Fields and Waves I

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Fields and Waves I

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

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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.
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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

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Examples of typical Gaussian surfaces
Ulaby
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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.

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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.

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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

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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

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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.

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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.

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Example 5
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Example 5
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Example 5
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Example 5 (last lecture)
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Electric Potential and Voltage

• Introducing the electric scalar potential

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Introduction to Voltage calculation

• Charge-Voltage Method

Maxwell first equation
Maxwell second equation
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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
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Local and integral form equivalence

• Stokess theorem

Surface integral on right is surface enclosed by
line on the left
Integral form
Local form
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Electric potential definition

• Form vector calculus
• Introducing the electric scalar potential

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Electric potential definition
Example Use case of point charge at origin and
obtain potential everywhere from E-field
Spherical Geometry
Reference V0 at infinity
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Electric potential definition
The integral for computing the potential of the
point charge is
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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

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Example 1
Hint
Use rb as the reference - Start here and move
away or inside rltb region
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Example 1
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Example 1
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Electric Potential and Voltage

• Field and voltage plots

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Potential surfaces
Potential is a SCALAR quantity
Graphs are done as Surface Plots or Contour Plots
Example - Parallel Plate Capacitor
Potential Surfaces
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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)
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Gradient physical example

• Current flow simulation

V10
V0
Current modulus in color shades And
arrows Potential in red equi lines
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Example 2

• Plot a set of equipotentials for the quadrupole

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Example 2
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Example 2 with Finite Element Computation
Plot of the electric field (direction and
magnitude)
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Example 2 V from Finite Element Computation
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Equipotentials
http//www.physics.utah.edu/p2220/examples.html
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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
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Equipotentials
http//www.adeptscience.co.uk/
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Contour Plots -- Matlab
From the Matlab Penny Demo
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Contour Plots Matlab
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Electric Potential and Voltage

• Direct calculation of potential from charges

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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
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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
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Direct calculus of potential from a charge
distribution
For a charge distribution
Volume charge distribution
Line charge distribution
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Example 2

• Determine the voltage and E field due to a finite
  length line charge in the z0 plane.

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Example 2
Line charge
Location of measurement of V
Line charge distribution
Integrate along charge means dl is dz
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Example 2
All at z0
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Example 2

• From Maple
• which is actually the same result when
  simplified.

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Numerical Simulation of Potential
Numerical Approximation
Break line charge into 4 segments
Charge for each segment
Segment length
Distance to charge
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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,

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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

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Example 2
All at z0
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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

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Electric Potential and Voltage

• Annexes

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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

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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
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POTENTIAL ENERGY
and
Since
We can define
Potential Energy
Also define Voltage Potential Energy/Charge