## 4 Coupling of Solutions in Regions \(R_{i}\) and \(R_{ii}\)  
Solutions (9) and (14) should be appropriately coupled by employing boundary conditions (6-8). First, taking account of the fact that \(\cos(k_{m}h/2)=0\), the following result for the MF on the upper surface of the tank cover can be obtained from Equation (14):  
\[H_{n,\varphi}^{(II)}\bigg{(}r,\frac{h}{2}\bigg{)}=\Big{(}\frac{A_{n}}{r}+B_{n} r\Big{)}\cosh\bigg{(}\frac{\beta_{n}h}{2}\bigg{)}. \tag{15}\]  
The substitution of (15) and (9) into boundary condition (8) yields:  
\[\Big{(}\frac{A_{n}}{r}+B_{n}r\Big{)}\cos h\bigg{(}\frac{\beta_{n}h}{2}\bigg{)}= \frac{I_{n}}{2\pi r}. \tag{16}\]Since the functions \(1/r\) and \(r\) are linearly independent, the following solution for the constants \(A_{n}\) and \(B_{n}\) can be obtained:  
\[A_{n}=\frac{I_{n}}{2\pi\cos\hbar\left(\frac{\theta_{n}n}{2}\right)},\quad B_{n}=0.\]  
Finally, the substitution of these results into Equation (14) yields the following solution to Equation (10) in region \(R_{II}\):  
\[ \begin{array}{ll}H_{n,\varphi}^{(II)}(r,z)&=\frac{I_{n}}{2\pi r}\frac{\cosh (\beta_{n}z)}{\cosh\left(\frac{\theta_{n}n}{2}\right)}\\ &+\frac{I_{n}}{2\pi}\sum\limits_{m=0}^{\infty}\frac{d\theta_{n}^{2}(-1)^{m}}{ k_{m}\lambda_{n,m}^{2}\lambda_{n,m}^{2}}\frac{\cos(k_{m}z)}{I_{1}(\lambda_{n,m},a)K_{1}(\lambda_{n,m},b)-I_{1}(\lambda_{n,m},b)K_{1}(\lambda_{n,m},a)}\\ &\times\{\frac{1}{a}[I_{1}(\lambda_{n,m},r)K_{1}(\lambda_{n,m},b)-I_{1}( \lambda_{n,m},b)K_{1}(\lambda_{n,m},r)]\\ &+\frac{1}{b}[I_{1}(\lambda_{n,m},a)K_{1}(\lambda_{n,m},r)-I_{1}(\lambda_{n,m },r)K_{1}(\lambda_{n,m},a)]\}.\end{array}  \tag{17}\]  
Both Equations (7) and (17) represent the solution to [PERSON]'s equations for the nth harmonic of the MF in the entire domain. The solution depends on the tank geometry (through the hole and disc radii a and b, the wall thickness h), amplitude of the nth harmonic of the electric current, and angular frequency (through the parameters \(\lambda_{n,m}\) and \(\beta_{n}\)). It can be noticed that the dependence of the solution on the disc radius \(b\) is weak enough to be neglected. Indeed, according to Equation (11), \(\beta_{n}=(1+j)/\delta_{n}\), where \(\delta_{n}=\sqrt{2/n\omega\mu\sigma}\) is the skin-effect depth for the nth harmonic of the MF. Therefore, the following estimation can be obtained:  
\[\left|\lambda_{n,m}\right|^{2}=\left|\left(\frac{\pi(2m+1)}{h}\right)^{2}+ \frac{2j}{\delta_{n}^{2}}\right|\geq\frac{2}{\delta_{n}^{2}}. \tag{18}\]  
Therefore, \(\left|\lambda_{n,m}\right|\geq\sqrt{2}/\delta_{n}\), which, due to the smallness of parameter \(\delta_{n}\), results in the estimation: \(\left|\lambda_{n,m}b\right|\geq\sqrt{2}b/\delta_{n}\gg 1\). This estimation leads to the following asymptotic behavior of the modified Bessel functions [25]:  
\[K_{1}(\lambda_{n,m}b)\propto\sqrt{\frac{\pi}{2\lambda_{n,m}b}}e^{-\lambda_{n, m}b}=\mathcal{O}\left(\sqrt{\frac{\delta_{n}}{b}}\;e^{-\frac{b}{\delta_{n}}} \right)\to 0\;\;\text{as}\;\frac{b}{\delta_{n}}\rightarrow\infty, \tag{19}\]  
where \(\mathcal{O}(x)\) is the Landau big O notation [26]. Thus, the terms with \(K_{1}(\lambda_{n,m}b)\) and \(1/b\) in solution (17) can be neglected.