Калибровка векторного потенциала

Last modified by Alexey Popov on 2019/03/07 11:32

$\mathbf E = - \frac{q_e}{m_e} (\mathbf A \cdot \nabla)\mathbf A-\frac{\partial \mathbf A}{\partial t}$

$\nabla \cdot \mathbf A = - \frac{1}{2}\frac{1}{c^2}\frac{q_e}{m_e}\frac{\partial (\mathbf A \cdot \mathbf A)}{\partial t}$

$\mathbf B = \nabla \times \mathbf A + \frac{1}{c^2}\frac{q_e}{m_e}(\mathbf A \times \mathbf E)$

$\nabla \cdot \mathbf E = - \frac{1}{2}\frac{q_e}{m_e} \nabla \cdot \nabla (\mathbf A \cdot \mathbf A) + \frac{q_e}{m_e} \nabla \cdot (\mathbf A \times (\nabla \times \mathbf A)) -\frac{\partial \nabla \cdot\mathbf A}{\partial t}$

$\nabla \cdot \mathbf E = - \frac{1}{2}\frac{q_e}{m_e} \nabla \cdot \nabla (\mathbf A \cdot \mathbf A) + \frac{1}{2}\frac{1}{c^2}\frac{q_e}{m_e}\frac{\partial^2 (\mathbf A \cdot\mathbf A)}{\partial t^2} + \frac{q_e}{m_e} \nabla \cdot (\mathbf A \times \mathbf B) - \frac{1}{c^2} \frac{q_e^2}{m_e^2} \nabla \cdot (\mathbf A \times (\mathbf A \times \mathbf E))$

$\mathbf {A} \times \left(\mathbf {A} \times \mathbf {E} \right)=\left(\mathbf {A} \cdot \mathbf {E} \right)\mathbf {A} -\left(\mathbf {A} \cdot \mathbf {A} \right)\mathbf {E}$

$\nabla \cdot ( \mathbf {A} \times \left(\mathbf {A} \times \mathbf {E} \right))=\nabla \cdot (\left(\mathbf {A} \cdot \mathbf {E} \right)\mathbf {A}) -\nabla \cdot (\left(\mathbf {A} \cdot \mathbf {A} \right)\mathbf {E}) =$

$\\= \ \left(\mathbf {A} \cdot \mathbf {E} \right) \,\nabla \cdot \mathbf {A} \,+\,\mathbf {A} \cdot \nabla \left(\mathbf {A} \cdot \mathbf {E} \right) - \ \left(\mathbf {A} \cdot \mathbf {A} \right) \,\nabla \cdot \mathbf {E} \,-\,\mathbf {E} \cdot \nabla \left(\mathbf {A} \cdot \mathbf {A} \right)$

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Created by Alexey Popov on 2019/03/07 10:15