Let me preface by saying that I am a freshman in an introductory level Electricity and Magnetism course. My professor has assigned this problem, as he briefly introduced the idea of vector potentials, along with curl and divergence operators. I am VERY much lacking in knowledge of any of these concepts and would appreciate guidance. I understand that this site is not supposed to "do the work" for you, but in this case, an explanation would prove very useful.(adsbygoogle = window.adsbygoogle || []).push({});

Problem Statement and Relevant Equations

Recall Ampere's Circuital Law

[tex]\oint{B\bullet \partial l}=\mu_0 I_{enc}=\mu_0 \int{J\bullet dA}[/tex]

But Stokes Theorem states:

[tex]\oint{B\bullet \partial l}=\int_{surface}{(\nabla \times B)\bullet dA}[/tex]

Therefore:

[tex]\nabla \times B=\mu_0 J[/tex]

We can apply similar reasoning to:

[tex]\oint{A\bullet \partial l}=\int{(\nabla \times B)\bullet dA}=\int_{surface}{B\bullet dA}=\phi_{enc}[/tex]

Apply this to a long solenoid.

[tex]\oint{A\bullet \partial l}=\phi_{enc}[/tex] is similar to [tex]\oint{B\bullet\partial l}=\mu_0 I_{enc}[/tex]

Pursue this analogy by drawin the appropriate Amperian loop (radius r) to show:

[tex]A=\frac{\mu_0 NI}{2}r\hat{\phi}[/tex] inside the solenoid, and:

[tex]A=\frac{\mu_0 NI}{2}\cdot\frac{R^2}{r}\hat{\phi}[/tex] outisde the solenoid.

Note thatAfollows the current direction. Also note that the1/rdependence givesB=0outside the coil. Explain and sketch the lines ofAin both cases.

Questions Regarding Problem

- Is the
Ain the problem referring to the vector potential?- What is all of this saying?

As I stated previously, I am very new to this and any help is greatly appreciated. Thanks

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# Homework Help: Vector Potential of Long Solenoid

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