# Magnetic and Electric Field Lines between Conductors

let's consider a system made with 3 conductors: two parallel wires (in this case rectangular wires) and a plane (GND) below them. Consider the following situation in which both wires are connected to the same potential (with respect to GND). This situation is called "even mode" and results in this field configuration:

How can I deduce those field lines of E and H? Do you know a criterion to find their distribution in space?

I think that E goes between points with different potentials. Since the two wires have the same potentials, lines start from each wires and go towards GND but not towards the other wire. Is it correct?

For the magnetic field, I would say it is equal to 0 at the centre of the space between the two wires, why is it not 0 in the picture?

• Are you really interested in how all this can be calculated? – Alex Trounev Dec 6 '19 at 11:37

How can I deduce those field lines of E and H? Do you know a criterion to find their distribution in space?

You can find the E field by solving the Poisson equation for the potential, and then taking the gradient to get the electric field. You will usually need to solve this numerically, rather than expect there to be a closed form solution.

For there to be a magnetic field, it's not enough to just know the potential of the conductors, you also need to know the current flowing in them. If you know the currents then you can solve the Biot-Savart equation, to find the magnetic field.

Since the two wires have the same potentials, lines start from each wires and go towards GND but not towards the other wire. Is it correct?

Yes.

For the magnetic field, I would say it is equal to 0 at the centre of the space between the two wires, why is it not 0 in the picture?

The picture doesn't show any field lines in the space between the wires. Why do you say it's inconsistent with zero field there?

• Generally speaking, in order to find the electric field in a system of conductors with given potentials, it is necessary to solve the Laplace equation, not Poisson's. – Alex Trounev Dec 6 '19 at 11:43

In the case of two conductors above the conducting plane, the Laplace equation for the electric field potential should be used $$\nabla ^2\phi=0$$ with boundary conditions $$\phi =U_0$$ on conductors and $$\phi=0$$ on a grounded surface (plane). Figure 1 shows the distribution of potential (left) and electric field for $$U_0 = 1$$. The solution to the equation is obtained using FEM.

In the case of two conductors with a current of the same direction, we use the Poisson equation for the vector potential $$\nabla \times(1/\mu \nabla \times \vec {A})=\vec j$$

In this problem $$\vec {j}=(0,0,j_z)$$ and so $$\vec {A}=(0,0,A_z)$$. Figure 2 shows the distribution of the vector potential (left) and the magnetic field for the case $$\mu = \mu_0$$