Electromagestism Exam

8/19/2019 Electromagestism Exam

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Links:

• http://in#o.ee.surrey.ac.uk/Teaching/5ourses/9T/dynamics/

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Links:

• https://www.cis.rit.edu/class/simg'2

'/notes/1asicprinciples).pd# • https://1ooks.google.de/1ooks3

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Question IV:

Links:

• http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/)'+electromagneticsandapplicationsspring2-/readings/MET)!'+S-!notes.pd#

• http://wwweng.l1l.go,/Ishuman/69@>@/H99H9@59S>T09H!MES5/paschen!report.pd#

&ate and Time:

2th March 2') #rom ):pm to *:+pm

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdfhttp://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_notes.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdfhttp://www-eng.lbl.gov/~shuman/XENON/REFERENCES&OTHER_MISC/paschen_report.pdf


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Question IV Solution:

E# we consider the static case Ji.e.K constant with time o#

Maxwells 9"uationsK we Nnd that the time derivatives o# the

electric Neld and magnetic Oux density are zero:

ThusK Maxwells e"uations #or static felds 1ecome:

or the static case J1ut just #or

the static caseK Maxwells e"uations QdecoupleR into two

independent pairs o# e"uations.

The Nrst set in,ol,es electric Neld EJ ŕ and

charge density

pv J ŕ only. These are called the

electrostatic equations in

ree-space:

These are the electrostatic equations #or #ree space Ji.e.K a,acuum.


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9ssentiallyK this is what the electrostatic e"uations tell us:

'. The static electric Neld is conservative.2. The source o# the static Neld is charge.

En other wordsK the static electric Neld EJ ŕ diverges #rom Jor

converges to charge.


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Question II Solution:

Divergence:

The mathematical deNnition o# di,ergence is:

where the sur#ace S is a closed sur#ace that completely surrounds a ,erysmall ,olume , at point r K and where ds points outward #rom the closed

sur#ace. rom the deNnition o# sur#ace integralK we see that di,ergence

1asically indicates the amount o# ,ector Neld ;J ŕ that is con,erging toK

or di,erging #romK a gi,en point.

or exampleK consider these ,ector Nelds in the region o# a speciNc point:

The Neld on the le#t is con,erging to a pointK and there#ore the di,ergence

o# the ,ector Neld at that point is negati,e. 5on,erselyK the ,ector Neld onthe right is di,erging #rom a point. ;s a resultK the di,ergence o# the

,ector Neld at that point is greater than Gero.

5onsider some other ,ector Nelds in the region o# a speciNc point:


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enerallyK the di,ergence o# a ,ector Neld results in a scalar Neld

Jdi,ergence that is positi,e in some regions in spaceK negati,e other

regionsK and Gero elsewhere. or most physical pro1lemsK the di,ergence

o# a ,ector Neld pro,ides a scalar Neld that represents the sources o# the

,ector Neld.

radient:

5onsider the topography o# the 9arths sur#ace.

7e use contours o# constant ele,ationcalled topographic contoursto

express on maps Ja 2dimensional graphic the third dimension o#

ele,ation Ji.e.K sur#ace height. 7e can in#er #rom these maps the slope o#

the 9arths sur#aceK as topographic contours lie closer together where the

sur#ace is ,ery steep.

Moreo,erK we can likewise in#er the direction o# these slopesa hillsidemight slope toward the southK or a cliU might dropoU toward the 9ast.

ThusK the slope o# the 9arths sur#ace has 1oth a magnitude Je.g.K Oat or

steep and a direction Je.g. toward the north. En other wordsK the slope o#

the 9arths sur#ace is a ,ector "uantity ThusK the sur#ace slope at e,ery

point across some section o# the 9arth Je.g.K &ouglas 5ountyK 5oloradoK or

@orth ;merica must 1e descri1ed 1y a ,ector Neld.

Say the topography o# some small section o# the 9arths sur#ace can 1e

descri1ed as a scalar #unction hJxK yK where h represents the heightJele,ation o# the 9arth at some point denoted 1y coordinates x and y.


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• The small sur#ace s o# i is centered at point ŕ K and oriented such

that it is normal to unit ,ector Wa o# i.• The contour 5 o# i is the closed contour that surrounds sur#ace s o#

i.

5url is a measurement o# the circulation o# ,ector Neld ; J ŕ around

point ŕ .

E# a component o# ,ector Neld ; J ŕ is pointing in the direction d; at

e,ery point on contour 5i Ji.e.K tangential to the contour. Then the line

integralK and thus the curlK will 1e positi,e. E#K howe,erK a component o# ,ector Neld ; J ŕ points in the opposite direction Jd; at e,ery point on

the contourK the curl at point r will 1e negati,e.

