Electromagnetic Vector Fields (Static)

In summary, the electric field produced by a charge density in different dimensions can be calculated using Maxwell's Equations and Gauss's law.
  • #1
tazzzdo
47
0

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



[itex]\vec{E}[/itex] and [itex]\vec{B}[/itex] are the electrical and magnetic vector fields, respectively

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

[itex]\vec{\nabla}[/itex] [itex]\times[/itex] [itex]\vec{E}[/itex] = -[itex]\partial[/itex][itex]\vec{B}[/itex]/[itex]\partial[/itex]t
[itex]\vec{\nabla}[/itex] [itex]\times[/itex] [itex]\vec{B}[/itex] = [itex]\partial[/itex][itex]\vec{E}[/itex]/[itex]\partial[/itex]t + [itex]\vec{j}[/itex]
[itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{E}[/itex] = [itex]\rho[/itex]
[itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{B}[/itex] = 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.
 
Last edited:
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  • #2
Gauss's law will figure heavily here. See if you can make headway with that.
 
  • #3
The different dimensions are what is confusing me.
 
  • #4
Think of distances in different dimensions. What is the length of a radial vector in 1,2, and 3 dimensions?
 
  • #5
So that would be the vector that connects the origin to whatever point charge I'm using (in whatever dimensions the problem defines)?
 
  • #6
Yes or the distance from the point charge to the field point in question.
 
  • #8
Here's what I've got so far for part 4:

Vn = [itex]\frac{R}{n}[/itex] [itex]\times[/itex] 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

[itex]\Rightarrow[/itex] Q = [itex]\int[/itex][itex]\int[/itex][itex]\int[/itex] ρ dV = ρ [itex]\times[/itex] volume = (4πR3/3)ρ

So then I think I would use the integral form:

∫∫ [itex]\vec{E}[/itex](r) [itex]\cdot[/itex] [itex]\hat{n}[/itex] dS

But [itex]\vec{E}[/itex](r) [itex]\cdot[/itex] [itex]\hat{n}[/itex] = 1, right?

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

[itex]\underbrace{∫...∫}_{n}[/itex] ([itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{E}[/itex]) dV = [itex]\underbrace{∫...∫}_{n}[/itex] 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.
 

Related to Electromagnetic Vector Fields (Static)

1. What is an electromagnetic vector field?

An electromagnetic vector field is a representation of the interaction between electric and magnetic fields. It combines the properties of both fields to describe the behavior of electromagnetic radiation and its effect on charged particles.

2. What is a static electromagnetic vector field?

A static electromagnetic vector field is one that does not change over time. This type of field is typically produced by stationary sources, such as charged particles at rest, and does not have any time-varying components.

3. How is a static electromagnetic vector field measured?

A static electromagnetic vector field can be measured using devices such as electromagnetic field meters, which detect the strength and direction of the electric and magnetic fields. These measurements can be used to map out the field and analyze its properties.

4. What are some practical applications of static electromagnetic vector fields?

Static electromagnetic vector fields have various practical applications, including in electronic devices such as antennas, transformers, and motors. They are also used in medical imaging techniques such as magnetic resonance imaging (MRI) and in telecommunications for wireless communication.

5. How is the behavior of a charged particle affected by a static electromagnetic vector field?

A charged particle placed in a static electromagnetic vector field will experience a force due to the interaction between the electric and magnetic fields. The direction and magnitude of this force depend on the charge and velocity of the particle, as well as the strength and direction of the field.

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