Using the gaussian law I want to determine the field strength of a charged stick on another load.
The base formula for that is
$$\epsilon_0 \oint {\vec{E} \cdot d \vec{A}} = q$$
A second formula for q is:
$$ q = \lambda \cdot h $$
$$\vec{E} = \frac{1}{4\cdot \pi \cdot \epsilon_0} \cdot \frac{q_1}{r^2} \cdot \hat{r}$$
In this case we use a cylinder to solve our problem and we only need the lateral surface.
$$A = 2\cdot \pi \cdot d \cdot h$$
My questions
I have a strong problem with a surface written as vector.
Assuming we have a situation like in the first picture of this question where the lateral surface of our cylinder touches the load.
How would I properly write the surface as vector (how can I determine the direction vector)?
Usually my approach would be $\vec{r} = (x_2 - x_1)\hat{x} + (y_2 - y_1)\hat{y}$. But as we have a solid here I have problems finding the start and end coordinates.Until now I only solved integrals in the form of $\int f(x) dx$ but now I have an equation like the one above. What would be my "x" here? How would I solve this integral?
As you can imagine this is a task from my electronics class. It's not a homework but it was an example.
Unfortunately we did explicitly not pay attention to the integral and we did not pay attention to the surface written as vector.
What we did is the following:
$$\epsilon_0 \oint {\vec{E} \cdot d \vec{A}} = q$$
$$\epsilon_0 \cdot |\vec{E}| \cdot 2 \cdot \pi \cdot h \cdot d = q$$
Because $\lambda$ is given in this particular task we can do
$$\epsilon_0 \cdot |\vec{E}| \cdot 2 \cdot \pi \cdot h \cdot d = \lambda \cdot h$$
$$|\vec{E}| = \frac{\lambda}{2\cdot \pi \cdot d \cdot \epsilon_0}$$