Solenoid with current carrying wire inside

[meaghan]

Homework Statement

I have a solenoid with a wire carrying current in the center. The wire has a radius of a, the solenoid has a radius of b. I need to find the magnetic field inside of each region. Inside of the wire, mu =/= muo.

Homework Equations

Wire B field = u_o I/2*pi*r
Solenoid B field = N/length *I

The Attempt at a Solution

When the r<a:
mu*n*I/length - mu*I*r/(2*pi*a^2)

When a<r<b
mu*n*I/length - mu*I/(2*pi*r

When r> b
then the solenoid will have no magnetic field,
I*mu/2*pi*r

I'm confused how the different mu values would factor into the equation. I'm 90% sure I did this correct

 

[Charles Link]

The magnetic field from the current carrying wire down the middle is in the $- \hat{a}_{\phi}$ direction. Meanwhile, the magnetic field from the solenoid turns is in the $\hat{z}$ direction.

 

[meaghan]

The magnetic field from the current carrying wire down the middle is in the $- \hat{a}_{\phi}$ direction. Meanwhile, the magnetic field from the solenoid turns is in the z-direction.

I forgot my directions. Does the wires having a different value for mu than air affect anything? I solved it first using H and then I got to getting the magnetic field and I was confused how it would affect it.

 

[Charles Link]

I forgot my directions. Does the wires having a different value for mu than air affect anything? I solved it first using H and then I got to getting the magnetic field and I was confused how it would affect it.

The wire down the middle could be made of e.g. iron which is a reasonably good conductor, and is also a magnetic material. In this case, the equation $B=\mu H$ applies, where $H=B/\mu_o$ is the $H$ field from the solenoid plus the $H$ field from the current in its own wire. (Use Biot-Savart and/or Ampere's law to compute it). Note: $\mu=\mu_o \mu_r$ where $\mu_r$ is the relative permeability of the (iron) wire. (A value for $\mu_r \approx 500$ is quite common for iron). The wire of the solenoid is assumed to be of a non-magnetic material. $\\$ Additional item: It may puzzle you how the equation $B=\mu H$ originates. The magnetic material (for uniform magnetization) has magnetic surface currents that generate a magnetic field ($B_m$) inside the magnetic material that is equal to $M$, so that $B=\mu_o H+M$ and $M= \mu_o \chi_m H$. This makes $B=\mu H$ with $\mu=\mu_o (1+\chi_m)$ . For an introduction to this concept, see https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/ You can also just use the formula $B=\mu H$, but it is good to have some idea of the origins of this formula. $\\$ Additional note: In the above, I've omitted the vector symbol, but I'm really meaning $\vec{B}=\mu \vec{H}$, etc. $\\$ And additional item: The more common problem is the long solenoid with an iron core. This problem adds the additional detail of running a DC current through that iron core. One thing the problem could specify more clearly is that there are two different currents: that of the solenoid $I_s$ , and that of the wire $I_w$ .