# Extended Field Interactions in [PERSON]'s Equation Revision
## 3 Potential Applications
### A Modification of the Spintronic Current Density  
Spintronics is an emerging field of technology that focuses on the manipulation and control of electrons' spin in solid-state devices [27]. The spintronic current density can be optimized to increase performance while using less energy and to better integrate with cutting-edge technology like quantum computers and neural networks. The total current density is derived from Equation (1) and the general constitutive relation \(\mathbf{J}=\sigma_{c}\mathbf{E}\):  
\[\mathbf{J}=\sigma_{c}\mathbf{E}_{0}-\sigma_{c}\frac{\partial\mathbf{A}}{ \partial t}-\sigma_{c}\,\boldsymbol{\widehat{\omega}}\,\Phi. \tag{13}\]  
In the context of spintronics, Equation (13) suggests that the current density \(\mathbf{J}\) can be decomposed into three terms: the first term \(\sigma_{c}\mathbf{E}_{0}\) is the ordinary Ohmic current, the second term \(-\sigma_{c}\frac{\partial\mathbf{A}}{\partial t}\) represents the contribution from the electromagnetic induction due to time-varying magnetic fields, and the third term \(-\sigma_{c}\boldsymbol{\widehat{\omega}}\Phi\) represents the contribution from the spin accumulation induced by the spin-orbit interaction.  
One possible manipulation of Equation (13) is to take its curl, which can give us insight into the behavior of the spintronic current density under certain conditions. By calculating the curl of both sides of Equation (13), the following is obtained:  
\[\
abla\times\mathbf{J}=\sigma_{c}\
abla\times\mathbf{E}_{0}-\sigma_{c}\frac{ \partial}{\partial t}(\
abla\times\mathbf{A})-\sigma_{c}\
abla\times( \boldsymbol{\widehat{\omega}}\,\Phi). \tag{14}\]  
By applying the Maxwell-Faraday equation \(\
abla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\), the second term on the right-hand side can be simplified:  
\[\
abla\times\mathbf{J}=\sigma_{c}\
abla\times\mathbf{E}_{0}+\sigma_{c}\frac{ \partial\mathbf{B}}{\partial t}-\sigma_{c}\
abla\times(\boldsymbol{\widehat{ \omega}}\Phi). \tag{15}\]  
If it is assumed that the magnetic field \(\mathbf{B}\) is negligible in comparison with the electric field \(\mathbf{E}_{0}\), the second term can be omitted for further simplification:  
\[\
abla\times\mathbf{J}=\sigma_{c}\
abla\times\mathbf{E}_{0}-\sigma_{c}\
abla \times(\boldsymbol{\widehat{\omega}}\Phi). \tag{16}\]  
This formula demonstrates how the axial vector term involving spin \(\boldsymbol{\widehat{\omega}}\) and the curl of the electric field relate to the curl of the spintronic current density. This connection may be leveraged to comprehend the dynamics of spin currents in intricate geometries and to develop new spintronic devices.  
Examining the behavior of the axial vector term \(\boldsymbol{\widehat{\omega}}\Phi\) in Equation (13) when other spin textures or geometries are present is one method that could be used. Consider, for instance, the situation of a magnetic domain wall in which the spin direction fluctuates from side to side. This may result in the spin current density having a nonzero value, which could have potential spintronics applications. This leads to a deeper understanding of spintronic current density and its applications [28; 29; 30; 31; 32].