- #1
phys9928
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- Homework Statement
- I am trying to solve for the magnetic field in and around a hollow, conductive cylinder that is placed in an axially directed external field. However I have more unspecified constants than unknowns. Under what conditions is such a problem solvable and what ?
- Relevant Equations
- Helmholtz and Laplace equations in polar coordinates.
My attempt at a solution:
Cylindrical coordinate system with ##r##, ##\theta##, ##z##. Conductivity ##\sigma## and permeability ##\mu_0##. Inner radius ##a## and outer radius ##b##. (##b>a##)
The external field is spatially uniform and driven at sinusoidally at frequency ##f##. The external field is given by ##\vec{B} = B_0 \exp(i 2 \pi f)##
I solve the Helmholtz equation within the rod ##(a<r<b)##, and the Laplace equation everywhere else ##(b<r<a)##. I assume ##z## derivatives are zero due to infinite rod and ##\theta## derivatives are zero due to rotational symmetry.
Therefore my general solution is:
$$
B(r) = \left.
\begin{cases}
a + b\ln{r}, & \text{for } 0 \leq r \leq a \\
c J_0(kr) + dY_0(kr), & \text{for } a \leq r \leq b \\
e + f\ln{r}, & \text{for } b \leq r \leq \infty
\end{cases}
\right\}
$$
Where ##J_0## and ##Y_0## are Bessel functions of the first and second kind.
However, my boundary conditions are:
- ##B(0)## = bounded
- ##B(r=\infty) = B_0##
- ##B(r = a^+) = B(r = a^-)##
- ##B(r = b^+) = B(r = b^-)##
Therefore I am left with four equations and five unknowns. Am I missing a condition (sommerfield radiation or something like that?) or have I made a mistake and the question is ill-posed?
In the case of a solid cylinder there are no issues with this approach as the Bessel function ##Y_0## is neglected due to the boundedness at ##r=0##.
Cylindrical coordinate system with ##r##, ##\theta##, ##z##. Conductivity ##\sigma## and permeability ##\mu_0##. Inner radius ##a## and outer radius ##b##. (##b>a##)
The external field is spatially uniform and driven at sinusoidally at frequency ##f##. The external field is given by ##\vec{B} = B_0 \exp(i 2 \pi f)##
I solve the Helmholtz equation within the rod ##(a<r<b)##, and the Laplace equation everywhere else ##(b<r<a)##. I assume ##z## derivatives are zero due to infinite rod and ##\theta## derivatives are zero due to rotational symmetry.
Therefore my general solution is:
$$
B(r) = \left.
\begin{cases}
a + b\ln{r}, & \text{for } 0 \leq r \leq a \\
c J_0(kr) + dY_0(kr), & \text{for } a \leq r \leq b \\
e + f\ln{r}, & \text{for } b \leq r \leq \infty
\end{cases}
\right\}
$$
Where ##J_0## and ##Y_0## are Bessel functions of the first and second kind.
However, my boundary conditions are:
- ##B(0)## = bounded
- ##B(r=\infty) = B_0##
- ##B(r = a^+) = B(r = a^-)##
- ##B(r = b^+) = B(r = b^-)##
Therefore I am left with four equations and five unknowns. Am I missing a condition (sommerfield radiation or something like that?) or have I made a mistake and the question is ill-posed?
In the case of a solid cylinder there are no issues with this approach as the Bessel function ##Y_0## is neglected due to the boundedness at ##r=0##.
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