Solving for Charge & Current Distributions

[Nusc]

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?

 

[pam]

Your E is wrong.
Apply the wave equation for A.

 

[Nusc]

Typo:

E=1/(4*pi*epsilon) q/r^2 rhat

 

[Nusc]

The answer is pho = q*delta(r)^3 r a vector

How do you get this?

 

[pam]

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 $${\vec E}\sim -\partial_t{\vec A}$$.

 

[Nusc]

This is question 10.3 in griffith

 

[Mindscrape]

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?

 

[Nusc]

I got it, problem solved.

 

[Nusc]

Given B = z hat

B=Curl A

How does one find A?

 

[Nusc]

A is dependent on t so denote A(t)

 

[Nusc]

We know that Div B = 0 but how does that help? I took the divergence of both sides and it gets me nowhere

 

[Mindscrape]

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

$$\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$

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

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

where tr is known as a retarded time

$$t_r \equiv t - \frac{\cal{R}}{c}$$

 

[Nusc]

Given B = Bo*t/tau z-hat where tau and Bo are constants?

 

[Mindscrape]

Will give you the current density.

 

[Nusc]

Using which formula?

 

[Nusc]

Sorry, the wave equation won't work here, because this A and phi are not in the Lorentz gauge.
Use $${\vec E}\sim -\partial_t{\vec A}$$.

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?

 

[pam]

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.

 

[Mindscrape]

Basically Maxwell's equations give you everything you ever wanted. It is imperative that you know how to manipulate them.

 

[Nusc]

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?

 

[Mindscrape]

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.

$$\nabla \times B = \mu_0 (\mathbf{J} + \mathbf{J_D})$$

 

[pam]

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.

 

[Nusc]

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.

$$\nabla \times B = \mu_0 (\mathbf{J} + \mathbf{J_D})$$

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.