Using a Transfer Matrix

K.E. Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ U.S.A.

Jackson problems 4.8, 5.8, and many similar problems can be solved using a transfer matrix. These problems are the shielding effect of a circular cross section dielectric or of a permeable sheath. For the magnetic case, there are no currents, so the curl of $\vec{H}$ and the divergence of $\vec{B}$ are both zero, and

$$\vec{H} \vec{\nabla} \Phi$$ =

$$\vec{B} \mu \vec{H}$$ = (1)

Except at the boundaries between media with different $\mu$ ,

$$\nabla^{2}_{} \Phi$$ (2)

and tangential $\vec{H}$ and normal $\vec{B}$ are continuous at boundaries. Continuity of tangential $\vec{H}$ implies that the potentials on either side of a surface can differ by at most a constant. We choose the constant to be zero so that that boundary condition is replaced by continuity of the potential. The dielectric case is identical to the magnetic case with the replacements

$$\mu \rightarrow \epsilon$$

$$\vec{H} \rightarrow \vec{E}$$

$$\vec{B} \rightarrow \vec{D}$$ (3)

For the two-dimensional case, where the fields go to a constant along $\hat{x}$ far from the shield, only solutions with this cos($\phi$) dependence will be produced. That is the solution in region i bounded by two circular cross section surfaces will have the form

$$\Phi_{i}^{} \rho \phi \left ( A_i \rho + \frac{C_i}{\rho} \right)\cos \phi$$ (4)

The boundary conditions between $\Phi_{i}^{}$ and $\Phi_{i+1}^{}$ at $\rho$ = a_i are then

$${\frac{C_i}{a_i}} {\frac{C_{i+1}}{a_i}}$$ =

$$\mu_{i}^{} \left ( A_i -\frac{C_i}{a_i^2} \right) \mu_{i+1}^{} \left ( A_{i+1} - \frac{C_{i+1}}{a_i^2} \right)$$ = (5)

with the solution  

$$\left ( \begin{array} {c} A_{i+1} \\ C_{i+1}\end{array} \right ... ...ght ) \left ( \begin{array} {c} A_{i} \\ C_{i}\end{array} \right )$$ (6)

We can now use this transfer matrix to solve for a multilayered sheath. At each boundary we apply Eq. 6 to get the coefficients of the potential in the next region. For the simple case of a sheath of permeability $\mu$ from $\rho$ = a to $\rho$ = b , we have the additional boundary conditions, C₁ = 0 and A₃ = - B₀ , and two boundaries. Identifying A₁ = - B_in , we can write

$$\left ( \begin{array} {c} -B_0 \\ C_3 \end{array} \right ) = \... ...ight ) \left ( \begin{array} {c} -B_{in} \\ 0\end{array} \right )$$ (7)

which gives immediately the result

$${\frac{(\mu+1)^2b^2-(\mu-1)^2 a^2}{4 \mu b^2}}$$ (8)

or

$${\frac{4 \mu b^2}{(\mu+1)^2b^2-(\mu-1)^2 a^2}}$$ (9)

The coeffients of the potential inside the sheath are

$${\frac{\mu+1}{2\mu}} {\frac{2 (\mu+1)b^2}{(\mu+1)^2b^2-(\mu-1)^2 a^2}}$$

$${\frac{a^2(\mu-1)}{2\mu}} {\frac{2 a^2 b^2 (\mu-1)}{(\mu+1)^2b^2-(\mu-1)^2 a^2}}$$ (10)

and outside

$${\frac{(b^2-a^2)(\mu^2-1)}{4\mu}} {\frac{b^2 (b^2-a^2)(\mu^2-1)}{(\mu+1)^2b^2-(\mu-1)^2 a^2}}$$ (11)

Jackson's example in three dimensions can also be written in terms of a transfer matrix. The potential within each spherical shell is

$$\Phi_{i}^{} \theta \left ( A_i r + \frac{C_i}{r^2} \right)\cos \theta$$ (12)

The boundary conditions are

$${\frac{C_i}{a_i^2}} {\frac{C_{i+1}}{a_i^2}}$$ =

$$\mu_{i}^{} \left ( A_i -\frac{2 C_i}{a_i^3} \right) \mu_{i+1}^{} \left ( A_{i+1} - \frac{2 C_{i+1}}{a_i^3} \right)$$ = (13)

Solving for the transfer matrix as in the two-dimensional case gives  

$$\left ( \begin{array} {c} A_{i+1} \\ C_{i+1}\end{array} \right ... ...ght ) \left ( \begin{array} {c} A_{i} \\ C_{i}\end{array} \right )$$ (14)

For the single shell from a to b ,

$$\left ( \begin{array} {c} -B_0 \\ C_3 \end{array} \right ) = \... ...ight ) \left ( \begin{array} {c} -B_{in} \\ 0\end{array} \right )$$ (15)

with the immediate result

$${\frac{(\mu+2)(2\mu+1)b^3-2(\mu-1)^2a^3}{9 \mu b^3}}$$ (16)

as given in Jackson.

These same techniques can be applied to higher l values too. For the spherical case, the equations are

$$\Phi_{i}^{} \theta \left ( A_i r^l + \frac{C_i}{r^{l+1}} \right)P \theta$$ (17)

The boundary conditions are

$${\frac{C_i}{a_i^{l+1}}} {\frac{C_{i+1}}{a_i^{l+1}}}$$ =

$$\mu_{i}^{} \left ( l A_i a_i^{l-1} -\frac{2 (l+1) C_i}{a_i^{l+2}} \right) \mu_{i+1}^{} \left ( A_{i+1} a_i^{l+1} - \frac{2 (l+1) C_{i+1}}{a_i^{l+2}} \right)$$ = (18)

or

$$\left ( \begin{array} {c} A_{i+1} \\ C_{i+1}\end{array} \right ... ...ght ) \left ( \begin{array} {c} A_{i} \\ C_{i}\end{array} \right )$$ (19)

For the single shell from a to b ,

$$\left ( \begin{array} {c} -B_0 \\ C_3 \end{array} \right ) = \... ...ight ) \left ( \begin{array} {c} -B_{in} \\ 0\end{array} \right )$$ (20)

with the result

$${\frac{(l+1+l\mu)(l+(l+1)\mu)b^{2l+1}-l(l+1)(\mu-1)^2a^{2l+1}}{(2l+1)^2 \mu b^{2l+1}}}$$ (21)

For $\mu$ $\rightarrow$ $\infty$ ,

$$\rightarrow \left ( 4+\frac{1}{l(l+1)} \right)\frac \mu {\frac{a^{2l+1}}{b^{2l+1}}}$$ (22)