EEE 431 Computational Methods in Electrodynamics Lecture 2

EEE 431Computational Methods inElectrodynamicsLecture 2

Rasime Uyguroglu

Boundary Conditions

Maxwell’s Equations are partial differentialequations,

Boundary conditions are needed to obtain aunique solution,

Maxwell’s differential equations do not applyon the boundaries because the fields arediscontinuous

Our target is to determine the electric andmagnetic fields in a certain region of space dueto excitations satisfying the problem’sboundary conditions

Finite Conductivity Case

Finite Conductivity Case (Cont.)

Applying Faraday’s Law we get

As the RHS vanishes and weget;

Or

Finite Conductivity Case (Cont.)

It follows that the tangential component ofthe electric field is continuous.

Finite Conductivity Case (Cont.)

Similarly, starting with the modifiedAmpere’s law

(no current J at the interface), we get:

Or

Finite Conductivity Case (Cont.)

It follows that the tangential componentof the magnetic field intensity iscontinuous if there are no boundaryelectric currents.

Assuming there are no surface charges, Gauss’s law

gives

It follows that

Finite Conductivity Case (Cont.)

Or alternatively

But as,

Finite Conductivity Case (Cont.)

Normal components of the electric field arediscontinuous across the interface.

Finite Conductivity Case (Cont.)

Similarly, by applying Gauss’s law formagnetic fields we get

It follows that

Finite Conductivity Case (Cont.)

But as

Normal components of the magneticfields are discontinuous.

Finite Conductivity Case (cont.)

no interface surface magnetic currents

no interface surface electric currents

Finite Conductivity Case (cont.)

no interface surface electric charges, whenperfect dielectric materials are assumed (zeroconductivity):

no interface magnetic surface charges

Boundary Conditions withSources

Boundary conditions must be changed totake into account the existence ofsurface currents and surface charges.

Boundary Conditions withSources

Applying the modified Ampere’s law weget:

As

Boundary Conditions withSources

 is the surface current density in (A/m)

Boundary Conditions withSources

We see that:

Tangential components of the magnetic fieldintensity are discontinuous if surface electriccurrent density exists.

Boundary Conditions withSources (Cont.)

If medium 2 is a perfect conductor:

Similarly, starting with Faraday’s Law:

Boundary Conditions withSources (Cont.)

For a perfect conductor

Boundary Conditions withSources (Cont.)

Applying Gauss’s law for the showncylinder we have

Boundary Conditions withSources (Cont.)

Gauss’s Law:

Summary of the BoundaryConditions

We have,

Time Harmonic ElectromagneticFields

If sources are sinusoidal and themedium is linear then the fieldseverywhere are sinusoidal as well. Thefield at each point is characterized by itsamplitude and phase (Phasor).