Question 4.20: Using the results of Example 4-19, determine the unknown pot...
Using the results of Example 4-19, determine the unknown potential V_{4}. The values of the other three potentials are known.
From Example 4-19 we obtained the S-matrix of the first triangle
[S^{(1)}] =\frac{\varepsilon }{24} \left[\begin{matrix}25 & -9& -16 \\ -9 & 9& 0 \\ -16 & 0 & 16 \end{matrix} \right]Following the same procedure, we obtain the S-matrix of the second triangle
[S^{(2)}] =\frac{\varepsilon }{24} \left[\begin{matrix}25 & -16& -9 \\ -16& 16& 0 \\ 9 & 0 & 9 \end{matrix} \right]To directly apply (4.93) for this particular case, we have to change the indices as follows: (1, 2, 3, 4, 5, 6) → (3, 2, 1, 3, 4, 2). This produces the following coupled S-matrix of the rectangle-a system of two triangles ( 1, 2, 3) and (2, 3, 4 ):
[S] = \left[\begin{matrix}S_{1,1}^{(1)}& S_{1,2}^{(1)} & S_{1,3}^{(1)}&0 \\ S_{2,1}^{(1)} & S_{2,2}^{(1)}+ S_{5,5}^{(2)} & S_{2,3}^{(1)}+ S_{5,6}^{(2)}& S_{5,4}^{(2)} \\ S_{3,1}^{(1)} & S_{3,2}^{(1)}+S_{6,5}^{(2)} & S_{3,3}^{(1)} + S_{6,6}^{(2)}& S_{6,4}^{(2)}\\ 0 & S_{4,5}^{(2)} &S_{4,6}^{(2)}& S_{4,4}^{(2)} \end{matrix} \right] (4.93)
[S^{(e)}] =\left[\begin{matrix} [S]_{kk}& [S]_{ku} \\ [S]_{uk}& [S]_{uu} \end{matrix} \right] = \frac{\varepsilon }{24} \left[\begin{matrix}S_{33}^{(1)}+ S_{33}^{(2)} & S_{32}^{(1)}+ S_{32}^{(2)} & S_{31}^{(1)}& S_{34}^{(2)} \\ S_{23}^{(1)}+ S_{23}^{(2)} & S_{22}^{(1)}+ S_{22}^{(2)} & S_{21}^{(1)}& S_{24}^{(2)} \\ S_{13}^{(1)} & S_{12}^{(1)} & S_{11}^{(1)} &0\\ S_{43}^{(2)} & S_{42}^{(2)} &0& S_{44}^{(2)} \end{matrix} \right] \\= \frac{\varepsilon }{24} \left[\begin{matrix} 25& -9 & -16 &0 \\ -9 & 25 & 0& -16 \\ -16 & 0 & 25 & -9 \\ 0 & -16 & -9 & 25\end{matrix} \right]Applying equation (4.97), we obtain the unknown potential V_{4}
[V]_{u} = -[S]_{u,u}^{-1}[S]_{u,k}[V]_{k} (4.97)
V_{4} = -\frac{24}{25\varepsilon } \frac{\varepsilon }{24} [0 -16 -9]\left[\begin{matrix} 8 \\ 0 \\ 0 \end{matrix} \right] = 0 VThis computed potential is used to calculate the potential profile in this triangular element.