Bob44
- 15
- 1
Maxwell’s equations imply the following wave equation for the electric field
$$\nabla^2\mathbf{E}-\frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}
= \frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf J}{\partial t}.\tag{1}$$
I wonder if eqn.##(1)## can be split into the following transverse part
$$\nabla^2\mathbf{E}_T-\frac{1}{c^2}\frac{\partial^2\mathbf{E}_T}{\partial t^2}
= \mu_0\frac{\partial\mathbf{J}_T}{\partial t}\tag{2}$$
and longitudinal part
$$\frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf{J}_L}{\partial t}=0.\tag{3}$$
Taking the divergence of eqn.##(3)## and substituting in the continuity equation ##\nabla \cdot \mathbf{J}_L=-\partial\rho/\partial t## we obtain a wave equation
$$\nabla^2\rho-\frac{1}{c^2}\frac{\partial^2\rho}{\partial t^2}=0.\tag{4}$$
Do these equations describe how transverse EM waves ##(2)## travel down the dielectric in a coaxial cable accompanied by electron density waves ##(4)## in the conductors?
$$\nabla^2\mathbf{E}-\frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}
= \frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf J}{\partial t}.\tag{1}$$
I wonder if eqn.##(1)## can be split into the following transverse part
$$\nabla^2\mathbf{E}_T-\frac{1}{c^2}\frac{\partial^2\mathbf{E}_T}{\partial t^2}
= \mu_0\frac{\partial\mathbf{J}_T}{\partial t}\tag{2}$$
and longitudinal part
$$\frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf{J}_L}{\partial t}=0.\tag{3}$$
Taking the divergence of eqn.##(3)## and substituting in the continuity equation ##\nabla \cdot \mathbf{J}_L=-\partial\rho/\partial t## we obtain a wave equation
$$\nabla^2\rho-\frac{1}{c^2}\frac{\partial^2\rho}{\partial t^2}=0.\tag{4}$$
Do these equations describe how transverse EM waves ##(2)## travel down the dielectric in a coaxial cable accompanied by electron density waves ##(4)## in the conductors?