# Extended Field Interactions in [PERSON]'s Equation Revision
## 3 Potential Applications
### Some Insight on Yukawa's Potential  
The Yukawa potential, which is an important rotational-invariant scattering potential used in atomic and nuclear physics, can be derived from this formalism. By imposing the constraint \(\mathbf{\
abla}\cdot\widetilde{\omega}=0\), this potential can be derived from the equation, and it is also possible to derive the Debye-Huckel equation for electrolytes. When considering a solenoidal vortex field with \(\mathbf{\
abla}\cdot\widetilde{\omega}=0\), and in the stationary case with \(\mathbf{\
abla}\Phi\cdot\mathbf{\widetilde{\omega}}=0\), the well-known Yukawa equation for a particle of charge \(\rho=Ze\) can be derived:  
\[\
abla^{2}\Phi+\kappa^{2}\Phi=-\frac{Ze}{\epsilon_{0}}\delta(\mathbf{r}- \mathbf{r}^{\prime}). \tag{5}\]  
Here, \(\kappa=\frac{\mathbf{\widetilde{\omega}}}{c^{2}}\) and the solution is  
\[G_{\omega}(\mathbf{r},\mathbf{r}^{\prime})=\frac{Ze}{4\pi\epsilon_{0}}\frac{e ^{\kappa|\mathbf{r}-\mathbf{r}^{\prime}|}}{|\mathbf{r}-\mathbf{r}^{\prime}|}. \tag{6}\]  
In Equations (5) and (6), the term \(\mathbf{r}^{\prime}\) represents the position vector of the source of the charge distribution, which, in this case, is associated with a point charge \(Ze\). It specifies the location in space where the charge \(Ze\) is situated. On the other hand, \(\mathbf{r}\) refers to the observation point in space where the potential \(\Phi\) and the Green's function \(G_{\omega}\) are being evaluated.  
Assuming a radially-dependent electric potential, Equation (5) simplifies to  
\[\begin{split}&\partial_{rr}^{2}\Phi+\frac{2}{r}\partial_{r}\Phi+ \bigg{[}\frac{1}{r^{2}}\partial_{r}(r^{2}\widetilde{\omega}_{r})\\ &+\frac{1}{r\sin\phi}\partial_{\theta}\ \widetilde{\omega}_{\theta}\bigg{]}\Phi+\widetilde{\omega}_{r}\partial_{r} \Phi=-\frac{Ze}{\epsilon_{0}}.\end{split} \tag{7}\]  
Consider simplifying to basic radial dependencies, such as \(\widetilde{\omega}=\widetilde{\omega}(r)\). Then,  
\[\partial_{rr}^{2}\Phi+\bigg{(}\frac{2}{r}+\widetilde{\omega}_{r}\bigg{)} \partial_{r}\Phi+\bigg{[}\frac{1}{r^{2}}\partial_{r}(r^{2}\widetilde{\omega}_ {r})\bigg{]}\Phi=-\frac{Ze}{\epsilon_{0}}. \tag{8}\]  
For the particular case \(\widetilde{\omega}=1\), Figure 1 shows the analytical solution of Equation (8). Consider the fact that the potential might be either attractive or repulsive.  
Analytical solutions to Equation (8) are essential in understanding the behavior of physical systems. Equation (8) is a special case of the general equation and its analytical solution is represented in Figure 1. The potential can be either repulsive or attractive.  
It is very similar to the Lennard-Jones potential. The Lennard-Jones potential is a mathematical model used to describe the interactions between atoms or molecules. It is a simple, effective model to describe the weak intermolecular forces that occur in nature. The Lennard-Jones potential is a combination of two terms: a repulsive term and an attractive term. The repulsive term is a function of the distance between the atoms, and the attractive term is a function of the inverse of the distance between the atoms. However, while it is similar to the Lennard-Jones potential, it introduces a novel aspect by incorporating the radial dependence of \(\widetilde{\omega}(r)\). This inclusion differentiates it from the standard Lennard-Jones model, which typically focuses on the interplay of repulsive and attractive forces without a specific radial dependence component.  
The study of Yukawa potentials in dusty plasmas has significant implications for real-world applications. For instance, it offers critical insights into the behavior of plasma under magnetic influences, which is crucial for advancements in controlled nuclear fusion technology [22]. Additionally, it contributes to our understanding of plasma thermodynamics, potentially aiding in the development of more efficient energy systems [23] or shedding light on phase transitions in plasma, a key aspect that could drive innovations in materials science [24].