Electromagnetic Waves Reading Shen and Kong Ch. 3 Outline Review of the Quasi-static Approximation Electric and Magnetic Components of Waves The Wave Equation (in 1-D) Uniform Plane Waves Phase Velocity and Intrinsic Impedance Wave-vector and Wave-frequency
Maxwells Equations (Free Space with Charges) Differential form Integral form E-Gauss: Faraday: H-Gauss: Ampere: In statics, both time derivatives are unimportant, Maxwells
Equations split into decoupled electrostatic and magnetostatic equations. In Electro-quasistatic (EQS) and magneto-quasitatic systems (MQS), one (but not both) time derivative becomes important. Quasi-static Maxwells Equations Electric Fields Magnetic Fields MQS
EQS For the error in the QS approximation to be small or EQS vs MQS for Time-Varying Fields Why did we not worry about the magnetic field generated by the time-varying electric field of a motor ?
animate A typical motor frequency of 2000 rpm satisfies EQS approximation for free-space As another example, note: At 60 Hz, the wavelength (typical length) in air is 5000 km, therefore, almost all physical 60-Hz systems in air are quasistatic (since they are typically smaller than 5000 km in size) Coupling of Electric and Magnetic Fields
Maxwells Equations couple the E and H fields: animate Uniform Electromagnetic Waves Ey varies along the z-direction and E is constant in the two other directions Uniform Electromagnetic Waves Ey varies along the z-direction and Ey is constant in the other two directions
animate Electromagnetic Waves Ey-field cannot vary in z-direction without a time-varying B-field and waves must have both electric and magnetic components ! Uniform Electromagnetic Plane Waves The y-component of E that varies across space is
associated with the x-component of B that varies in time Uniform Electromagnetic Plane Waves Source free: The Wave Equation Time-varying Ey generates spatially
varying Bz Time-varying Bz generates spatially varying Ey The temporal and spatial variations in E y are coupled together to yield the Wave Equation. The Wave Equation via Differential Equations
Faraday: Ampere: Substitution yields the wave equation: Uniform Plane Wave Solutions The 1-D wave equation
Ey(z,t) is any function for which the second derivative in space equals its second derivative in time, times a constant. The solution is therefore any function with the same dependence on time as on space, e.g. The functions f+(z-ct) and f- (z+ct) represent uniform waves propagating in the +z and -z directions respectively. Speed of Light
The velocity of propagation is determined solely by the dielectric permittivity and magnetic permeability: The functions f+ and f- are determined by the source and the other boundary conditions. animate Magnetic Field of a Uniform Plane Wave In vacuum
A Uniform Plane Wave Inside a material where is known as the phase velocity of the wave The Characteristic Impedance
is the intrinsic impedance of the medium given by Like the velocity of propagation, the intrinsic impedance is independent of the source and is determined only by the properties of the medium. Sinusoidal Uniform Plane Waves
where is known as the wave-number Sinusoidal Uniform Plane Waves Sinusoidal Uniform Plane Waves Spatial quantities:
Temporal quantities: How Are Uniform EM Plane Waves Launched? Generally speaking, electromagnetic waves are launched by time-varying charge distributions and currents, that together must satisfy: Man-made systems that launch waves are often called antennas. Uniform plane waves are launched by current sheets:
mage is in the public domain. . K Dipole Antenna Quarter wavelength vertical antenna has one connection to the vertical element and uses earth connection to provide an image
for the other quarter wave. The voltage and current waveforms are out of phase. The antenna generates (or receives) the omni-directional radiation pattern in the horizontal plane. The antenna does not have to be re-orientated to keep the signals constant as, for example, a car moves its position. Voltage Electric fields (blue) and magnetic Quarter
fields (gray) radiated by a dipole wave antenna veritcal antenna Current Coaxial Cable feeder Connection to earth
KEY TAKEAWAYS Time-varying Ey generates spatially varying Bz Time-varying Bz generates spatially varying Ey The1-D Wave Equation has solutions of the form
with propagation velocity: (speed of light) and more generally: (phase velocity) where is known as the intrinsic impedance
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