I Estimating Particle Number in an Electric Field

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In an electric field how the ion numbers are calculated?
In an electric field ##E## if the energy of charged particle (ion) is formulated as ##W=qEr## where ##r## is the distance, how the number of particles is estimated?
 
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There is no relationship between 'the number of particles' and anything else you write. Or is there ?

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Thread 'Gauss' law seems to imply instantaneous electric field'

Thread 'Gauss' law seems to imply instantaneous electric field'

Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
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Thread 'Do electron density waves accompany EM waves in coaxial cables?'

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...
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