Solving for Charge & Current Distributions

In summary, the conversation discusses finding the charge and current distributions for a given magnetic field, using Maxwell's equations and potentials. The solution involves manipulating the equations and finding the current density from the given magnetic field, then using it to find the vector potential. The conversation also mentions the Lorentz gauge and the electrostatic potential, and suggests using the formula A = (B x r)/2 for a constant B.
  • #1
Nusc
760
2

Homework Statement



Find the charge and current distributions for

V(r,t)=0 A(r,t) = -1/(4*pi*epsilon) q*t/r^2 r-hat

Homework Equations

We know
E=1/(a*pi*epsilon) q/r^2 rhat
B = 0

The Attempt at a Solution


What formula do I use?

We know grad x B = mu*J +mu*epsilon dE/dt

Would this suffice to solve for J?

What about rho?
 
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  • #2
Your E is wrong.
Apply the wave equation for A.
 
  • #3
Typo:

E=1/(4*pi*epsilon) q/r^2 rhat
 
  • #4
The answer is pho = q*delta(r)^3 r a vector

How do you get this?
 
  • #5
pam said:
Apply the wave equation for A.
Sorry, the wave equation won't work here, because this A and phi are not in the Lorentz gauge.
Use [tex]{\vec E}\sim -\partial_t{\vec A}[/tex].
 
Last edited:
  • #6
This is question 10.3 in griffith
 
  • #7
What is the whole point of using potentials? I'll answer this one for you, it's because two of maxwell's equations are automatically solved with potentials. The no magnetic monopoles equation and Faraday's Law are solved, bam, done.

So, how do E and B relate to the potentials?
 
  • #8
I got it, problem solved.
 
  • #9
Given B = z hat

B=Curl A

How does one find A?
 
  • #10
A is dependent on t so denote A(t)
 
  • #11
We know that Div B = 0 but how does that help? I took the divergence of both sides and it gets me nowhere
 
  • #12
It will depend on what kind of gauge you are using, of which there are infinite. The Lorentz gauge is the most standard since it can handle special relativity, and it has a vector equation of

[tex]\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}[/tex]

which you should recognize as being similar to the electrostatic potential, with a time term in there, so you can just guess that the solution will be

[tex]\mathbf{A} = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}(\mathbf{r'},t_r)}{\cal{R}} d\tau'[/tex]

where tr is known as a retarded time

[tex]t_r \equiv t - \frac{\cal{R}}{c}[/tex]
 
  • #13
Given B = Bo*t/tau z-hat where tau and Bo are constants?
 
  • #14
Will give you the current density.
 
  • #15
Using which formula?
 
  • #16
pam said:
Sorry, the wave equation won't work here, because this A and phi are not in the Lorentz gauge.
Use [tex]{\vec E}\sim -\partial_t{\vec A}[/tex].

ARe you sure this is true, I was told that this is not a requirement.

Now, given B. How do I find A and Which formula do I use?
 
  • #17
Nusc said:

Homework Statement



Find the charge and current distributions for

V(r,t)=0 A(r,t) = -1/(4*pi*epsilon) q*t/r^2 r-hat
You are given A. I showed you how to get E.
B=curl A=0.
You should find that E just equals Coulomb field of a point charge.
 
  • #18
Basically Maxwell's equations give you everything you ever wanted. It is imperative that you know how to manipulate them.
 
  • #19
This is a different question, I had already solved the one in the first post.

Given B = Bo*t/tau z-hat where tau and Bo are constants how does one find A?

B = grad A What rule do I need to apply to solve for A?
 
  • #20
Practically, it's kind of stupid to convert B into a potential because B is what you are always after, well usually H, but all you have to do is find the current density from B, using maxwell's equations, then go to A.

[tex]\nabla \times B = \mu_0 (\mathbf{J} + \mathbf{J_D})[/tex]
 
  • #21
Nusc said:
This is a different question, I had already solved the one in the first post.

Given B = Bo*t/tau z-hat where tau and Bo are constants how does one find A?

B = grad A What rule do I need to apply to solve for A?
B=curl A. For a constant B, A=(BXr)/2.
 
  • #22
Mindscrape said:
Practically, it's kind of stupid to convert B into a potential because B is what you are always after, well usually H, but all you have to do is find the current density from B, using maxwell's equations, then go to A.

[tex]\nabla \times B = \mu_0 (\mathbf{J} + \mathbf{J_D})[/tex]

Although the question never specified whether E is 0, that's Ampere's law for the static case. Is that a reasonable assumption?

"For a constant B, A=(BXr)/2."

How do you show this? Never heard of it and what's r ?

Here's the original question:

In a certain region, the magnetic field as a linear function of time is given by

B = Bo (t/tau) z hat

Bo and tau constans.

FInd a simple expression for the vector potential which will yield this field.
 

1. What is the difference between charge and current distributions?

Charge distribution refers to the spatial distribution of electric charge, while current distribution refers to the spatial distribution of electric current. This means that charge distribution describes the amount and location of electric charge in a given space, while current distribution describes the flow of charge over time.

2. How do you solve for charge and current distributions?

To solve for charge and current distributions, you will need to use the appropriate mathematical equations and principles, such as Coulomb's law, Ohm's law, and Kirchhoff's laws. These equations allow you to calculate the distribution of charge or current based on known values and physical properties of the system.

3. What factors affect the distribution of charge and current?

The distribution of charge and current can be affected by a variety of factors, including the geometry of the system, the material properties of the conductors, and the presence of other electric fields or sources. These factors can influence the flow of charge and current through a system and can impact the resulting distribution.

4. Can charge and current distributions change over time?

Yes, charge and current distributions can change over time due to a variety of factors. For example, in a circuit with changing currents, the distribution of charge and current will also change as the electric charge flows through the circuit. Additionally, external factors such as temperature or other environmental conditions can also affect the distribution of charge and current.

5. How are charge and current distributions related to each other?

Charge and current distributions are closely related, as they both describe the behavior and movement of electric charge. However, they are not the same - charge distribution describes the amount and location of charge, while current distribution describes the flow of charge over time. In many systems, the distribution of charge and current will be interdependent and will change together.

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