# Boundary conditions of the Electric field of a conducting transmission line

Solving Maxwell's Equations for the Electric Field of a linear-isotropic cylindrical conductor leads to the electric field being proportional to the Bessel function of the first kind (see below).

Maxwell's Equations:

$Div[E] = \frac{\rho}{\epsilon_{ps}}$-> 0 (reasonable approximation of no free charge in conductor)

$Div[B] = 0$

$Curl[E] = -\frac{d}{dt}[B]$ ... (i)

$Curl[B] = \mu J + \mu\epsilon_{ps}\frac{d}{dt}[E]$ ... (ii) Take Curl of (i):

$Curl[Curl[E]] = Grad[Div[E]] - Laplacian[E] = -\frac{d}{dt}[Curl[B]]$

Since Div[E] ~ 0,

$Laplacian[E] = \frac{d}{dt}[Curl[B]] = \frac{d}{dt}[\mu J + \mu espilon_{ps} E]$ The assumption of a linear, isotropic material prompts us to treat eps, mu, and rho as frequency-dependent scalars (rather than tensors). Also, we make the assumption of J = rho*E, a rough approximation of Ohm's law that follows from the kinetic theory of charges in the context of the Drude model.

So: $Laplacian[E] = d/dt[\mu\rho E + \mu\epsilon\frac{d}{dt}[E]]$ which is a basic wave-equation with a dampening terms via $\mu\rho \frac{d}{dt}[E]$....

Representing Laplacian[E] in cylindrical coordinates to suit the geometry of the line, and solving the PDE by separation of variables (E[r,t] = R[r]*T[t]*Z[z]) leads to:

$R[r] = BesselJ[0,\lambda r]$ (which is the r-spatial component of E).

In math/physics, it is here that we would typically quantize lambda using a boundary condition on the E-field at r = a (a being the radius of the line), and then employ some initial conditions to derive the required Bessel-Fourier series representation of the solution. Does this act of assigning boundary condition(s) apply to such wire systems, and if so, what form of boundary condition(s) on E[a] should we use? >> Are the standard boundary conditions of electromagnetism sufficient?

In mathematics, a boundary condition is typically specified and a Bessel-fourier series is then used to determine the nature of the field in the line. In electrical engineering, what type of boundary conditions are applied to a conductor carrying a current in the direction of the E-field?

If you assume the conductive elements are perfect ($\rho=0$), then the boundary condition is that the E field tangent to the surface goes to 0. This is often called "perfect conductor boundary conditions".
If you want to model a real conductive material ($\rho > 0$), then you will have to model the fields and currents inside the conductive region also. The boundary condition will be that the tangential component of $\vec{E}$ is continuous across the boundary.