The problem is originally taken from a Physic problem, but I have some issue to compute an integral, and that's why I am here.
Context
I need to compute the electric field from a Sphere of radius $a$ charged in surface $\sigma = \sigma_0\cdot cos(\theta)$. We want to compute the electric field
- Use two spheres uniformly volumic charged
- Compute the integral directly.
I succeed with the first method, and now, for fun, I am trying to do the second method. Some reminder : the electrical field in a point is:
$$\vec{E}(\mathbf{1}) = \frac{1}{4\pi\epsilon_0}\int\int\int\frac{\rho \vec{r}}{r^3}dV_{\mathbf{2}}=\frac{1}{4\pi\epsilon_0}\int \frac{\vec{r}}{r^3}dQ$$ With $\vec{r}$ the vector from 2 to 1. 2 Moving on the charge distribution, 1 is fixed.
Also, the electrical field can be written as.
$$\vec{E} = -\nabla\phi$$ $$\phi(\mathbf{1})= \frac{1}{4\pi\epsilon_0}\int\frac{dQ}{r}$$
The attempt
Let's try to compute the $dQ$ value since it appears on both equations. Since the distribution is spherical, I decided to use spherical coordinates. Since it is surface and not volumic, I got:
$$dQ=\sigma_0a^2cos(\theta)sin(\theta)d\theta d\phi$$.
Now we can try to deduce the expression of $\vec{r}$, and I think it is here that I have some issues.
$$\vec{r}=\mathbf{1} - \mathbf{2}$$
With
$$\mathbf{2} = \left( \begin{array}{c} a \cdot sin(\theta)cos(\phi) \\ a \cdot sin(\theta)sin(\phi) \\ a \cdot cos(\theta) \end{array} \right)$$
$$\mathbf{1} = \left( \begin{array}{c} x\\ y\\ z\\ \end{array} \right) = \left( \begin{array}{c} R\cdot sin(\theta_2)cos(\phi_2)\\ R\cdot sin(\theta_2)sin(\phi_2)\\ R\cdot cos(\theta_2)\\ \end{array} \right)$$
And now, I am not able to continue the computation... If we put the thing together, we obtain something not easy to compute, for both $\vec{E}$ or $\phi$.
$$\vec{E}(\mathbf{1}) = \frac{1}{4\pi\epsilon_0} \int_0^{2\pi}\int_0^{\pi}\frac{\left( \begin{array}{c} x - a \cdot sin(\theta)cos(\phi)\\ y - a \cdot yin(\theta)sin(\phi)\\ z - a \cdot cos(\theta) \end{array} \right) \sigma_0a^2cos(\theta)sin(\theta)d\theta d\phi}{(a^2 + R^2 - 2a(x \cdot sin(\theta)cos(\phi) + y\cdot sin(\theta)sin(\phi) + z\cdot cos(\theta)))^{\frac{3}{2}}}$$
and
$$\phi(\mathbf{1})=\frac{1}{4\pi\epsilon_0} \int_0^{2\pi}\int_0^{\pi}\frac{\sigma_0a^2cos(\theta)sin(\theta)d\theta d\phi}{\sqrt{a^2 + R^2 - 2a(x \cdot sin(\theta)cos(\phi) + y\cdot sin(\theta)sin(\phi) + z\cdot cos(\theta))}}$$
For $(x, y, z) = \vec{0}$, the result for the electric field is easy and we obtain $\vec{E}(\vec{0})=-\frac{\sigma_0}{3\epsilon_0}\vec{z}$. However, I am not able to compute it in the general case...
I tried to use something like $\vec{E}$ is $-\nabla \phi$ and "hope" that the integral will disappear, but without a lot of success (because I think it is not legal to just "remove" the integral thanks to the gradient ^^).
Do you have any hints??