Воздействие векторного поля М на плотность зарядов

Last modified by Alexey Popov on 2019/07/23 17:49

$\rho\mathbf M = \rho{\frac {D{\mathbf {B}}}{Dt}}= {\frac {D{\rho\mathbf {B}}}{Dt}} - {\frac {D{\rho}}{Dt}}\mathbf {B}$

${\frac {D{\rho\mathbf {B}}}{Dt}}={\frac {\partial {\rho\mathbf {B}}}{\partial t}}+\frac{1}{2} \left ( \nabla \left(\mathbf {B} \cdot \mathbf {J} \right)\,+\,\mathbf {u} \left(\nabla\rho \cdot \mathbf {B} \right) \,+\,\nabla \times \left(\mathbf {B} \times \mathbf {J} \right) - \mathbf {B} \times \left(\nabla \times \mathbf {J} \right) + \mathbf {B} \times \left(\nabla\rho \times \mathbf {u} \right)-\mathbf {J} \times \left(\nabla \times \mathbf {B} \right)-\mathbf {u} \times \left(\nabla\rho \times \mathbf {B} \right) \right)$

$\left(\nabla\rho \cdot \mathbf {B} \right) \mathbf {u} = (\nabla\rho \times \mathbf u) \times \mathbf B + (\mathbf u\cdot \mathbf B)\nabla\rho = (\nabla\rho \times \mathbf u) \times \mathbf B + \nabla(\mathbf J\cdot \mathbf B) - \rho\nabla(\mathbf u\cdot \mathbf B)$

${\frac {D{\rho\mathbf {B}}}{Dt}}={\frac {\partial {\rho\mathbf {B}}}{\partial t}} + \nabla \left(\mathbf {B} \cdot \mathbf {J} \right)+\frac{1}{2} \left ( - \rho\nabla(\mathbf u\cdot \mathbf B)\,+\,\nabla \times \left(\mathbf {B} \times \mathbf {J} \right) - \mathbf {B} \times \left(\nabla \times \mathbf {J} \right) -\mathbf {J} \times \left(\nabla \times \mathbf {B} \right)-\mathbf {u} \times \left(\nabla\rho \times \mathbf {B} \right) \right)$

$\mathbf {u} \times (\nabla\rho \times \mathbf {B} )=(\mathbf {u} \cdot \mathbf {B} )\nabla\rho -(\nabla \cdot \mathbf {J} )\mathbf {B}$

${\frac {D{\rho\mathbf {B}}}{Dt}}={\frac {\partial {\rho\mathbf {B}}}{\partial t}}+\frac{1}{2} \left ( \nabla(\mathbf J\cdot \mathbf B) \,+\,\nabla \times \left(\mathbf {B} \times \mathbf {J} \right) - \mathbf {B} \times \left(\nabla \times \mathbf {J} \right) -\mathbf {J} \times \left(\nabla \times \mathbf {B} \right) +(\nabla \cdot \mathbf {J} )\mathbf {B} \right)$

$\rho\mathbf M = \rho{\frac {D{\mathbf {B}}}{Dt}}={\rho\frac {\partial {\mathbf {B}}}{\partial t}} + \frac{1}{2} \left ( \nabla(\mathbf J\cdot \mathbf B) \,+\,\nabla \times \left(\mathbf {B} \times \mathbf {J} \right) - \mathbf {B} \times \left(\nabla \times \mathbf {J} \right) -\mathbf {J} \times \left(\nabla \times \mathbf {B} \right) -(\nabla \cdot \mathbf {J} )\mathbf {B} \right) ={\rho\frac {\partial {\mathbf {B}}}{\partial t}} + (\mathbf J \cdot \nabla)\mathbf B$

$\rho\mathbf M = {\rho\frac {\partial {\mathbf {B}}}{\partial t}} + (\mathbf J \cdot \nabla)\mathbf B$

$\rho\mathbf M = {\rho\frac {\partial {\mathbf {B}}}{\partial t}} + \rho (\mathbf v \cdot \nabla)\mathbf B$

$\rho\mathbf M = {\rho\frac {\partial {\mathbf {B}}}{\partial t}} + \rho \nabla (\mathbf v \cdot \mathbf B) - \mathbf J \times (\nabla \times \mathbf B)$

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Created by Alexey Popov on 2019/07/23 14:45