Electric Field Due to Infinite Line Charge

[SimbaTheLion]

Homework Statement

"Compute the electric field due to an infinite line charge by integrating the expression obtained from the inverse square law."

Homework Equations

I think that the equation required is:

E(\bold{r}) = \frac{1}{4 \pi \epsilon_0} \int_{-\infty}^{+\infty} \frac{(\bold{r} - \bold{r}') \sigma}{|\bold{r} - \bold{r}'|^3} dx

The Attempt at a Solution

Well, I don't know what to put in the above integral for r and r', or even if the above integral is even right...

Thanks for any help :) .

 

[Phrak]

Your equation looks correct.

Build a coordinate system. Include the wire in some convenient place, like along the x-axis. The wire will extend to plus and minus infinite x. As presented, the integral already assumes the wire lies parallel to the x-axis.

r is a displacement vector that varies with x, and locates points within the wire.

r' is a displacement vector of the point for which we wish to calculate the electric field. It is a constant with respect to the variable of integration, x.

 

[kreil]

im working on the same problem. it appears that if infinite limits of integration are used the expression blows up, is it better to set the limits from 0-> L and then take the limit as L goes to infinity?

 

[SimbaTheLion]

Mmm, I'm getting an integral of something similar to 1/x² between -infinity and infinity, that doesn't look good...

 

[kreil]

\int_{-\infty}^{+\infty} \frac{dx}{(x^2+y^2)^{3/2}}= \left( \frac{x}{y^2 \sqrt{y^2+x^2}} \right) |_{-\infty}^{+\infty}

The main observation you need to make to solve this is:

\lim_{x \rightarrow \infty} \frac{x}{y^2 \sqrt{y^2+x^2}} = \frac{1}{y^2}

Since x is a first order variable on both the numerator and denominator.