Warpper → Новые уравнения Максвелла

$$\mathbf A = \mathbf E_q \times\mathbf B_q$$

$$\frac{\rho}{\varepsilon_0} = \square \varphi= \frac{1}{2}(\Delta \mathbf E_q^2 - \frac{1}{c^2}\frac{\partial^2 \mathbf E_q^2}{\partial t^2} +\Delta \mathbf B_q^2 - \frac{1}{c^2}\frac{\partial^2 \mathbf B_q^2}{\partial t^2})$$

$$\mu_0 \mathbf J = \square \mathbf A = \Delta (\mathbf E_q \times\mathbf B_q) - \frac{1}{c^2}\frac{\partial^2 (\mathbf E_q \times\mathbf B_q)}{\partial t^2}$$

$$\nabla \cdot \mathbf E = \frac{\rho}{\varepsilon_0} + (\nabla \times\mathbf E_q)^2 + (\nabla \times\mathbf B_q)^2$$

$$\nabla \cdot \mathbf B = 0$$

$$\nabla\times \mathbf E = - \frac{\partial \mathbf B}{\partial t}$$

$$\nabla\times \mathbf B = \mu_0 \mathbf J + (\nabla \times\mathbf E_q)\times(\nabla \times\mathbf B_q)+ \frac{1}{c^2}\frac{\partial \mathbf E}{\partial t}$$

$$ \nabla\cdot (\frac{\rho}{\varepsilon_0} \mathbf E) + \mu_0 \nabla \cdot (\mathbf J\times \mathbf B) = $$

$$ = \frac{1}{2}\Delta \mathbf E^2 + \frac{1}{2}\Delta \mathbf B^2 + \mu_0\nabla \cdot \frac{\partial (\mathbf E\times \mathbf B)}{\partial t} + \mu_0(\nabla \times \mathbf B)^2+\frac{1}{c^2}(\nabla \times \mathbf E)^2 $$

$$\mathbf r \times (\nabla \times \mathbf E) = - \frac{\partial (\mathbf r \times \mathbf B)}{\partial t}$$

$$\mathbf r \times (\nabla \times \mathbf B) = \mathbf r \times \mathbf J + \frac{\partial (\mathbf r \times \mathbf E)}{\partial t}$$

$$ \rho_g = \mu_0 \nabla \cdot \nabla (\mathbf r \times \mathbf J)^2 + ... $$

$$ \rho_q =\varepsilon_0\nabla \cdot \nabla ( \mathbf r \times \mathbf A)^2 + ...$$

$$ \rho =\frac{1}{c^2}{ \left ( \frac{1}{2}\varepsilon_0\nabla \cdot \nabla \left ( \mathbf r \times \mathbf E \right )^2+\frac{1}{2}\frac{1}{\mu_0}\nabla \cdot \nabla \left ( \mathbf r \times \mathbf B \right )^2+ \varepsilon_0\nabla \cdot \frac{\partial (\left (\mathbf r \times\mathbf E\right )\times (\mathbf r \times\mathbf B))}{\partial t}-\varepsilon_0\left ( \nabla \times \left ( \mathbf r \times \mathbf E \right )\right)^2-\frac{1}{\mu_0}\left ( \nabla \times \left ( \mathbf r \times \mathbf B \right )\right)^2 \right)} $$