Calculating the Electric Field of a Long, Thin Wire Using Gauss' Law

[howsockgothap]

Homework Statement

Calculate the electric field at a point 2.79 cm perpendicular to the midpoint of a 2.02 m long thin wire carrying a total charge of 5.04 uC.

You could integrate BUT if the wire is very long compared to the distance from the wire to where you are calculating the electric field, then the electric field will be radial and Gauss's law might be easier.

Homework Equations

E_R=λ/2piε₀R

The Attempt at a Solution

First I tried doing it by integral (getting E=k_exQ/(x²+R²)^3/2) but then it gave me the hint to use Gauss' Law, so I treated it like a cylinder, using charge divided by length for λ and 0.0279 m for R. I came out with 5.76x10⁷ N/C which I am told is wrong.

THEN I worked that to E=2k_eλ/R after having a "duh" moment but I still seem to be getting the wrong answer (2.58x10⁵)

Edit: calculator/math error. I solved it.

 

[Tarti]

The situation is very easy if you use assume really an infinitely long wire. Then, the electric field can only depend on the distance to the wire, say ρ in cylindrical coordinates.
Gauss's law $\int_{\partial V}\mathbf{E}d\mathbf{A}=\frac{1}{\varepsilon_0} \int \rho dV$
becomes:

$2\pi \rho E_\rho(\rho)\int dz=\frac{1}{\varepsilon_0} \lambda \int dz$​

with charge density $\lambda$ [C/m]. This is a rather dirty trick to see that

$E_\rho(\rho)=\frac{\lambda}{2 \pi \varepsilon_0 \rho }\ .$​

This result is correct in the infinite-length-limit of a line charge. Here you can find the full calculation.