Вектор Умова Пойнтинга в функциях потенциальных полей

Last modified by Alexey Popov on 2019/02/21 15:43

$\mathbf J = \mathbf E_q \times \mathbf B_q$

$\mathbf E_q = - \nabla \varphi_q-\frac{\partial \mathbf A_q}{\partial t} + 2(\mathbf A_q\times \mathbf B_q)$

$\mathbf B_q = \nabla \times \mathbf A_q$

$\mathbf J = - \nabla \varphi_q \times (\nabla \times \mathbf A_q)-\frac{\partial \mathbf A_q\times (\nabla \times \mathbf A_q)}{\partial t}+\mathbf A_q\times \frac{\partial \nabla \times \mathbf A_q}{\partial t} - 2\mathbf B \times (\mathbf A_q\times \mathbf B_q)=$

$=-\nabla\times(\varphi_q(\nabla \times \mathbf A_q))+\varphi_q(\nabla \times (\nabla \times \mathbf A_q))-\frac{\partial \mathbf A_q\times \mathbf B_q}{\partial t}-\mathbf A_q\times (\nabla \times \mathbf E_q)-2\mathbf B \times (\mathbf A_q\times \mathbf B_q)=$

$\nabla \cdot \mathbf A_q = -\frac{1}{c_q^2}\frac{\partial \varphi_q}{\partial t}$

$\mathbf J =-\nabla\times(\varphi_q \mathbf B_q)-\varphi_q\nabla^2\mathbf A_q+\varphi_q\frac{1}{c_q^2}\frac{\partial \mathbf E_q}{\partial t} + \frac{1}{c_q^2}\varphi_q\frac{\partial^2 \mathbf A_q}{\partial t^2}-\frac{\partial \mathbf A_q\times \mathbf B_q}{\partial t}-\mathbf A_q\times (\nabla \times \mathbf E_q)-2\mathbf B \times (\mathbf A_q\times \mathbf B_q)$

$\mathbf S = - \nabla \mathbf E^2 - c^2\nabla \mathbf B^2 - \frac{\partial \mathbf J}{\partial t}- 2(\mathbf J \times \mathbf B)$

$\mathbf W = \nabla \times \mathbf J$

$\nabla \times \mathbf S = - \frac{\partial \mathbf W}{\partial t}$

$\nabla \cdot \mathbf J = \frac{\partial \mathbf E^2}{\partial t} + c^2\frac{\partial \mathbf B^2}{\partial t} + \mathbf E \cdot \square \mathbf A$

$\nabla \times\mathbf W = \nabla \times\nabla \times \mathbf J = \nabla(\nabla \cdot \mathbf J) - \nabla^2 \mathbf J$

$\mathbf a = -\nabla \mathbf v^2 - \frac{\partial \mathbf v}{\partial t} - 2\omega \times \mathbf v-\frac{\partial \nabla( \mathbf v \cdot \mathbf r)}{\partial t}-\frac{\partial \nabla \times( \mathbf v \times \mathbf r)}{\partial t}-$

Дипольный момент

$-\Delta \mathbf v^2\mathbf r+\frac{1}{c^2}\frac{\partial^2 \mathbf v^2\mathbf r}{\partial t^2}-$

$-\Delta (\mathbf v\times(\mathbf v \times \mathbf r)) + \frac{1}{c^2}\frac{\partial^2 (\mathbf v\times(\mathbf v \times \mathbf r))}{\partial t^2}-$

Квадрупольный момент

$-\Delta\nabla \mathbf v^2\mathbf r^2 + \frac{1}{c^2}\frac{\partial^2 \nabla \mathbf v^2 \mathbf r^2}{\partial t^2}-$

$-\Delta\nabla ( \mathbf v \times \mathbf r)^2 + \frac{1}{c^2}\frac{\partial^2 \nabla (\mathbf v \times \mathbf r)^2}{\partial t^2} -$

Октупольный момент

$-\Delta^2\mathbf v^2 \mathbf r^3+ \frac{2}{c^2}\frac{\partial^2 \Delta\mathbf v^2 \mathbf r^3}{\partial t^2} - \frac{1}{c^4}\frac{\partial^4 \mathbf v^2 \mathbf r^3}{\partial t^4}-$

$-\Delta^2(\mathbf v \times \mathbf r)^2 \mathbf r+ \frac{2}{c^2}\frac{\partial^2 \Delta(\mathbf v \times \mathbf r)^2 \mathbf r}{\partial t^2}-\frac{1}{c^4}\frac{\partial^4 (\mathbf v \times \mathbf r)^2 \mathbf r}{\partial t^4}$

$\nabla \times (A \times (\nabla \times A))=-\frac{\partial \nabla \times A}{\partial t}$

$-\Delta A + \frac{1}{c^2}\frac{\partial^2 A}{\partial t^2} = \frac{1}{c^2}\frac{A\times (\nabla \times A)}{\partial t}$

