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Home , Charge density, Current density, Electric flux, Magnetic flux, Vacuum permeability, Vacuum permittivity

EE334 Electromagnetic Theory I Todd Kaiser

Maxwell’s Equations: ’s equations were developed on experimental evidence and have been found to govern all classical electromagnetic phenomena. They can be written in differential or integral form.

r r r 'sLaw ∇ ⋅ D = ρ D ⋅ dS = ρ dv = Q ∫∫ enclosed SV r r r Nomagneticmonopoles ∇ ⋅ B = 0 ∫ B ⋅ dS = 0 S r r ∂B r r ∂ r r Faraday'sLaw ∇× E = − E ⋅ dl = − B ⋅ dS ∫∫S ∂t C ∂t r r r ∂D r r r r ∂ r r Modified 'sLaw ∇× H = J + H ⋅ dl = J ⋅ dS + D ⋅ dS ∫ ∫∫SS ∂t C ∂t where: E = (V/m) D = Electric Density (C/m2) H = Intensity (A/m) B = Density (T) J = Electric (A/m2) ρ = Electric (C/m3)

The for current is consistent with Maxwell’s Equations and the conservation of charge. It can be used to derive Kirchhoff’s Current Law:

r ∂ρ ∂ρ r ∇ ⋅ J + = 0 if = 0 ∇ ⋅ J = 0 implies KCL ∂t ∂t

Constitutive Relationships: The field intensities and flux densities are related by using the constitutive equations. In general, the (ε) and the permeability (µ) are (different values in different directions) and are functions of the material. In simple materials they are scalars.

r r r r D = ε E ⇒ D = ε rε 0 E

r r r r B = µ H ⇒ B = µ r µ0 H where:

εr = ε0 = permittivity µr = Relative permeability µ0 =

Boundary Conditions: At abrupt interfaces between different materials the following conditions hold:

r r r r nˆ × (E1 − E2 )= 0 nˆ ⋅(D1 − D2 )= ρ S r r r r r nˆ × ()H1 − H 2 = J S nˆ ⋅ ()B1 − B2 = 0 where: n is the normal vector from region-2 to region-1 Js is the surface current density (A/m) 2 ρs is the density (C/m )

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Electrostatic Fields: When there are no time dependent fields, electric and magnetic fields can exist as independent fields. The electric fields are produced by charge distributions governed by:

r ρ r r 1 Q ∇ ⋅ E = E ⋅ dS = ρ dv = enclosed Gauss's Law ε ∫∫ε ε SV r r r ∇× E = 0 ∫ E ⋅ dl = 0 C Gauss’s Law can be used to find the electric field of highly symmetric problems where only a single vector component is required.

Using the principle of superposition, the field and the potential due to an arbitrary charge distribution are:

r ρ()r' ρ(r') E()r = Rˆdv′ V = dv′ ∫∫VV4π ε R 2 4π ε R where r is the position vectorof electricfield r'is theposition vectorof thechargedistribution R = r − r' is the distance between the electric field and charge distribution

Since the curl of the electric field is zero the field can be written as the of a potential:

r a r r r r E = −∇V Va −Vb = − E ⋅ dl E ⋅ dl = 0 ∫∫b

The last equation implies Kirchhoff’s Law.

From Gauss’s Law the potential must satisfy Poisson’s Equation or Laplace’s equation for charge free regions: ρ ∇ 2V = − ∇ 2V = 0 ε Surfaces of constant potentials are called equipotential surfaces, and are found to intersect electric fields at right angles. Perfect conducting surfaces are equipotential surfaces.

The force existing between point charges was found to obey ’s Law:

q q F = 1 2 rˆ 4π ε r 2

Where r is the distance between the charges and ε is the permittivity of the material between the charges, and force direction is on the line connecting the point charges. Like charges repel and opposites attract.

Charge storage devices can be created using multiple conductors separated by a . The and energy storage in are given by:

r r ε E ⋅dS Q 1 r r 1 C = ∫ = W = D⋅ E dv = CV 2 r r V e 2 ∫ 2 ∫ E ⋅dl V

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Electrostatic forces can be calculated using the gradient of the energy function:

F = ∇We

Moving Charges:

A charge (q) moving with a (u) in an electric and magnetic field will experience a force: r r r Law F = q(E + ur × B) This was how the electric field and magnetic fields were discovered and defined.

