(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Give an expression for the magnetic field and show that a magnetic vector exists such as [tex]\vec{A}(P) = A(r)\hat{z}[/tex] and [tex]\vec{B}(P) = \vec{\nabla} \times \vec{A}[/tex]

For the infinite wire shown in figure 1.

Here is a link to the figure and problem statement. The problem is the second problem on the first page.

https://docs.google.com/fileview?id...UtOGNkYi00ZGQyLTkxOTktNWVjYzM2MGViNDg3&hl=en"

2. Relevant equations

[tex] \vec{\nabla} \times \vec{A} = \left(\frac{1}{r}\frac{\partial A_{z}}{\partial \theta} - \frac{\partial A_{\theta}}{\partial z}\right)\hat{r} + \left(\frac{\partial A_{r}}{\partial z} - \frac{\partial A_{z}}{\partial r}\right)\hat{\theta} + \frac{1}{r}\left(\frac{\partial(rA_{\theta})}{\partial r} - \frac{\partial A_{r}}{\partial \theta}\right)\hat{z}[/tex] curl in polar coordinates

[tex]\oint \vec{B} \cdot d\vec{l} = \mu_{0}I_{enc}[/tex] Ampere's Law

3. The attempt at a solution

First we must solve for the magnetic field at point P. Using Ampere's Law we get,

[tex] \vec{B}(P) = \frac{\mu_{0}I}{2\pi r}\hat{\theta} [/tex] where the positive theta direction is into the page.

Now in order to findAwe can set the curl equations equal to the magnetic field equation by curl(A) =B.

Doing so gives,

[tex]\frac{1}{r}\frac{\partial A_{z}}{\partial \theta} - \frac{\partial A_{\theta}}{\partial z} = 0 [/tex]

[tex]\frac{\partial A_{r}}{\partial z} - \frac{\partial A_{z}}{\partial r} = \frac{\mu_{0}I}{2\pi r}[/tex]

[tex]\frac{\partial(rA_{\theta})}{\partial r} - \frac{\partial A_{r}}{\partial \theta} = 0[/tex]

The answer to this problem (given to us by our instructor) is,

[tex]\vec{A} = \left(-\frac{\mu_{0}I}{2\pi}ln(r) + K \right)\hat{z}[/tex]

and the only way I can get there is by saying that because A points most often in the direction of the current (which is the z direction in this case) then,

[tex]-\frac{dA_{z}}{dr} = \frac{\mu_{0}I}{2\pi r}.[/tex]

Then solving this differential equation gives the desired result.

Can someone please point me in the right direction as to how solve this problem in a more rigorous manner?

Do I need to use the equation [tex]\vec{A} = \frac{\mu_{0}}{4\pi}\int \frac{\vec{J}(\vec{r'})}{r}d\tau '[/tex]?

Many Thanks in advance,

KEØM

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# Homework Help: Magnetostatics: Magnetic Vector Potential

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