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I have been assigned a problem that I can't solve.

I have a rotating axis to which a thread is connected in one end. The thread is perpendicular to the rotating axis. The thread has a charge density [tex]\lambda[/tex] and a length [tex]L[/tex].

First of all, I need a mean value of the charge density of the circular disc described by the rotating thread with respect to time. I have interpreted this as the charge density:

[tex]\sigma(r) = \frac{\lambda \mathrm{d}r}{2\pi r\mathrm{d}r} = \frac{\lambda}{2\pi r}

[/tex]

The charge density is supposed to be a function of [tex]r[/tex]; the distance to the center of the disc. However, in the density above, the charge density is infinite close to the center. I can't interpret this conceptually.

Second, I am supposed to determine the electrical field a distance [tex]r_0[/tex] from the center of the disc along the rotational axis. Coulombs law yields:

[tex]E = \frac{1}{4\pi\epsilon_0}\iint_\Omega \frac{\mathrm{d}q}{R^2} = \frac{1}{4\pi\epsilon_0}\iint_\Omega\frac{\sigma\mathrm{d}x\mathrm{d}y}{r_0^2 + x^2 + y^2} = [\mathrm{Polar\ coordinates}] = \frac{\lambda}{4\pi\epsilon_0}\int_0^L\frac{\mathrm{d}r}{r_0^2 + r^2} = \ldots = \frac{\lambda\theta}{4\pi\epsilon_0r_0}[/tex]

This result is a bit strange, if you consider the extreme values. For instance:

[tex]

\begin{array}{ll}

\lim_{\theta\rightarrow 0}E = 0 & \mathrm{Ok!} \\

\lim_{\theta\rightarrow \frac{\pi}{2}}E = k & \mathrm{Ok?} \\

\lim_{r_0\rightarrow 0}E = \infty & \mathrm{Not\ Ok??} \\

\lim_{r_0\rightarrow\infty}E = 0 & \mathrm{Ok}

\end{array}

[/tex]

From the third extreme value, I must conclude that the result is wrong, as it should be 0 in the disc (the forces cancel eachother)

What is wrong? Is it the mathematics or the physics that fail?

Please Help!

Nille

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# Homework Help: Electric Field

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