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Math Help - Integrating vectors (NOT vector fields)

  1. #1
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    Integrating vectors (NOT vector fields)

    This is a problem in electromagnetics applying coulomb's law, in brief, we have a disc(wire) with charges on it. We want to calculate the electric field at a point h on the axis of the disc. The book had a rather annoying way of solving the problem, so I wanted to go with something simpler and went with the following:

    (1) \, d\vec{E} = \int \frac{1}{4\pi \epsilon R^2}  \vec{R} dq. R is a unit vector
    (2) \, \vec{R} = \frac{1}{\sqrt{r^2 + h^2}} (-r\cos \theta \vec{i} - r\sin \theta \vec{j} + h\vec{k}).
    (3) \, dq = \rho_s dA = \rho_s rdrd\theta .
    when we put all of that together, we get:
    \vec{E} = \int\int _s \frac{\rho_s}{4\pi \epsilon (r^2 + h^2)^{3/2} }   (-r\cos \theta \vec{i} - r\sin \theta \vec{j} + h\vec{k}) rdrd\theta .
    But unless we take away the 2 rs from the vector (-r\cos \theta \vec{i} - r\sin \theta \vec{j} + h\vec{k}) we won't get to the right answer which is \frac{\rho_s}{2\epsilon}( \, \frac{h}{\sqrt{a^2 + h^2}} - 1 \,) \, \vec{k}.
    Note: integral limits  0 < r < a and 0 < \theta < 2\pi.

    What did I do wrong?
    Last edited by rebghb; October 16th 2010 at 04:14 AM.
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  2. #2
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    It is not clear without a picture. It takes much time.
    Using the symmetry along the z axis

    <br />
\displaystyle { dE=\frac{1}{4 \pi \epsilon _0} \; \frac{dq}{R^2} \; cos {\alpha}} }<br />

    <br />
dq=\rho \; 2 \pi r dr<br />

    where
    <br />
\displaystyle {  cos {\alpha}= \frac {h}{\sqrt{r^2+h^2}} }<br />
    is the projection of electric field to z axis
    and
    <br />
R^2=r^2+h^2<br />

    <br />
\displaystyle {  E=\frac{\rho \; h}{2 \epsilon _0}  \int_0^a \frac{r \; dr} {(r^2+h^2)^{3/2}}=\frac{\rho}{2 \epsilon _0}  (1-\frac{h}{\sqrt{a^2+h^2}})<br />
}<br />
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  3. #3
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    Dear zzzoak,

    Thanks for the reply

    I know it takes a lot of time... My professor used your way but I didn't like it, I was testing a new way. Now my question is, why doesn't my way work?

    Is there something we must know when integrating vectors? (integrating vectors as in sum of a very large number of vectors).

    Best,
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