Outside the origin circular loop current density

In summary: When solving for ##J##, remember to use the Laplace transform, and remember to use the right hand side as the numerator, and the left hand side as the denominator.
  • #1
Mr. Rho
15
1
Hi, I'm trying to write the current density for such circular loop in spherical coordinates. For a circular loop of radius [itex]a[/itex] that lies in the [itex]XY[/itex] plane at the origin, the current density it's simply:

[itex]\mathbf{J}= \frac{I}{2\pi\sin\theta}\delta(\theta-\frac{\pi}{2})\frac{\delta(r-a)}{a}\hat{\phi}[/itex]​

I want the current density of the circular loop of radius [itex]a[/itex] displaced a distance [itex]c[/itex] towards the [itex]y[/itex] axis.

Any suggestions?
 
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  • #2
I can't make sense of this. Your units don't balance. What is the relationship between a, c, and r? Could you try again?
 
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  • #3
stedwards said:
I can't make sense of this. Your units don't balance. What is the relationship between a, c, and r? Could you try again?
Sorry I wrote the equation wrong, just fixed it. I'm using this kind of spherical coordinates:

250px-Spherical_polar.png
 
  • #4
No, really. Think about it a bit and restate the entire question. 'cuse, now current and current density have the same units, and nobody knows what ##c## is. I'd sleep on it.
 
  • #5
stedwards said:
No, really. Think about it a bit and restate the entire question. 'cuse, now current and current density have the same units, and nobody knows what ##c## is. I'd sleep on it.

I don't know what I was thinking, the correct current density is:

[itex]\mathbf{J}=I\delta(\theta-\frac{\pi}{2})\frac{\delta(r-a)}{a}\hat{\phi} = I\sin\theta\delta(\cos\theta)\frac{\delta(r-a)}{a}\hat{\phi}[/itex]​

it satisfies [itex]I=\int\mathbf{J}\cdot{d\mathbf{S}}=\int_{0}^{\pi}\int_{0}^{\infty}\mathbf{J}\cdot{\hat{\phi}}rdrd\theta[/itex], where [itex]\mathbf{S}[/itex] is a surface perpendicular to the current direction.

Sorry for not making myself clear for what I'm asking. I hope this image makes things clear:

Untitled.png
The current density I present is case (i) and the current density I need is case (ii).
 
  • #6
To begin with, take the origin-centered solution for a circle of radius ##a##, change to Cartesian coordinates, translate to the right (##x \leftarrow x' = x + c##), then back to spherical coordinates.

It will give the equation for the current path you want.
 
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1. What is "Outside the origin circular loop current density"?

The outside the origin circular loop current density refers to the distribution of electric current around a circular loop that is centered at a point outside of the loop itself. This current density can be calculated using the Biot-Savart Law, which describes the magnetic field created by a current-carrying wire.

2. How is the outside the origin circular loop current density different from the inside loop current density?

The main difference between the outside and inside loop current density is their respective positions relative to the current-carrying wire. The inside loop current density is located within the circular loop, while the outside loop current density is located outside of the loop.

3. What factors affect the outside the origin circular loop current density?

The outside the origin circular loop current density is affected by the distance from the origin point to the loop, the radius of the loop, the magnitude of the current, and the permeability of the medium surrounding the loop. These factors impact the strength and direction of the magnetic field created by the current.

4. How does the outside the origin circular loop current density impact magnetic fields?

The outside the origin circular loop current density is directly related to the magnetic field created by the current. As the current density increases, the magnetic field strength also increases. The direction of the current also affects the direction of the magnetic field lines, which can be determined using the right-hand rule.

5. What are some practical applications of understanding the outside the origin circular loop current density?

Knowledge of the outside the origin circular loop current density is important in many fields, including electrical engineering, physics, and geophysics. Some practical applications include designing efficient electrical circuits, analyzing the behavior of magnetic materials, and studying Earth's magnetic field.

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