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Homework Statement
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Charge is distributed uniformly over a large square plane of side l, as shown in the figure. The charge per unit area (C/m^2) is \sigma. Determine the electric field at a point P a distance z above the center of the plane, in the limit l \to \infty.
[Hint: Divide the plane into long narrow strips of width dy, and use the result of Example 11]
Homework Equations
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Result of Example 11: \frac{2k\lambda }{x} (electric field at a distance x due to an infinitely long wire)(that point is symmetric about the x-axis, so there is no y component of the electric field.)
k = \frac{1}{4\pi\epsilon_0}
The Attempt at a Solution
Charge densities:
\sigma = \frac{dq}{dy*l} (an infinitely small q over an infinitely small surface)
\lambda = \frac{dq}{l} (total charge of the strip / total length)
dE = \frac{2k\lambda}{\sqrt{y^2+z^2}}
(electric field due to a long strip)
dE_z = dE sin\theta = \frac{2k\lambda y}{{(y^2+z^2)}^{3/2}}
(its z component is what we need)
dE_z = dE sin\theta = \frac{2k\sigma y}{{(y^2+z^2)}^{3/2}}dy
(dy is necessary, so replace lambda with sigma)
The following is what I get after the integration,
{-2\sigma k} \frac{1}{\sqrt {y^2+z^2}}
The limits are zero and infinity, so I end up with;
\frac{2\sigma k}{z}
There is an example of uniformly charged disk in my textbook. The formula for electric field for that disk does not depend on the distance. That's why I believe I've done this question wrong. What do you think about my solution? I am not sure if I wrote charge densities correct, so that may be the mistake.
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