Fields and Waves I Lecture 9 K. A. Connor Introduction to Electrostatics and Gauss’ Law K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY Welcome to Fields and Waves I Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

J. Darryl Michael – GE Global Research Center, Niskayuna, NY These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following: Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY J. Darryl Michael – GE Global Research Center, Niskayuna, NY Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY Lale Ergene – ITU Informatics Institute, Istanbul, Turkey Jeffrey Braunstein – Chung-Ang University, Seoul, Korea Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook. 18 September 2018 Fields and Waves I

Review of Math for Fields Overview Review of Math for Fields Vector notation Coordinate systems Line, area and volume integrals Gradient, divergence and curl Electrostatics Sources – Charges and Charge Distributions Coulomb’s Law Gauss’ Law 18 September 2018 Fields and Waves I

Maxwell’s Equations Integral Form Differential Form Electrostatics 18 September 2018 Fields and Waves I

Electrostatics Integral Form Differential Form Note the Math Operations Integral Form Differential Form 18 September 2018 Fields and Waves I

Electrostatics – What is its value? Capacitance I-V Characteristics of Devices Useful when the system size is small compared to & 18 September 2018 Fields and Waves I

Capacitance Charge-Voltage Method Energy Method 18 September 2018 Fields and Waves I

Electrostatics Basic Configurations Ulaby 18 September 2018 Fields and Waves I

SOURCE of Electrostatic E-Field is CHARGE ELECTRICAL CHARGES SOURCE of Electrostatic E-Field is CHARGE Examples of various charge distributions: 1. Point charge - Q (units of Coulomb) - model individual particle (eg. Electron) or a well-localized group of charge particles 2. Volume Charge Density - r or rv (units of Coulomb/m3) - large # of particles - ignore discrete nature to smooth out distribution Eg. Doped Region of Semiconductor, e-beam in a cathode ray tube ( Beam has finite radius ) 18 September 2018 Fields and Waves I

Other examples of Charge Distribution…….. ELECTRICAL CHARGES Other examples of Charge Distribution…….. 3. Surface Charge Density - r or rs (units of Coulomb/m2) Eg. Very thin charge layer on conductor surface 4. Line Charge Density - r or rl (units of Coulomb/m) - not as physically realizable Eg. Model for a wire, electron beam from far 18 September 2018 Fields and Waves I

ELECTRICAL CHARGES Maxwell’s equation: More generally, Derived from: 18 September 2018 Fields and Waves I

Example 1 Charge density Find the charge in the spherical volume containing a charge distribution . Evaluate the result analytically. Then find a numerical result when meters and . b. A surface charge on a disk increases linearly from in the center to at the outer edge where r = 2 meters. Find the total charge on the disk. 18 September 2018 Fields and Waves I

Example 1 18 September 2018 Fields and Waves I

Fields and Waves I Force and fields

(force), between point charges COULOMB’S LAW (force), between point charges Q1 Unit vector in r-direction Force on Charge 2 by Charge 1 R Q2 18 September 2018 Fields and Waves I

COULOMB’S LAW - E Field ,of Q1 is Then, Unit vector pointing away from Q1 Then, - we work with E-Field because Maxwell’s equations written in those terms 18 September 2018 Fields and Waves I

Proportional to Magnitude or strength of E-Field E-FIELDS , is a VECTOR Field How do we represent it? - Field points in the direction that a +q test charge would move Represent using Arrows : Direction and Length Point charge Proportional to Magnitude or strength of E-Field 18 September 2018 Fields and Waves I

Figure from Ulaby Ulaby 18 September 2018 Fields and Waves I

Computation of E-fields from multiple charges: Example: DIPOLE - 2 separated opposite polarity point charges +Q Apply superposition of Fields x y Planes of symmetry: Horizontal axis: Ex cancels, Ey adds -Q +Q vector Vertical axis: only Ey component -Q Resulting vector 18 September 2018 Fields and Waves I

E-FIELDS - Some examples http://shinliang.blogspot.com/2009/04/21-coulombs-law.html 18 September 2018 Fields and Waves I

E-FIELDS - Some examples Note, in the upper right figure, that four times as many field lines leave the +4 positive charge as leave the +1 charge. All of the field lines end at infinity, as they do with a single positive charge. http://people.seas.harvard.edu/~jones/cscie129/nu_lectures/lecture6/field_vis/e_vis.html 18 September 2018 Fields and Waves I

E-FIELDS - Dipole Lines of Symmetry 18 September 2018 Fields and Waves I

E-FIELDS How would the DIPOLE field lines change if the charges were the same polarity? 18 September 2018 Fields and Waves I

E-FIELDS 18 September 2018 Fields and Waves I

Example 2 Sketch the electric field lines for the electric quadrupole shown. Sketch the planes for which you expect the field to be symmetric. After completing your sketch, verify your result with the applet at http://hibp.ecse.rpi.edu/~crowley/java/Efield/App.htm . Dipole results can be seen with the applet or with the Mathcad worksheet for Sect. 3.6.2 18 September 2018 Fields and Waves I

Example 2 18 September 2018 Fields and Waves I

Example 2 with Finite Element Computation Plot of the electric field (direction and magnitude) 18 September 2018 Fields and Waves I

Calculating the electric field thanks to the Gauss’ Law Fields and Waves I Calculating the electric field thanks to the Gauss’ Law

