Vector Potential: Infinite Wire and Infinite Solenoid

[DrPapper]

Homework Statement

Homework Equations

Provided in the questions I believe. Here's the triangle from question two.

The Attempt at a Solution

QUESTION SET 1 TOP OF PICTURE
A.) I didn't know how to just "guess" what the constant should be so I actually worked it out. I found the constant to be

c=\mu ₀ I ₀ / 4 \pi

But how would I just know as they want me to?

B.) I'm fine on this one.

C.) I think I'm fine on this one, and we're told to skip it. However, I believe it will grow linearly until it hits the edge of the wire then drop to zero.

Extra Question not shown: Is the answer unique?

I don't know how I'm supposed to know that with what's given. I know we can add any arbitrary constant we want for any vector potential (or so I was told) but I'm not really sure how to approach this part.

QUESTION SET 2 BOTTOM OF PICTURE
A.) I used the most general form for Biot-Savart law:

I'll wait to work on B since I think my part A is wrong.

Thanks in advance to any an all who give some help on this. :D

 

[Charles Link]

Your Part A is almost correct. The two equations you are comparing are $curl A=B$ and $curl B=\mu_oJ$ . $A$ will have an almost identical solution (Biot-Savart form), but will not have the $\mu_o$ in it. (It will have the $1/(4 \pi))$. (This instruction set is pointing out something that previously was somewhat absent from the textbooks. Usually they have used Stokes theorem to get Amperes law in integral form to solve for $B$ when given $curl B=u_oJ$. In general, another solution to the $curl B =\mu_o J$ equation is in fact a Biot-Savart form for $B$. In this case, you have $curl A=...$ so that you can write down a Biot-Savart form for $A$.)

 

[Charles Link]

(Please also see my previous response). On another problem, they give you a magnetic field B that has the same characteristics as the "current" of a long (infinite) solenoid... The B field for an infinite solenoid is uniform in the z-direction everywhere inside the solenoid and zero outside. This can be computed by the integral form of Ampere's law and (and Stokes theorem) with a rectangular line integral, upon the assumption that B=0 outside of the cylinder. (More detailed computations with Biot-Savart validate this assumption.)... Anyway, a B field that has the characteristics of a surface flow around a cylinder is a very hypothetical problem, (you will probably never encounter a B field with these characteristics), but assuming it were to occur, the mathematics can be readily applied to tell you what the function for the vector potential $A$ would need to look like. Remember $curl A=B$. We could even apply Stokes theorem to get the line integral $\int A*dl =$ flux of B through the (narrow) rectangle to compute $A$.

 

[DrPapper]

Your Part A is almost correct. The two equations you are comparing are $curl A=B$ and $curl B=\mu_oJ$ . $A$ will have an almost identical solution (Biot-Savart form), but will not have the $\mu_o$ in it. (It will have the $1/(4 \pi))$. (This instruction set is pointing out something that previously was somewhat absent from the textbooks. Usually they have used Stokes theorem to get Amperes law in integral form to solve for $B$ when given $curl B=u_oJ$. In general, another solution to the $curl B =\mu_o J$ equation is in fact a Biot-Savart form for $B$. In this case, you have $curl A=...$ so that you can write down a Biot-Savart form for $A$.)

Does this mean that the only thing wrong with my solution for 2.A is the \mu₀? Or do I need to think this over a bit more? :D

Also, if you don't mind, could you help me out with my questions on the first portion? I believe I have the correct answers, but that was from brute force working it out.

 

[Charles Link]

Suggestion is to write $\mu_o J$ as a single term. e.g. $curl B=(\mu_o J)$ and follow the $( \mu_o J)$ to the position in the Biot-Savart solution for $B$. All you need to do to solve for $A$ when $B$ is on the right side of $curl A =B$ is replace $\mu_o J$ by $B$ in the Biot-Savart equation. $curl B =\mu_o J$ is an inhomegeneous differential equation that has a Biot-Savart type solution for $B$. It is possible that there could be a homogeneous solution that needs to be added (solution of curl B=0), but in this case that is not necessary. (This solution to the curl equation doesn't seem to be in widespread use, but it really is quite useful. A more common solution that is seen quite often is the solution for $div E=\rho/ \epsilon _o$. This one has a well known integral solution for $E$ with the inverse square law for the distribution of electric charge.) And yes, I'd be glad to look over your solutions to your other questions here.

 

[DrPapper]

This is what my work ended up becoming for the second problem's part A.

 

[Charles Link]

Very good except it would be good to be careful with the unprimed and primed coordinates in Biot-Savart's law.
e.g. $B(x)=(\mu_o/(4\pi)) \int (J(x') \times (x-x'))/|x-x'|^3) dx'$. You can let $r=x-x'$ and $\hat{r}=(x-x')/|x-x'|$ but note, in particular, the $x$ in $B(x)$ and the $x'$ in $J(x')$. I think you might be more familiar with Biot-Savart's law for a point charge $q$, but in integral form this is how it is written. The reason for this detail should be clear=you are finding the magnetic field $B$ at some location $x$, and any moving charges at location $x'$ make a contribution. (Remember current density $J=nqv$ where $n$ is the density of the particles, etc.). This readily transfers to the form you need for $A(x)$. Would enjoy seeing your corrected form.