(adsbygoogle = window.adsbygoogle || []).push({}); How can I not do this??

I am trying to determine the electric field due to an arc of charge which spans the angles 0 to [tex] \pi [/tex] with a total charge q. Does anyone know why I cannot integrate this directly? The only way I get the right answer is to convert [tex] a_{\rho}[/tex] (unit vector in radial direction) to x,y coordinates and then integrate. I was just wondering if I could integrate with everything still expressed in polar form:

Begin with the following formula for the electric field:

[tex]\vec{E} = \int_{0}^{\pi} \frac{\rho_{L} d\ell (-a_{\rho})}{4 \pi \epsilon R^{2}} [/tex]

(Where R is the radius of the arc and [tex] \rho_{L} [/tex] is the charge density of the arc.)

Then convert [tex] d\ell [/tex] to [tex] R d\phi [/tex] and integrate over the limits of the arc. This is all fine and dandy, (as [tex] \int^{\pi}_{0} d\phi [/tex] is obviously [tex] \pi [/tex] in this case) but when I convert from the radial unit vector to the rectangular ones [tex] a_{\rho} = \cos \phi a_{x} + \sin \phi a_{y} [/tex] I find myself very confused by the introduction of the angle [tex] \phi [/tex] again. I can do this and get the correct answer if I convert the radial unit vector to the rectangular ones first, but I want to know if it is possible to integrate without the introduction of rectangular coordinates (converting the polar electric field vector to rectangular coords after the integration is done). Thanks for any help. If I am not clear enough I can try to rephrase my problem as I enjoyed learning how to use the latex typesetting

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# Homework Help: How can I not do this?

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