R^2)^-5/2 is never zero for finite z and R.

As a check of your final answer, you might wish to plot your E(z) vs z.

 

[robphy]

Regardless, I'll give it a shot. If I factor that out, my guess is I get something like...

(z^2 + R^2)^-5/2[(z^2 + R^2) - 3z^2]

is that correct?

Wait, ah-ha! Now I've got..

(R^2 - 2z^2) / (z^2 + R^2)^-5/2

Set equal to zero, z would have to be R/2^1/2. Since I know R, I can just plug it in now. Rmax = 1.70cm which lines up with the answer given in the book.

Thanks Doc Al and robphy. I apologize for trying your patience and am thankful for your help.

Great!
...although a typo:
Not
(R^2 - 2z^2) / (z^2 + R^2)^-5/2
but either
(R^2 - 2z^2) * (z^2 + R^2)^-5/2
or
(R^2 - 2z^2) / (z^2 + R^2)^5/2

 

[frankfjf]

Oh, that was indeed a typo, forgot that once it's "back down there" so to speak, the exponent need not be negative.

 

[Doc Al]

So should I abandon the equation I got in post #29?

Just for the record, here's how I would finish this starting from the expression in post #29.

(1) Set it equal to zero:
(z^2 + R^2)^-3/2 [1 - 3z^2(z^2 + R^2)^-1] = 0

(I've color-coded the factors to make them easy to refer to.)

(2) Now solve for the value of z that makes each factor equal to zero. Start with the first (red) factor: the only way that can equal zero is for z to be +/- infinity; that's not the answer we want.

Now the second (blue) factor:
[1 - 3z^2(z^2 + R^2)^-1] = 0
1 = 3z^2/(z^2 + R^2)
z^2 + R^2 = 3z^2
z = R/2^1/2

That's the one you want. Easy!

 

[robphy]

I think #40 is algebraically simpler, with fewer fractions.
I think #29 has some appeal since it arranges terms into dimensionless quantities, which could be useful for scaling arguments and making approximations.