Electrical Field around Spherical Ball at origin

[Philosophaie]

Homework Statement

Spherical Ball centered at origin uniform $\rho$ with a radius a. Find E along x-axis.

Homework Equations

$E = \frac{\rho}{4\pi\epsilon_0}\int\int\int\frac{r^2*sin\theta}{r_\rho^2} d\phi d\theta dr$

The Attempt at a Solution

Evaluate E spherically along the x-axis:
$(x_1, 0, 0): r_\rho^2=(x - x_1)^2 + y^2 + z^2=r^2 + x_1^2 - 2*x_1*cos\theta*cos\phi$

$r_\rho =\sqrt{r^2 + x_1^2 - 2*x_1*cos\theta*cos\phi}$

It seems I am missing a component.

 

[GregoryS]

Surely you are! $r_ρ=\sqrt{r^2+x^2_1−2∗r*x_1∗cosθ∗cosϕ}$

But you can solve your problem without any troubles using the Gauss theorem:
$$
E × 4πx^2 = \frac{ρ × 4/3πa^3}{ε_0}
$$

 

[Philosophaie]

I need to understand this the long way.

The equation for E I wrote incorrectly
$E = \frac{\rho}{4*\pi*\epsilon_0}*\int\int\int\frac{\vec{e_{r_{\rho}}}}{r_{\rho}^2}*r^2*sin^2\theta d\phi d\theta dr$

How do I find:

$\vec{e_{r_\rho}}$