How Is the Electric Field Around an Infinitely Long Wire Derived?

[Uku]

Homework Statement

I know that around an infinitely long wire with even charge distribution the electric field is expressed as:

$$E=\frac{1}{2\pi\epsilon_{0}}\frac{\lambda}{r}$$ (1)

Where $$\lambda$$ can be expressed as $$\lambda=\frac{dq}{dl}$$

Right, but I want to know where I get this formula from, I mean the E field.

The Attempt at a Solution

So I know that in general:

$$E=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho}{r^{2}}\widehat{r}dV$$

In my case I don't have volume, I have a thread. I can also forget about the unit vector, since the field is radially pointed outward. The charge is evenly distributed so I can write:

$$E=\frac{1}{4\pi\epsilon_{0}}\lambda\int\frac{1}{r^{2}}dL$$

Okay, but now... I can integrate the expression from minus infinity to infinity, but how do I get to that formula (1)

 

[stevenb]

Okay, but now... I can integrate the expression from minus infinity to infinity, but how do I get to that formula (1)

You can find a number of sites that work this out for you. A google search, or even a search here at PF will give you many looks at this problem. Here is one, for example.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html