4. electrostatics applied em by ulaby, michielssen and ravaioli
TRANSCRIPT
![Page 1: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/1.jpg)
4. ELECTROSTATICSApplied EM by Ulaby, Michielssen and Ravaioli
![Page 2: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/2.jpg)
Chapter 4 Overview
![Page 3: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/3.jpg)
Maxwell’s Equations
God said:
And there was light!
![Page 4: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/4.jpg)
Charge Distributions
Volume charge density:
Total Charge in a Volume
Surface and Line Charge Densities
![Page 5: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/5.jpg)
Current Density
For a surface with any orientation:
J is called the current density
![Page 6: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/6.jpg)
Convection vs. Conduction
![Page 7: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/7.jpg)
Coulomb’s Law
Electric field at point P due to single charge
Electric force on a test charge placed at P
Electric flux density D
![Page 8: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/8.jpg)
Electric Field Due to 2 Charges
![Page 9: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/9.jpg)
Electric Field due to Multiple Charges
![Page 10: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/10.jpg)
Electric Field Due to Charge Distributions
Field due to:
![Page 11: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/11.jpg)
Cont.
![Page 12: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/12.jpg)
![Page 13: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/13.jpg)
Cont.
![Page 14: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/14.jpg)
Example 4-5 cont.
![Page 15: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/15.jpg)
Gauss’s Law
Application of the divergence theorem gives:
![Page 16: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/16.jpg)
Applying Gauss’s Law
Construct an imaginary Gaussian cylinder of radius r and height h:
![Page 17: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/17.jpg)
Electric Scalar Potential
Minimum force needed to move charge against E field:
![Page 18: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/18.jpg)
Electric Scalar Potential
![Page 19: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/19.jpg)
Electric Potential Due to Charges
In electric circuits, we usually select a convenient node that we call ground and assign it zero reference voltage. In free space and material media, we choose infinity as reference with V = 0. Hence, at a point P
For a point charge, V at range R is:
For continuous charge distributions:
![Page 20: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/20.jpg)
Relating E to V
![Page 21: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/21.jpg)
Cont.
![Page 22: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/22.jpg)
(cont.)
![Page 23: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/23.jpg)
Poisson’s & Laplace’s Equations
In the absence of charges:
![Page 24: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/24.jpg)
![Page 25: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/25.jpg)
Conduction Current
Conduction current density:
Note how wide the range is, over 24 orders of magnitude
![Page 26: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/26.jpg)
Conductivity
ve = volume charge density of electronshe = volume charge density of holese = electron mobilityh = hole mobilityNe = number of electrons per unit volumeNh = number of holes per unit volume
![Page 27: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/27.jpg)
![Page 28: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/28.jpg)
Resistance
For any conductor:
Longitudinal Resistor
![Page 29: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/29.jpg)
G’=0 if the insulating material is air or a perfect dielectric with zero conductivity.
![Page 30: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/30.jpg)
Joule’s Law
The power dissipated in a volume containing electric field E and current density J is:
For a resistor, Joule’s law reduces to:
For a coaxial cable:
![Page 31: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/31.jpg)
Tech Brief 7: Resistive Sensors
An electrical sensor is a device capable of responding to an applied stimulus by generating an electrical signal whose voltage, current, or some other attribute is related to the intensity of the stimulus.
Typical stimuli : temperature, pressure, position, distance, motion, velocity, acceleration, concentration (of a gas or liquid), blood flow, etc.
Sensing process relies on measuring resistance, capacitance, inductance, induced electromotive force (emf), oscillation frequency or time delay, etc.
![Page 32: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/32.jpg)
Piezoresistivity
The Greek word piezein means to press
R0 = resistance when F = 0F = applied forceA0 = cross-section when F = 0 = piezoresistive coefficient of material
![Page 33: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/33.jpg)
Piezoresistors
![Page 34: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/34.jpg)
Wheatstone Bridge
Wheatstone bridge is a high sensitivity circuit for measuring small changes in resistance
![Page 35: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/35.jpg)
Dielectric Materials
![Page 36: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/36.jpg)
Polarization Field
P = electric flux density induced by E
![Page 37: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/37.jpg)
Electric Breakdown
Electric Breakdown
![Page 38: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/38.jpg)
Boundary Conditions
![Page 39: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/39.jpg)
Summary of Boundary Conditions
Remember E = 0 in a good conductor
![Page 40: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/40.jpg)
![Page 41: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/41.jpg)
Conductors
Net electric field inside a conductor is zero
![Page 42: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/42.jpg)
Field Lines at Conductor Boundary
At conductor boundary, E field direction is always perpendicular to conductor surface
![Page 43: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/43.jpg)
![Page 44: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/44.jpg)
![Page 45: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/45.jpg)
Capacitance
![Page 46: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/46.jpg)
Capacitance
For any two-conductor configuration:
For any resistor:
![Page 47: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/47.jpg)
![Page 48: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/48.jpg)
Application of Gauss’s law gives:
Q is total charge on inside of outer cylinder, and –Q is on outside surface of inner cylinder
![Page 49: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/49.jpg)
Tech Brief 8: Supercapacitors
For a traditional parallel-plate capacitor, what is the maximum attainable energy density?
= permittivity of insulation materialV = applied voltage = density of insulation materiald = separation between plates
Mica has one of the highest dielectric strengths ~2 x 10**8 V/m. If we select a voltage rating of 1 V and a breakdown voltage of 2 V (50% safety), this will require that d be no smaller than 10 nm. For mica, = 60 and = 3 x 10**3 kg/m3 .
Hence:
W = 90 J/kg = 2.5 x10**‒2 Wh/kg.
By comparison, a lithium-ion battery has W = 1.5 x 10**2 Wh/kg, almost 4 orders of magnitude greater
Energy density is given by:
![Page 50: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/50.jpg)
A supercapacitor is a “hybrid” battery/capacitor
![Page 51: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/51.jpg)
Users of Supercapacitors
![Page 52: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/52.jpg)
Energy Comparison
![Page 53: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/53.jpg)
Electrostatic Potential Energy
Electrostatic potential energy density (Joules/volume)
Total electrostatic energy stored in a volume
Energy stored in a capacitor
![Page 54: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/54.jpg)
Image Method
Image method simplifies calculation for E and V due to charges near conducting planes.
1.For each charge Q, add an image charge –Q2.Remove conducting plane3.Calculate field due to all charges
![Page 55: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/55.jpg)
![Page 56: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/56.jpg)
Tech Brief 9: Capacitive Sensors
![Page 57: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/57.jpg)
Humidity Sensor
![Page 58: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/58.jpg)
Pressure Sensor
![Page 59: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/59.jpg)
Planar capacitors
![Page 60: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/60.jpg)
Fingerprint Imager
![Page 61: 4. ELECTROSTATICS Applied EM by Ulaby, Michielssen and Ravaioli](https://reader033.vdocuments.us/reader033/viewer/2022061608/56649f415503460f94c60e2e/html5/thumbnails/61.jpg)