# Electric Field -- Rod from far away

1. Mar 1, 2017

### Arman777

1. The problem statement, all variables and given/known data
Theres a rod and has $a$ lenght $L$ and we want to calculaate the electric field far from rod like $L<<a$

2. Relevant equations

3. The attempt at a solution
I found the equation for a normal case and it was $\frac{1}{4\pi e_0}\frac{2Q}{L^2}\left(ln\left(\frac{a}{L+a}\right)+\frac{L}{a}\right)$ .Itn non-uniform thats why theres $2$ anyway in the $L<<a$ case it will be just $\frac{1}{4\pi e_0}\frac{2Q}{L^2}$ .
You might ask you did so far how you cant do that well I am not sure about the $\frac {L} {a}$ term it seem like it will be $\frac {1} {a}$ but in that case the units will not match so it must be 0.When something like this happens we will think $a→∞$ ?

2. Mar 1, 2017

### haruspex

I don't know why you think it would be like 1/a. As you say, it will tend to zero as a tends to infinity, but the same is true of the log term, leaving you with zero overall. Of course, that is trivially true, but we need the first nonzero term in the approximation.
So you need to start by expanding the ln as a power series.

3. Mar 1, 2017

### Arman777

It was used as a hint in our homework but I didnt give too much attantion cause $ln\left(\frac{a}{L+a}\right)$ would be just zero.Why do I need to write in as power series..?
In our homework as a hint it says $ln(1+ε)=ε-\frac {e^2} {2}+\frac {e^3} {2}..$ so $ln\left(\frac{a}{L+a}\right)$ can be written as $-ln\left(\frac{L+a}{a}\right)$
and that is $-ln\left(\frac{L}{a}+1\right)$ so $ε=\frac {L} {a}$
so simply
$-ln\left(\frac{L}{a}+1\right)+\frac {L} {a}=-ε+ε=0$ (cause $\frac {L} {a}=ε$)

4. Mar 1, 2017

### Arman777

Should ı write like this ?

5. Mar 1, 2017

### Staff: Mentor

Presumably the rod is charged in some way? Uniformly?
How is the point at distance $a$ positioned with respect to the rod? For clarity, suppose that the rod is lying along the y-axis with its center at the origin. Is the point $a$ located somewhere along the x-axis, or perhaps the y-axis?

6. Mar 1, 2017

### Arman777

I wrote the question so silly way sorry for that.Charge distrubiton is not uniform and its like λ(x)=bx (https://www.physicsforums.com/threads/electric-field-dimension-analysis.906014/ this thread also the first 2 parts of my question)
Here a picture to make things clear
I choose the left end of the rod is origin.

7. Mar 1, 2017

### Arman777

its positivly charged

8. Mar 1, 2017

### haruspex

That's the right approach, but you need to keep another term from the expansion. This a general rule in problems like this... keep just enough terms that the result is not zero.

9. Mar 1, 2017

### Arman777

Why are leaving another term ? Dont we want it to be a zero ?
So we will write $-e+\frac {e^2} {2}+e$ and $\frac {e^2} {2}=0$ again ? (e means epsilon)

10. Mar 1, 2017

### Arman777

$\frac {ε^2} {2}$ will not be zero its $\frac {L^2} {2a^2}$ so my E will be
$\frac {1} {4πε_0} \frac {Q} {a^2}$ which thats what we want.
so the other terms would be zero ?

11. Mar 1, 2017

### Staff: Mentor

No, the other terms would be considered to be negligible.

12. Mar 1, 2017

### Arman777

Oh I see your point ok