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Review of Electromagnetic Theory

- Maxwell’s Equations:
- Constitutive Relations:

is magnetic conductive current density (in volts/square meter)

- Boundary Conditions:
- Constitutive parameters are σ, ε,μ.
- Linear Medium: σ, ε,μ are independent of E and H.
- Homogeneous Medium: σ, ε,μ are not functions of space variables or.
- IsotropicMedium: σ, ε,μ are independent of direction (scalars).

is magnetic resistivity

Review of Electromagnetic Theory

- When a medium is source-free: J = 0, ρv = 0
- In practice, only two of Maxwell’s equations are used:
- Since other two are implied.
- Also, in practice, it is sufficient to make tangential components of fields satisfy necessary boundary conditions.
- Since normal components implicitly satisfy their corresponding boundary conditions.

- Wave Equations:
- Altogether there are six scalar equations for Ex, Ey, Ez, Hx, Hy, Hz the form of:
- Time-varying Potentials:

Review of Electromagnetic Theory

- Time-Harmonic Fields:
- In sinusoidal steady state:
- Source-free wave equation in phasor representation:
- General wave equation in phasor representation:
- Special Case 1: Poisson’s equation for static case (ω = 0):
- Special Case 2: Laplace’s equation for static case and source-free:

- Classification of EM Problems:
- ThisClassification help to answer the question of “What method is best for solving a problem”.
- Three independent items define a problem uniquely:
- (1)the solution region (problem domain) of the problemas R:
- (2) the nature of the equation describing the problem,
- (3) the associated boundary conditions as S.
- Classification of Solution Regions:
- There are two classifications:
- Solution region R is interior (inner, closed, or bounded)
- Solution region R is exterior (outer, open, or unbounded)
- If part or all of S is at infinity, R is exterior otherwise R is interior.
- For example, wave propagation in a waveguide is an interior problem.
- For example, wave propagation in free space (scattering of EM waves by raindrops, and radiation from a dipole antenna) are exterior problems.
- Solution region R could be linear,homogeneousandisotropic.

Ris the solution region

Sis the boundary condition

- Classification of Differential Equations:
- EM problems are classified in terms of equations describing them.
- Equations could be differential or integral or both defined as:
- For example:
- Another example:
- A second-order partial differential equation (PDE):
- or simply:
- PDE operator:

- In non-linear PDEs, coefficients are function of quantity

- Any linear second-order PDE can be classified as elliptic, hyperbolic, or parabolic:
- An elliptic PDE usually models an interior problem such as:
- A Hyperbolic PDE usually models an exterior problem as:
- A ParabolicPDE usually models an exterior problem such as diffusion (or heat) equation:

Laplace’s equation:

Poisson’s equations:

Elliptic problem

parabolic, or hyperbolic problem

- Nondeterministic Problems:
- Previous problems are deterministic, since quantity of interest can be determined directly.
- Another type of problem where quantity is found indirectly is called nondeterministicor eigenvalue.
- StandardEigen problem is of the form of:
- A more general version is generalized Eigen-problem having the form of:
- Only some particular values of λcalled eigenvaluesare permissible.
- Eigen-problems are usually encountered in vibration and waveguide problems.
- In these problems eigenvalues λcorrespond to physical quantities such as resonance and cutoff frequencies.

Where source term has been replaced by λ

Where M, like L, is a linear operator

- Classification of Boundary Conditions:
- Usually boundary conditions are of the Dirichlet and Neumann types.
- Dirichlet boundary condition:
- A good example is the charged metal plate.
- Because all points on a metal are at same potential, a metal plate can readily be modeled by a region of points with some fixed voltage.
- Neumannboundary condition:
- Mixed boundary condition:
- These conditions are called homogeneous boundary conditions.
- General ones are inhomogeneous:
- Dirichlet:
- Neumann:
- Mixed:

i.e., the normal derivative of vanishes on S

h(r) is a known function

- Some Important Theorems:
- Superposition Principle:
- If each member of a set of functions φn , n=1,2,…,Nis a solution to PDE:
- Then a linear combination of them also satisfies the PDE as:
- Uniqueness Theorem:
- This guarantees that solution a PDE with some prescribed boundary conditions is only one possible.
- If a set of fields (E,H) is found which satisfies simultaneously Maxwell’s equations and prescribed boundary conditions, this set is unique.
- Therefore, a field is uniquely specified by sources (ρv,J) within medium plus tangential components of E or H over boundary.
- To prove uniqueness theorem, suppose there exist two solutions:

- Uniqueness Theorem (cont.):
- We denote the difference of the two fields as:
- These must satisfy the source-free Maxwell\'s equations:
- Dotting both sides with ΔEgives:
- Integrating over volume:
- Therefore ΔE and ΔH satisfy the Poynting theorem just as E1,2and H1,2
- Only Etand Htcontribute to surface integral on the left side.
- Therefore, if E1tand E2tor H1tand H2t are equal over S, ΔEtand ΔHtvanish on S.
- Consequently, surface integral is identically zero, and hence right side must vanish also.
- It follows that ΔE=0 due to second integral on right side and hence also ΔH=0 throughout the volume.
- Thus E1=E2 and H1=H2, confirming that the solution is unique.

Using:

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