MAXWELL’S FIRST EQUATION : GAUSS’ LAW Enclosed Charge Differential Form Integral Form - ‘dv’ integral over volume enclosed by ‘ds’ integral For vacuum and air - think of D and E as being the same D vs E depends on materials constant 18 September 2018 Fields and Waves I

Example 3 Gauss’ Law Show that the electric field of a point charge satisfies Gauss’ Law by integrating over the surface of a sphere of radius a. 18 September 2018 Fields and Waves I

Example 3 18 September 2018 Fields and Waves I

Use Gauss’ Law to find D and E in symmetric problems GAUSS’ LAW - strategy Use Gauss’ Law to find D and E in symmetric problems Get D or E out of integral Always look at symmetry of the problem - and take advantage of this 18 September 2018 Fields and Waves I

GAUSS’ LAW - use of symmetry Example: A sheet of charge - charges are infinite in extent on say x,y plane , is sum due to all charges Arbitrary Point P , points in all other components cancel only a function of z (not x or y) x y z Surface of infinite extent of charge Can write down: 18 September 2018 Fields and Waves I

Example 4 Setting up the Problem For the three charge distributions, find the direction of the electric field, the surface on which the field is a constant and the flux is nonzero, and sketch the surfaces. 18 September 2018 Fields and Waves I

Example 4 18 September 2018 Fields and Waves I

For example a planar sheet of charge, where z is constant GAUSS’ LAW ,is constant. For example a planar sheet of charge, where z is constant To use GAUSS’ LAW, we need to find a surface that encloses the volume GAUSSIAN SURFACE - takes advantage of symmetry - when r is only a f(r) - when r is only a f(z) 18 September 2018 Fields and Waves I

Use Gaussian surface to “pull” this out of integral GAUSS’ LAW Use Gaussian surface to “pull” this out of integral Integral now becomes: Usually an easy integral for surfaces under consideration 18 September 2018 Fields and Waves I

Example of using GAUSS’ law to find Z = a -a < z < a Z > a ; z< -a Z = -a “a slab of charge” By symmetry: From symmetry If r0 > 0, then Z=0 18 September 2018 Fields and Waves I

in region |z| < a and create a surface at arbitrary z GAUSS’ LAW First get in region |z| < a and create a surface at arbitrary z Use Gaussian surface with top at z = z’ and the bottom at -z’ Note: Gaussian Surface is NOT a material boundary 18 September 2018 Fields and Waves I

GAUSS’ LAW =0, since Evaluate LHS: These two integrals are equal 18 September 2018 Fields and Waves I

Key Step: Take E out of the Integral GAUSS’ LAW Key Step: Take E out of the Integral Computation of enclosed charge 18 September 2018 Fields and Waves I

GAUSS’ LAW (drop the prime) 18 September 2018 Fields and Waves I

GAUSS’ LAW Need to find , for |z| > a Z = z’ r0 Z = -z’ 18 September 2018 Fields and Waves I

Computation of enclosed charge GAUSS’ LAW As before, Computation of enclosed charge Note that the z-integration is from -a to a ; there is NO CHARGE outside |z|>a 18 September 2018 Fields and Waves I

For the region outside |z|>a GAUSS’ LAW Once again, For the region outside |z|>a 18 September 2018 Fields and Waves I

Note: E-field is continuous GAUSS’ LAW -a z a Note: E-field is continuous Plot of E-field as a function of z for planar example 18 September 2018 Fields and Waves I

Example 5 Full Gauss’ Law Solution A charge distribution with cylindrical symmetry is shown. The inner cylinder has a uniform charge density . The outer shell has a surface charge density such that the total charge on the outer shell is the negative of the total charge in the inner cylinder. Ignore end effects. 18 September 2018 Fields and Waves I

Find the electric field for all r. Example 5 Find the electric field for all r. Check your answer by evaluating the divergence and curl of the electric field. What is the closed line integral of the electric field around the contour shown? Express the surface charge density in terms of the volume charge density. 18 September 2018 Fields and Waves I

Example 5 18 September 2018 Fields and Waves I

Example 5 18 September 2018 Fields and Waves I

Example 5 18 September 2018 Fields and Waves I

Example 5 18 September 2018 Fields and Waves I

Ulaby 18 September 2018 Fields and Waves I

Using Gauss’ Law to find E Recognize the coordinate system. Using symmetry, determine which components of the field exist. Identify a Gaussian surface for which the sides are either parallel to or perpendicular to the field components. This surface is arbitrary in size. Determine the total charge within that surface. The charges can be distributed on lines, surfaces or in volumes. 18 September 2018 Fields and Waves I

Using Gauss’ Law to find E Evaluate the electric flux passing through the Gaussian surface. If the field is parallel to the surface If the field is perpendicular to the surface, where the subscript refers to the direction of the surface. Note that a high level of symmetry is necessary to make these simplifications. 18 September 2018 Fields and Waves I

Using Gauss’ Law to find E Now equate the two sides of Gauss’ Law to find E: Remember that the Gaussian surface is arbitrary in position so the surface area is a function. For example for a spherical surface 18 September 2018 Fields and Waves I

Using Gauss’ Law to find E There is a short write-up on this topic in the Supplementary Materials: http://hibp.ecse.rpi.edu/%7Econnor/education/Fields/gauss_law.pdf 18 September 2018 Fields and Waves I