Уравнение непрерывности

$\nabla \cdot \mathbf v = - \frac{1}{c^2}\mathbf a \cdot \mathbf v - \frac{1}{2}\frac{1}{c^2}\frac{\partial \mathbf v\cdot \mathbf v}{\partial t} - \frac{1}{2}\frac{1}{c^2}\frac{\partial^2 \mathbf v \cdot \mathbf r}{\partial t^2}$

Дипольный момент

$2\left(\mathbf {r} \cdot \nabla \right)\mathbf {v} \ =\ \mathbf {r} \left(\nabla \cdot \mathbf {v} \right)\,-\,\mathbf {v} \left(\nabla \cdot \mathbf {r} \right)\,+\,\nabla \left(\mathbf {r} \cdot \mathbf {v} \right) -\mathbf {r} \times \left(\nabla \times \mathbf {v} \right) - \mathbf {v} \times \left(\nabla \times \mathbf {r} \right) \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {v} \right)$

$2\left(\mathbf {r} \cdot \nabla \right)\mathbf {v} \ =\ \mathbf {r} \left(\nabla \cdot \mathbf {v} \right)\,-\,3 \mathbf {v} \,+\,\nabla \left(\mathbf {r} \cdot \mathbf {v} \right) -\mathbf {r} \times \boldsymbol \omega \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {v} \right)$

$\frac{d \mathbf r}{dt} = \frac{\partial \mathbf r}{\partial t} + \ \frac{1}{2}\mathbf {r} \left(\nabla \cdot \mathbf {v} \right)\,-\,\frac{1}{2} 3 \mathbf {v} \,+\,\frac{1}{2}\nabla \left(\mathbf {r} \cdot \mathbf {v} \right) -\frac{1}{2}\mathbf {r} \times \boldsymbol \omega \,-\,\frac{1}{2}\nabla \times \left(\mathbf {r} \times \mathbf {v} \right)$

$\frac{d \mathbf r}{dt} + \frac{3}{2} \mathbf {v}= \frac{\partial \mathbf r}{\partial t} + \ \frac{1}{2}\mathbf {r} \left(\nabla \cdot \mathbf {v} \right)\, \,+\,\frac{1}{2}\nabla \left(\mathbf {r} \cdot \mathbf {v} \right) -\frac{1}{2}\mathbf {r} \times \boldsymbol \omega \,-\,\frac{1}{2}\nabla \times \left(\mathbf {r} \times \mathbf {v} \right)$

$\nabla \cdot \frac{d \mathbf r}{dt} + \frac{3}{2}\nabla \cdot \mathbf {v} =\ \frac{1}{2}\nabla \cdot (\mathbf {r} \left(\nabla \cdot \mathbf {v} \right))\, \,+\,\frac{1}{2}\nabla \cdot \nabla \left(\mathbf {r} \cdot \mathbf {v} \right) - \frac{1}{2}\nabla \cdot(\mathbf {r} \times \boldsymbol \omega) \,$

$\nabla ^{2}(\mathbf {r} \cdot \mathbf {v} )=\mathbf {r} \cdot \nabla ^{2}\mathbf {v} -\mathbf {r} \cdot \nabla ^{2}\mathbf {r} +2\nabla \cdot ((\mathbf {v} \cdot \nabla )\mathbf {r} +\mathbf {v} \times (\nabla \times \mathbf {r} ))$

$\nabla ^{2}(\mathbf {r} \cdot \mathbf {v} )=\mathbf {r} \cdot \nabla ^{2}\mathbf {v} +2\nabla \cdot ((\mathbf {v} \cdot \nabla )\mathbf {r}$

$\nabla \cdot \frac{d \mathbf r}{dt} =\ \frac{1}{2} \mathbf {r} \cdot \nabla^2 \mathbf {v}\, \,+\,\frac{1}{2}\nabla \cdot \nabla \left(\mathbf {r} \cdot \mathbf {v} \right) - \frac{1}{2}\nabla \cdot(\mathbf {r} \times \boldsymbol \omega) \,$

$2\left(\mathbf {E} \cdot \nabla \right)\mathbf {A} \ =\ \mathbf {E} \left(\nabla \cdot \mathbf {A} \right)\,-\,\mathbf {A} \left(\nabla \cdot \mathbf {E} \right)\,+\,\nabla \left(\mathbf {E} \cdot \mathbf {A} \right) -\mathbf {E} \times \mathbf {B} - \mathbf {A} \times \left(\nabla \times \mathbf {E} \right) \,-\,\nabla \times \left(\mathbf {E} \times \mathbf {A} \right)$

$2\left(\mathbf {B} \cdot \nabla \right)\mathbf {A} \ =\ \mathbf {B} \left(\nabla \cdot \mathbf {A} \right)\,+\,\nabla \left(\mathbf {B} \cdot \mathbf {A} \right) - \mathbf {A} \times \left(\nabla \times \mathbf {B} \right) \,-\,\nabla \times \left(\mathbf {B} \times \mathbf {A} \right)$

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

$\boldsymbol \omega = \nabla \times \mathbf v + \frac{1}{c^2}\mathbf v \times \mathbf a$

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Created by Alexey Popov on 2019/02/06 12:12