Calculate magnetic potential of a given current density

In summary: 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?Yes, that is correct.
  • #1
TheBlueDot
16
2
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!
 

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  • #2
Since it is magnetostatics, everything is static and we can ignore the t and t'. What it says is that A at the point [itex]\vec{r}[/itex] is given by adding up all of the currents [itex]\vec{J}(\vec{r'})[/itex] at all points [itex]\vec{r'}[/itex] and dividing by the distance between [itex]\vec{r}[/itex] and [itex]\vec{r'}[/itex] 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?
 
  • #3
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?
 
  • #4
TheBlueDot said:
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 [itex] \vec{r}[/itex]. You haven't included [itex] \vec{r}[/itex] anywhere. Note you need two position vectors, [itex] \vec{r'}[/itex] which describes the current distribution, and which you have replaced by s, but you also need the second position vector [itex] \vec{r}[/itex] 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?
 

1. How is magnetic potential related to current density?

The magnetic potential at a given point in space is directly proportional to the current density passing through that point. This means that as the current density increases, the magnetic potential also increases.

2. What is the formula for calculating magnetic potential?

The formula for calculating magnetic potential is Φ = μ₀I/4πr, where Φ is the magnetic potential, μ₀ is the permeability of free space, I is the current passing through the given point, and r is the distance from the point to the current.

3. Can magnetic potential be negative?

Yes, magnetic potential can be negative. This indicates that the direction of the magnetic field is in the opposite direction of the current flow.

4. How does the shape of the current affect the magnetic potential?

The shape of the current does not affect the magnetic potential as long as the current is passing through the given point. The formula for calculating magnetic potential takes into account the distance from the point to the current, regardless of its shape.

5. What are the units for magnetic potential?

The units for magnetic potential are joules per ampere (J/A) or equivalently, tesla-meter (Tm). Both of these units represent the energy per unit current.

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