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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 [itex]\sigma [/itex]. Determine the electric field at a point P a distance

*z*above the center of the plane, in the limit [itex]l \to \infty[/itex].

[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: [itex]\frac{2k\lambda }{x}[/itex] (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.)

[tex]k = \frac{1}{4\pi\epsilon_0}[/tex]

## The Attempt at a Solution

Charge densities:

[tex]\sigma = \frac{dq}{dy*l}[/tex] (an infinitely small q over an infinitely small surface)

[tex]\lambda = \frac{dq}{l}[/tex] (total charge of the strip / total length)

[tex]dE = \frac{2k\lambda}{\sqrt{y^2+z^2}}[/tex]

(electric field due to a long strip)

[tex] dE_z = dE sin\theta = \frac{2k\lambda y}{{(y^2+z^2)}^{3/2}}[/tex]

(its z component is what we need)

[tex] dE_z = dE sin\theta = \frac{2k\sigma y}{{(y^2+z^2)}^{3/2}}dy[/tex]

(dy is necessary, so replace lambda with sigma)

The following is what I get after the integration,

[tex]{-2\sigma k} \frac{1}{\sqrt {y^2+z^2}}[/tex]

The limits are zero and infinity, so I end up with;

[tex]\frac{2\sigma k}{z}[/tex]

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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