Magnitude of electric field due to line of charge

[CentreShifter]

Homework Statement

A charge (uniform linear density = 9.4 nC/m) lies on a string that is stretched along an x-axis from x = 0 to x = 3.0 m. Determine the magnitude of the electric field at x = 5.5 m on the x axis.

Homework Equations

$$\stackrel{\rightarrow}{dE}=\frac{\lambda dx}{4 \pi \epsilon_{0} r^2}\hat{r}$$

The Attempt at a Solution

I believe the unit vector can eliminated since the point is on the same axis as the line of charge (cos(0)=1). My concern is with my limits of integration and with my "r" argument. My final expression was:

$$E = \frac{\lambda}{4 \pi \epsilon_{0} 5.5^2} \int^{3}_{0} dx$$ which got me an 8.38244 N/C, a wrong answer.

 

[LowlyPion]

Looks like your r is a function of x.

For any dq isn't r = (5.5 - x) ?

 

[CentreShifter]

Yes! I knew the influence of dq would vary with distance from it. I just wasn't sure how to incorporate that. I guess I'm having trouble visualizing r as a function of x that way. Is the x in (5.5 - x) the position of dq?

 

[LowlyPion]

Yes! I knew the influence of dq would vary with distance from it. I just wasn't sure how to incorporate that. I guess I'm having trouble visualizing r as a function of x that way. Is the x in (5.5 - x) the position of dq?

As I see it, yes. That should be the r for a particular dq located at x, and relative to the point 5.5.