Calculate Electric Flux from Point Charge to Plate:

[Faiq]

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
$$ dydz = \frac{\partial(y,z)}{\partial (\phi,\theta)}d\phi d\theta = -\sin^2\phi \cos \theta ~d\phi d\theta$$
$$ \oint_s \vec{E}^\ \cdot \vec{n}^\ dS= \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}i\cdot \vec{n}^\ = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}(\cos \theta \sin \phi ) = -\iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ d\phi d\theta}{|\mathbf{r}|^2}(\cos^2 \theta \sin^3 \phi ) = \iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ dS}{|\mathbf{r}|^2}(\cos \theta \sin \phi ) = -\iint \frac{\lambda}{4\pi \varepsilon_0} \frac{ d\phi d\theta}{x_0^2}(\cos^4 \theta \sin^5 \phi ) $$

 

[haruspex]

$\int \sin^5(\phi).d\phi$ is fairly straightforward.
What would you do with $\int \cos^2$?

(Is the plate centred on the x axis?)

 

[Faiq]

Convert into a sine function?
Yes

 

[haruspex]

Convert into a sine function?
Yes

No. Cos(2x)=?