Electromagnetic Vector Fields (Static)

[tazzzdo]

Homework Statement

Use the integral form and symmetry arguments to compute the electric field produced by the following charge densities:

(i) Point charge q, placed at the origin, in 3 dimensions;
(ii) Point charge q, placed at the origin, in 2 dimensions;
(iii) Point charge q, placed at the origin, in 1 dimension;
(iv) Sphere of charge Q, with center at the origin, in 3 dimensions;
(v) Sphere of charge Q, with center at the origin, in 6 dimensions;

Homework Equations

$\vec{E}$ and $\vec{B}$ are the electrical and magnetic vector fields, respectively

Maxwell's Equations (with all constants set to 1):

$\vec{\nabla}$ $\times$ $\vec{E}$ = -$\partial$$\vec{B}$/$\partial$t
$\vec{\nabla}$ $\times$ $\vec{B}$ = $\partial$$\vec{E}$/$\partial$t + $\vec{j}$
$\vec{\nabla}$ $\cdot$ $\vec{E}$ = $\rho$
$\vec{\nabla}$ $\cdot$ $\vec{B}$ = 0

The Attempt at a Solution

No idea how to even set it up. I'm a Math major taking Vector Calculus, Physics is not my cup of tea lol.

 

[Antiphon]

Gauss's law will figure heavily here. See if you can make headway with that.

 

[tazzzdo]

The different dimensions are what is confusing me.

 

[hexiphysics]

Think of distances in different dimensions. What is the length of a radial vector in 1,2, and 3 dimensions?

 

[tazzzdo]

So that would be the vector that connects the origin to whatever point charge I'm using (in whatever dimensions the problem defines)?

 

[hexiphysics]

Yes or the distance from the point charge to the field point in question.

 

[sean_mp]

You need to use one, two and three dimensional dirac delta functions for your charge densities for the first three parts. Check out page 4 of these notes:
http://theory.uchicago.edu/~sethi/Teaching/P221-F2008/DiracandGreenNotes(08).pdf

 

[tazzzdo]

Here's what I've got so far for part 4:

V_n = $\frac{R}{n}$ $\times$ 4R², where n is the dimensions and R is the radius

ρ(r) = Q/(4πR³/3), where Q is the total charge of the sphere

$\Rightarrow$ Q = $\int$$\int$$\int$ ρ dV = ρ $\times$ volume = (4πR³/3)ρ

So then I think I would use the integral form:

∫∫ $\vec{E}$(r) $\cdot$ $\hat{n}$ dS

But $\vec{E}$(r) $\cdot$ $\hat{n}$ = 1, right?

Then apply the divergence theorem to get (in whatever n dimensions):

$\underbrace{∫...∫}_{n}$ ($\vec{\nabla}$ $\cdot$ $\vec{E}$) dV = $\underbrace{∫...∫}_{n}$ ρ dV

I feel like I'm on the right track. And the approach would be the same for the point charges, except I would be using a radius r > R since the points charge is essentially the limit of the sphere charge as it approaches 0.