Electric Field Due to a Line of Charge Problem

[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.