Bound Bulk Current Density and Surface Current Density

In summary: Therefore, the total bound surface current is given by:##I_S = \int \vec K_b \cdot d \vec l = -\oint K_{b, z} dl = -\oint \vec M \cdot d\vec {l} = -\int \vec J_b \cdot d\vec a ##Similarly, for the total bound bulk current, we have:##I_B = \int \vec J_b \cdot d\vec a = \int \left( \vec \nabla \times \vec M \right) \cdot d\vec a = \oint \vec M \cdot d\vec {l} ##where the last equality follows from Stokes' theorem. Therefore
  • #1
Mr_Allod
42
16
Homework Statement
A steady current I flows down a long cylindrical wire of radius a. Assume that the wire has magnetic susceptibility, ##\chi_m##, and that the current is distributed in such a way that ##J## is proportional to ##s##, the distance from the wire axis. Determine the magnetic field B both inside and outside the wire. Calculate the bound bulk current density ##J_b## in the wire and bound surface current density ##K_b## on the surface of the wire, and show that the total bound bulk current is equal and opposite to the total bound surface current.
Relevant Equations
##\vec H = \frac 1 \mu_0 \vec B - \vec M ##

##\vec M = \chi_m \vec H##

##\vec J_b = \vec \nabla \times \vec M##

##\vec K_b = \vec M \times \hat s##
Hi there, I've worked through most of this question but I'm stuck on the final part, showing that total bulk current ##I_B## is equal and opposite to total surface current ##I_S##. I calculated ##\vec H## the normal way I would if I was looking for ##\vec B## in an infinitely long cylindrical wire where ##\vec J## is proportional to ##s## and found:
##
\vec H =
\begin{cases}
I \frac {s^2} {2\pi a^3} & \text{inside} \\
I\frac 1 {2\pi s} & \text{outside }
\end{cases}
##

Then rearranging the relationships ##\vec H = \frac 1 \mu_0 \vec B - \vec M ## and ##\vec M = \chi_m \vec H## I got:
##\vec B = \mu_0 \vec H (1+ \chi_m)##
##\mu = (1 + \chi_m)##

Using this I found ##\vec B## inside and outside the wire to be:
##
\vec B =
\begin{cases}
\mu I \frac {s^2} {2\pi a^3} & \text{inside} \\
\mu_0 I\frac 1 {2\pi s} & \text{outside }
\end{cases}
##
Then using the two cross-products I found ##\vec J_b## and ##\vec K_b##:
##\vec J_b = \chi_m I \frac {3s} {2\pi a^3} \hat z##
##\vec K_b = -\chi_m I \frac 1 {2\pi a} \hat z##

Now I am unsure of the right way to find the total bulk current ##I_B## and surface current ##I_S##. I'd appreciate it if someone could point me in the right direction.
 
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  • #2
Mr_Allod said:
##\vec J_b = \chi_m I \frac {3s} {2\pi a^3} \hat z##
##\vec K_b = -\chi_m I \frac 1 {2\pi a} \hat z##
I think these are correct.

Now I am unsure of the right way to find the total bulk current ##I_B## and surface current ##I_S##. I'd appreciate it if someone could point me in the right direction.
Consider a cross section of the cylinder (perpendicular to the axis of the cylinder). The total bound bulk current is the total bound current through the cross section. You can calculate this using your expression for ##J_b##.

The total bound surface current is the total bound surface current through the circular boundary of the cross section. You can calculate this using ##K_b##.
 
  • #3
Do you mean something like this?

##\int \vec J_b \cdot d\vec a = \chi_m \frac 3 {2\pi a^3} \int_0^a s(2\pi s) ds ##
## \oint \vec K_b \cdot d \vec l = -\chi_m \frac 1 {2\pi a} \int_0^{2\pi} a d\phi##

They give me the expected result but I wasn't sure if I approached it correctly.
 
  • #4
Mr_Allod said:
Do you mean something like this?

##\int \vec J_b \cdot d\vec a = \chi_m \frac 3 {2\pi a^3} \int_0^a s(2\pi s) ds ##
## \oint \vec K_b \cdot d \vec l = -\chi_m \frac 1 {2\pi a} \int_0^{2\pi} a d\phi##

They give me the expected result but I wasn't sure if I approached it correctly.
Yes, that looks good.

------------------------------

If you want to, you can show the result more generally (without using the explicit expressions for ##\vec J_b## and ##\vec K_b##):

##\int \vec J_b \cdot d\vec a = \int \left( \vec \nabla \times \vec M \right) \cdot d\vec a = \oint \vec M \cdot d\vec {l} ##

where the last equality follows from Stokes' theorem.

For the case of the circular cross section, ##\vec M \cdot d\vec {l} = M_\phi dl = -K_{b, z} dl##, where the last equality follows from ##\vec K_b = \vec M \times \hat s##.
 

Related to Bound Bulk Current Density and Surface Current Density

1. What is bound bulk current density?

Bound bulk current density refers to the flow of electric current within a material, such as a conductor or semiconductor. It is caused by the movement of free electrons or ions within the material, and is typically measured in amperes per square meter.

2. How is bound bulk current density related to surface current density?

Bound bulk current density and surface current density are related in that the surface current density is the component of the bound bulk current density that flows along the surface of a material. This is due to the fact that the surface of a material has a higher concentration of free electrons and ions, making it easier for current to flow along it.

3. What factors affect the bound bulk current density and surface current density?

The bound bulk current density and surface current density are affected by several factors, including the material's conductivity, the applied voltage, and the dimensions of the material. Changes in these factors can alter the flow of current within the material.

4. How are bound bulk current density and surface current density measured?

Bound bulk current density and surface current density can be measured using various techniques, such as current probes, Hall effect sensors, and electromagnetic field measurements. These methods allow scientists to accurately measure the strength and direction of the current flow within a material.

5. What are some practical applications of understanding bound bulk current density and surface current density?

Understanding bound bulk current density and surface current density is important in many fields, including electrical engineering, materials science, and electronics. It can help in the design and optimization of electronic devices, as well as in the development of new materials with desired electrical properties.

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