Electrical Field around Spherical Ball at origin

  • #1
461
0

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.
 
Last edited:

Answers and Replies

  • #2
12
0
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:
$$
E × 4πx^2 = \frac{ρ × 4/3πa^3}{ε_0}
$$
 
  • #3
461
0
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}}##
 
Last edited:

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