Fields and Waves I - PowerPoint PPT Presentation

About This Presentation
Title:

Fields and Waves I

Description:

Fields and Waves I Lecture 10 Electric Potential and Voltage K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute ... – PowerPoint PPT presentation

Number of Views:93
Avg rating:3.0/5.0
Slides: 56
Provided by: KA75
Category:

less

Transcript and Presenter's Notes

Title: Fields and Waves I


1
Fields and Waves I
  • Lecture 10
  • Electric Potential and Voltage
  • K. A. Connor
  • Electrical, Computer, and Systems Engineering
    Department
  • Rensselaer Polytechnic Institute, Troy, NY

2
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 Ulabys textbook.
3
Overview
  • Review Using Gauss Law to find E
  • General Approach
  • Example 5
  • Electric Potential and Voltage
  • Field and voltage plots
  • Direct calculation of V from Q

4
Examples of typical Gaussian surfaces
Ulaby
5
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.

6
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.

7
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

8
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/Fiel
    ds/gauss_law.pdf

9
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.

10
Example 5
  1. Find the electric field for all r.
  2. Check your answer by evaluating the divergence
    and curl of the electric field.
  3. What is the closed line integral of the electric
    field around the contour shown?
  4. Express the surface charge density in terms of
    the volume charge density.

11
Example 5
12
Example 5
13
Example 5
14
Example 5 (last lecture)
15
Electric Potential and Voltage
  • Introducing the electric scalar potential

16
Introduction to Voltage calculation
  • Charge-Voltage Method

Maxwell first equation
Maxwell second equation
17
Maxwells second equation
In the last lecture we looked at Maxwells 1st
equation
Today, we will use Maxwells 2nd equation
Importance of this equation is that it allows the
use of Voltage or Electric Potential
18
Local and integral form equivalence
  • Stokess theorem

Surface integral on right is surface enclosed by
line on the left
Integral form
Local form
19
Electric potential definition
  • Form vector calculus
  • Introducing the electric scalar potential

20
Electric potential definition
Example Use case of point charge at origin and
obtain potential everywhere from E-field
Spherical Geometry
Reference V0 at infinity
21
Electric potential definition
The integral for computing the potential of the
point charge is
22
Example 1
  • Using the E field from Example 5 of the last
    lecture
  • Find the voltage as a function of r for rgtb and
    bgtrgta
  • Check the result using
  • Evaluate the voltage at the origin

23
Example 1
Hint
Use rb as the reference - Start here and move
away or inside rltb region
24
Example 1
25
Example 1
26
Electric Potential and Voltage
  • Field and voltage plots

27
Potential surfaces
Potential is a SCALAR quantity
Graphs are done as Surface Plots or Contour Plots
Example - Parallel Plate Capacitor
Potential Surfaces
28
E-field from Potential Surfaces
V4
V3
V2
V1
From
Gradient points in the direction of largest change
Therefore, E-field lines are perpendicular
(normal) to constant V surfaces
(add E-lines to potential plot)
29
Gradient physical example
  • Current flow simulation

V10
V0
Current modulus in color shades And
arrows Potential in red equi lines
30
Example 2
  • Plot a set of equipotentials for the quadrupole

31
Example 2
32
Example 2 with Finite Element Computation
Plot of the electric field (direction and
magnitude)
33
Example 2 V from Finite Element Computation
34
Equipotentials
http//www.physics.utah.edu/p2220/examples.html
35
Hand drawing of equipotentials
http//ocw.mit.edu/NR/rdonlyres/Electrical-Enginee
ring-and-Computer-Science/6-013Electromagnetics-an
d-ApplicationsFall2002/922D1A06-9AC9-4076-B1F5-066
EE896043C/0/Rec11Notes.pdf
36
Equipotentials
http//www.adeptscience.co.uk/
37
Contour Plots -- Matlab
From the Matlab Penny Demo
38
Contour Plots Matlab
39
Electric Potential and Voltage
  • Direct calculation of potential from charges

40
Direct calculus of potential
As we have seen
Given r or Q
E-field
V
derive
derive
Now
Looking for techniques so that
V
, given r or Q
derive
41
Direct calculus of potential from a single charge
For the case of a point charge
x
, is field point where we are measuring/calculatin
g V
, is location of charge
42
Direct calculus of potential from a charge
distribution
For a charge distribution
Volume charge distribution
Line charge distribution
43
Example 2
  • Determine the voltage and E field due to a finite
    length line charge in the z0 plane.

44
Example 2
Line charge
Location of measurement of V
Line charge distribution
Integrate along charge means dl is dz
45
Example 2
All at z0
46
Example 2
  • From Maple
  • which is actually the same result when
    simplified.

47
Numerical Simulation of Potential
Numerical Approximation
Break line charge into 4 segments
Charge for each segment
Segment length
Distance to charge
48
Numerical Simulation of Potential
Get V at two nearby points
Use
So..use 2 points to get DV and Dr
  • V is a SCALAR field and easier to work with
  • In many cases, easiest way to get E-field is to
    first find V and then use,

49
Example 2
  • Approximate the line of charge with 4 point
    charges. Calculate the potential due to the 4
    charges in the z0 plane.
  • When and
    evaluate
  • at and
  • Estimate the electric field at
  • Compare your results with the direct calculation

50
Example 2
All at z0
51
HW3
  • Some comments on homework assignment 3
  • http//hibp.ecse.rpi.edu/7Econnor/education/Field
    s/HW3-s05.pdf
  • http//hibp.ecse.rpi.edu/7Econnor/education/Field
    s/vdata.xls

52
Electric Potential and Voltage
  • Annexes

53
POTENTIAL ENERGY
Work done by a force is given by
If vectors are parallel, particle gains energy -
Kinetic Energy
If,
Conservative Force
Example GRAVITY
  • going DOWN increases KE, decreases PE
  • going UP increases PE, decreases KE

54
POTENTIAL ENERGY
If we are dealing with a conservative force, we
can use concept of POTENTIAL ENERGY
For gravity, the potential energy has the form mgz
Define the following integral
Potential Energy Change
55
POTENTIAL ENERGY
and
Since
We can define
Potential Energy
Also define Voltage Potential Energy/Charge
Write a Comment
User Comments (0)
About PowerShow.com