Current Density and Charge Density in a loop of Wire

[Philosophaie]

I am trying to formulate the Current Density for a Loop of wire with a diameter,d, current,I, and an cross-sectional Area of the wire ##\pi(d/2)^2##. With spherical coordinates (radial, azimuthal, polar)

##\bar j## = ##\frac{I}{\pi(d/2)^2}*cos \theta *sin \phi \hat x##
##+\frac{I}{\pi(d/2)^2}*sin \theta *sin \phi \hat y##
##+\frac{I}{\pi(d/2)^2}*cos \phi \hat z##

Charge Density, ##\rho=-grad . \bar j##

 

[NFuller]

Is the loop in the x-y plane?

You will need to use the Heaviside step function ##H(x)## to properly define ##\mathbf{j}##. Once you clarify the orientation of the wire, we can give more input on how to formulate the problem.

Your definition of charge density is also incorrect. I think you meant
$$\frac{\partial \rho}{\partial t}=-\nabla\cdot\mathbf{j}$$

 

[Philosophaie]

The loop is on the x-y plane radius a to the center of the wire away from the origin. The charge density starts at zero then a DC current is induced ramping up to a steady-state value which can be emulated by Heaviside step function H(x). Will the above Current and Charge Density be calculated as above?

 

[NFuller]

No. You need to just build up ##\mathbf{j}## in spherical coordinates using step functions such that ##\mathbf{j}## is zero outside of the wire. Since the wire is cylindrical in shape this will be somewhat tricky. Am I correct in assuming that you are trying to define ##\mathbf{j}## in spherical coordinates given a known current ##I## through the wire?

 

[Philosophaie]

Since the wire is cylindrical in shape this will be somewhat tricky

How would I go about accounting for the distance,a, from the center of the coordinate system and the center of the circular loop accounting for ##\bar j## has to have the units of Amps/Area.

 

[NFuller]

How would I go about accounting for the distance,a, from the center of the coordinate system and the center of the circular loop accounting for ##\bar j## has to have the units of Amps/Area.

If the vector ##\mathbf{a}## points from the origin to the center of the wire and vector ##\mathbf{r}## points to a coordinate in space where you are evaluating ##\mathbf{j}(\mathbf{r})##, then the current density can be written as
$$\mathbf{j}(\mathbf{r})=\frac{4I}{\pi d^{2}}\;H(d/2-|\mathbf{r}-\mathbf{a}|)$$
Now you just need to replace these vectors with their spherical coordinate representations.

 

[Philosophaie]

This seems right except the units will be Amps/(unit length) instead of Amps/(unit length)^2 as is for Current Density. Does Heaviside function take care of this?

H=0 before induced current
H = (##\frac{d}{2}(\hat{\vec r -\vec a})+\vec a-\vec r##) after induced current

 

[NFuller]

This seems right except the units will be Amps/(unit length) instead of Amps/(unit length)^2 as is for Current Density. Does Heaviside function take care of this?

##H(\mathbf{r})## is dimensionless even though the argument does have dimensions of length. So the formula in post #6 has dimensions of Current/(length)^2.

H = (##\frac{d}{2}(\hat{\vec r -\vec a})+\vec a-\vec r##) after induced current

I'm not sure how you got this. What do you get for ##|\mathbf{r}-\mathbf{a}|## in spherical coordinates?