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Faiq
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Homework Statement
A ##10 cm## (on y axis) by ##10 cm## (on z axis) flat plate is located ##5 cm## away (on x axis) from a point charge ##q##. Calculate the electric flux from the point charge to the plate.
Can somebody solve it using surface integral using both spherical and cartesian coordinates. I did solve it using symmetry but I don't know how to solve it using integrals?
Cartesian
$$ \oint_s \vec{E}^\ \cdot \vec{n}^\ dS= \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{\mathbf{r-r'}}{|\mathbf{r-r'}|^3}\cdot \vec{n}^\ dS = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{x-x_{0}}{((x-x_0)^2+y^2+z^2)^\frac{3}{2}} dS = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{x-x_{0}}{((x-x_0)^2+y^2+z^2)^\frac{3}{2}} dydz$$
where ##x_0## is ##5cm##
Spherical
$$ dydz = \frac{\partial(y,z)}{\partial (\phi,\theta)}d\phi d\theta = -\sin^2\phi \cos \theta ~d\phi d\theta$$
$$ \oint_s \vec{E}^\ \cdot \vec{n}^\ dS= \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}i\cdot \vec{n}^\ = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}(\cos \theta \sin \phi ) = -\iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ d\phi d\theta}{|\mathbf{r}|^2}(\cos^2 \theta \sin^3 \phi ) = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}(\cos \theta \sin \phi ) = -\iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ d\phi d\theta}{x_0^2}(\cos^4 \theta \sin^5 \phi ) $$
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