Moving charges give rise to a current density:

r J = ρ ur Convection current density

This also applies to charges moving in conductors, however the scatter in the material complicating the model. Experimentally, it was found the net of electrons is proportional to the applied electric field in the conductor.

r r J = σ E Point form of Ohm's Law The conductivity (σ (S/m)) is a property of the material. Any excess of charge in a conductor will redistribute itself to an equilibrium state with zero net charge with a time constant τd=ε/σ which is the dielectric relaxation time. The total current is related to the current density by:

r r I = ∫ J ⋅ dS S

The DC resistance of a uniform wire of conducting material is given by:

r r r r ∫ E ⋅ dl ∫ E ⋅ dl 1 L R = r r = r r = ∫ J ⋅ dS ∫σ E ⋅ dS σ A where L is the of the wire and A is the cross-sectional .

Magnetostatic Fields: Static magnetic fields are governed by:

r r r No magnetic monopoles ∇ ⋅ H = 0 ∫ H ⋅ dS = 0 S r r r r r r Ampere's Law ∇× H = J H ⋅ dl = J ⋅ dS = I enclosed ∫∫S C

Static magnetic fields are produced by steady currents (DC). Ampere’s Law can be used to find magnetic fields of highly symmetric configurations when only one component of the field is required.

Biot-Savart Law is used to compute the magnetic field for an arbitrary current distribution:

r r Idl × Rˆ r J (r ')× Rˆ H r = dl' H r = dv′ () ∫ 2 () ∫ 2 L 4π R V 4π R

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For a long wire:

r I H = φˆ 2π r

A wire carrying a DC current experiences a force in a magnetic field given by:

r r r F = I ∫ dl × B when the wire is straight, this reduces to a force per length, the force will be C perpendicular to both the current and the field.

r r The magnetic flux is given by: Ψ = ∫ B ⋅ dS r r The magnetic flux across a closed surface is always zero: Ψ = ∫ B ⋅ dS = 0 S

The ratio of the total magnetic flux and the current is called the . For a circuit with N loops the inductance and the stored magnetic energy are given by:

r r N B ⋅ dS ∫ NΨ 1 r r 1 L = S = W = B ⋅ H dv = LI 2 r r I m 2 ∫ 2 ∫ J ⋅ dS V S Since the induced magnetic fields are also proportional to the number of turns the inductance usually varies as N2.

Magnetostatic forces are computed as the gradient of the energy: F = ∇Wm

Time-varying Fields Maxwell’s equations are general and apply for all electromagnetic phenomena. However, sinusoidal varying fields are most commonly used with the time dependence of (cos ωt). The fields are written in a phasor form:

r r E()r.t = Re{E(r )e jω t }

Maxwell's Equations and continuity equations in phasor - form are :

r r ∇× E = -jω B r r r ∇× H = J + jω D r ∇ ⋅ B = 0 r ∇ ⋅ D = ρ r ∇ ⋅ J = − jω ρ

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Energy, , and Poynting Theorem:

The power density carried by the electric and magnetic fields is given by the : r r r S = E × H ()W / m 2 The direction of the Poynting vector is the direction of power flow. Using Maxwell’s equations, the flux of the Poynting vector out of a closed surface (S) is Poynting’s Theorem.

r r r r r r r ⎡ ∂ ⎛ B ⋅ H ⎞ ∂ ⎛ D ⋅ E ⎞ r r⎤ ()E × H ⋅ dS = − ⎢ ⎜ ⎟ + ⎜ ⎟ + E ⋅ J ⎥dv ∫∫t ⎜ 2 ⎟ ∂t ⎜ 2 ⎟ SV⎣⎢∂ ⎝ ⎠ ⎝ ⎠ ⎦⎥

This represents three forms of energy:

1 r r 1 2 Stored Electric Energy Density w = D ⋅ E = ε E e 2 2 1 r r 1 2 Stored Magnetic Energy Density w = B ⋅ H = µ H m 2 2 r r 2 Ohmic Losses (Heat) pσ = E ⋅ J = σ E

r r r In phasor-form the complex Poynting vector is: S = E × H ∗ This gives the instantaneous power density, usually the average power density is more important and that is given by:

1 r r Time - averaged Power Density P = Re{E × H ∗ } (W / m2 ) AVE 2

Terminology:

Homogeneous: A medium in which the material constants are uniform in the area of interest.

Inhomogeneous: The material varies in composition such that the material properties are functions of position. Examples: ε(x) µ(x) σ(x)

Isotropic: The material properties do not depend on the polarization of the electromagnetic fields.

Anisotropic: Material properties depend on the polarization of the electromagnetic fields and must be written as tensors.

Source-free: Region where no electromagnetic sources are found therefore ρ=J=0

Non-magnetic: The material has µ=µ0

Dispersive: The material properties are a function of frequency. Examples: ε(ω), µ(ω)

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