Calculate magnetic potential of a given current density

[TheBlueDot]

Hello,

I'm current studying magnetostatics and I'm struggling to understand the equation for A. Here is the equation from Griffith in the attachment. My confusion is, if the object is a cylinder and the current density is a function of s, J(s), then how would I write r-r'?

Thank you!

 

[phyzguy]

Since it is magnetostatics, everything is static and we can ignore the t and t'. What it says is that A at the point $\vec{r}$ is given by adding up all of the currents $\vec{J}(\vec{r'})$ at all points $\vec{r'}$ and dividing by the distance between $\vec{r}$ and $\vec{r'}$ Exactly how you write this depends on where the currents are flowing. When you say J(s), what is s? Do you mean a cylinder where current is only flowing on the surface of the cylinder?

 

[TheBlueDot]

Hi phyzguy,

Thank you for replying.
For the problem I'm looking at, J(s) = ks, where k is a constant and s is the radius of the cylinder.
With that J(s), I got $$A = \frac {\mu_{o}} {4\pi}\int \frac{a*s(sdsd\phi dz)}{s}\hat{z}$$. Is this correct?

 

[phyzguy]

Hi phyzguy, Thank you for replying.
For the problem I'm looking at, J(s) = ks, where k is a constant and s is the radius of the cylinder.
With that J(s), I got $$A = \frac {\mu_{o}} {4\pi}\int \frac{a*s(sdsd\phi dz)}{s}\hat{z}$$. Is this correct?

No, you're still missing a couple of things. First, what is the direction of J? Is it in the z direction as you have indicated, or is it in the φ direction? Second, A is a function of the position vector $\vec{r}$. You haven't included $\vec{r}$ anywhere. Note you need two position vectors, $\vec{r'}$ which describes the current distribution, and which you have replaced by s, but you also need the second position vector $\vec{r}$ which describes where you are calculating A. Also, you need limits on your integration. Does s run from 0 up to some R? What about the limits on φ and z?