LikewiseK these ,ector Nelds will result in a curl with Gero ,alue at point

ŕ :


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enerallyK the curl o# a ,ector Neld result is in another ,ector Neld whose

magnitude is positi,e in some regions o# spaceK negati,e in other regionsK

and Gero elsewhere.

The curl o# ,ector Nelds expressed using our coordinate systems.

5artesian:

5ylindrical:

Spherical:

"aplacian:


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;nother diUerential operator used in electromagnetics is the Laplacian

operator. There is 1oth a scalar Laplacian operatorK and a ,ector Laplacian

operator. Doth operationsK howe,erK are expressed in terms o# deri,ati,e

operations.

The Scalar Laplacian:

The scalar Laplacian is simply the di,ergence o# the gradient o# a scalar

Neld:

V X Vg J ŕ

The scalar Laplacian there#ore 1oth operates on a scalar Neld and results

in a scalar Neld. >#tenK the Laplacian is denoted as Q ∇2

RK i.e.

∇2g( ŕ) 4 VXV g J ŕ

rom the expressions o# di,ergence and gradientK we Nnd that the scalar

Laplacian is expressed in 5artesian coordinates as:

The Yector Laplacian:

The ,ector LaplacianK denoted as J ∇2 A( ŕ) K 1oth operates on a ,ector Neld

and results in a ,ector NeldK

and is deNned as:

E# we e,aluate the a1o,e expression #or a ,ector expressed in the5artesian coordinate systemK we Nnd that the ,ector Laplacian is:

En other wordsK we e,aluate the ,ector Laplacian 1y e,aluating the scalar

Laplacian o# each 5artesian scalar component.


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7hile the LorentG #orce law deNnes how electric and magnetic Nelds can

1e o1ser,edK Maxwells #our e"uations explain how these Nelds can 1e

created directly #rom charges and currentsK or indirectly and e"ui,alently

#rom other time ,arying Nelds. >ne o# those #our e"uations is ausss Law

#or chargeK which states that the total charge B Z5oulom1s[ within ,olumeY e"uals the integral o# the normal component o# the electric

displacement ,ector ⎯ & o,er the sur#ace area ; o# that ,olume:

w\\ J& ] nWda 4 \\\ ^ d, 4 B

En ,acuum:

& 4∈

0 9

where the permitti,ity o# ,acuum _ 4 *.*($`''2 o arads/m. 9"uation J'.+.'

re,eals the dimensions o# ⎯

&: 5oulom1s/m2 K o#ten a11re,iated here as Z5/m2 [.

Question I Solution:

Two closely spaced layers o# chargeK e"ual in magnitude and opposite in

signK comprise a charge dou1le layer. Such dou1le layers occur in the

mem1ranes o# all li,ing cells. ;n understanding o# their electrical

properties is essential in studying the mechanism o# ner,e transmissionand cell meta1olism. 0ere we consider the simplest type o# dou1le layerK

where the layers o# charge are on parallel plane conductors. Dy supposing

that the areas o# the two plane conductors are ,ery largeK on the scale o#

their separationK the mathematical description 1ecomes ,ery simpleK 1ut

the physical ideas in,ol,ed apply e"ually well to other shapes and areas

o# dou1le charge sheets. 5onsider two conductors ha,ing plane parallel

#aces a small distance l apart and charged with sur#ace charge density o#

magnitude s. The material in the space 1etween the conductors has

permitti,ity e.


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The conductors are suciently close so that the eUects on the non

uni#orm Neld at the edges may 1e neglected. Then the dou1le layer has

the #ollowing properties.

Detween the charge layersK the magnitude o# the electric Neld is

uni#ormK directed #rom the positi,e towards the negati,e chargesK

and o# E 4

s/e E terminates on the charges according to aussbs law.

7ithin the conductorsK the electric Neld is necessarily Gero. 9ach

conductor is an e"uipotential region.

There is a potential diUerence 1etween the two conductors gi,en 1y

V 4 El 4

sle.• >#ten the spacing l is ,ery smallK and we are not ,ery concerned

with the region inside the dou1le layer. En eUect the charge dou1le

layer represents a potential discontinuity 1etween the two

conductors. En #actK whene,er a potential diUerence exists 1etweentwo contiguous pieces o# matter a charge dou1le layer is in,ol,ed.

9xamples include li,ing cellsK 1atteriesK thermocouplesK

semiconductor CunctionsK etc.

Electromagestism Exam

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