lecture note #2a chapter 2. electrostatics...

Inha University 1

Chapter 2. Electrostatics (정전기학)

2.1 The Electric Field

2.2 Divergence & Curl of Electrostatic Fields

2.2.1 Field Lines, Flux, and Gauss’s Law

2.2.2 The Divergence of E

2.2.3 Applications of Gauss’s Law

2.2.4 The Curl of E

2.3 Electric Potential

2.4 Work and Energy in Electrostatics

2.5 Conductors

Lecture Note #2A

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2.1 The Electric Field (전기장)

2.1.1 Electrostatics vs. Electrodynamics

“Test” charge

v,a 0and/or 0 v If

“Electrodynamics”

,a 0and/or 0 v If

“Electrostatics”

Total forces acting on the test charge Q =

321 FFFF

(전하의속도, 가속도)

(속도) (가속도)

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2.1.2 Coulomb’s Law

“Test” charge

0 10858mN

The force acting on the test charge Q due to a single point charge q, which is at

rest, =

: the permittivity of free space.

: the separation vector (거리벡터) from the source q’s position r’ to the test charge Q’s position r.

If q and Q have the same sign, the Coulomb force is repulsive (반발력)

If q and Q have the opposite sign to each other, the Coulomb force is attractive (인력)

A “point”

charge

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2.1.3 The Electric Field

Total forces acting on the test charge Q =

QqQqQq

ˆˆˆ4

“Test” charge

: the Electric Field

(전기장)

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2.1.3 The Electric Field

zEEEEE zz

23 220

Total electric field at the point P =

[Example 2.1] Find the electric field at a distance z above the midpoint between two equal charges

(q), a distance d apart.

(Solution)

qzzqqE z

021 xx EE

When z >> d,

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2.1.4 Continuous Charge DistributionsFor a continuous charge distribution, the electric field at the point P =

'dldq'dl

For a line charge distribution, the electric field at the point P =

dlrE rr

Line-charge density :

'da'rdq'da

For a surface charge distribution, the electric field at the point P =

rE ' ˆ'

Surface-charge density :

'dV'rdq'dV

For a volume charge distribution, the electric field at the point P =

rE ' ˆ'

Volume-charge density :

: Coulomb’s LawBack

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2.1.4 Continuous Charge Distributions

For points far from the line (z >> L), the electric field at the points:

'ˆ ˆ

222220

232223220

[Example 2.2]

,ˆ ˆ ' xxzzrr

(Solution) ,ˆ zzr

,22 xz r 22

ˆ ˆ ˆ

In the limit L , the electric field of an infinite straight wire:z

<적분과정은 다음 쪽에>

Inha University 8

Mathematical Information

xcosxsec

dxxsecxtand

xtanxcos

xcosdxxsin

xsindxxcos

xsecxcosxcos

xsinxcos

xsinxtan

xcosxcot

xsinxtan

222222

sindcos

dseczdxtanzx 2

dydxxyxz 2 22

232322

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2.2 Divergence and Curl of Electrostatic Fields

The Electric field from a point charge q located at the origin : rr

High flux (선속) density Strong electric field

Low flux density Weak electric field The Electric field flux through

a surface S :

E cosdaEadE

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2.2.1 Gauss’s Law

The Electric field flux through any closed surface = the total electric charge inside.

Gauss surface

The charge outside the surface has no contribution to the total flux

through the Gauss surface

ˆ sinˆ4

qrddrr

r2 –terms cancels out.

Thus, the integral has no radius dependency.

When there are a bunch of scattered charges inside the

Gauss surface,

qadEadE

Total charge

Gauss’s Law (integral form)

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2.2.1 Gauss’s Law

The Gauss’s Law (a differential form) :

dVEadE 1

where is the charge density, i.e.,

(continued)

Q VdVQ

dVdVE 0

: Gauss’s Law (a differential form)

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2.2.2 The Divergence of the Electric Field

From the Coulomb’s law (see page 8)

dVrrE ''ˆ

From the 3-dimensional Delta function theorem (chapter 1)

rr'dV'r'rrrE

To recover the integral form of the Gauss’s law from the differential form

QdVadEdVE

[Appendix]

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Find the field outside a uniformly charged solid sphere of radius R and

total charge q.

qrEoutside 2

For a Gaussian spherical surface of a radius r > R, the Gauss’s law becomes

[Example 2.3]

qrEddrE

daErdarEadE

(Solution)

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How about the field inside a uniformly charged solid sphere of radius R and

total charge q ? (Assume that the total charge is uniformly distributed over the

entire volume of the sphere.)

Einside

For a Gaussian spherical surface of a radius r < R, the Gauss’s law becomes

VSdVadE

[Example 2.3]

(Solution)

r dd'drsin'rddsinrE

2 dφθdθsin'dr'rdφθdθsinrEr

2 r'dr'rrE

Since ,

rrEinside 3

rqrEinside

qrEoutside

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Application Guides for Gaussian Surfaces

1) Spherical symmetry:

Make your Gaussian surface a concentric sphere.

Use the symmetry property in choosing the Gaussian surface

2) Cylindrical symmetry:

Make your Gaussian surface a coaxial cylinder.

3) Plane symmetry:

Make a thin Gaussian “rectangular box” that straddles

the surface.(표면의아래와위를감싸는직사각형가우스면을선택)

(Assume infinitely long cylinders.)

(Assume infinitely large planes.)

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A long cylinder carries a charge density that is proportional to the distance from

the axis: = ks, for some constant k. Find the electric field inside this cylinder.

sksE ˆ3

For a Gaussian cylinder of length l and radius s, the Gauss’s law becomes

0enclosed

[Example 2.4]

slEdzsdE

daEdaEsdasEadE

22 klsslEs

(Solution)

klslskdzddssk

dzds'ds'ks'dVq

VVenlosed

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An infinite plane carriers a uniform surface charge . Find its electric field.

For a Gaussian rectangular box extending equal distances above and below the plane,

the Gauss’s law becomes

0enclosed

[Example 2.5]

AAEAEAEadE

(Solution)

Aqenlosed

where A is the are of the box surface.

Note that there is no distance dependence

for this infinite plane charge distribution.

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Two infinite parallel planes carry equal but opposite uniform charge densities

. Find its electric field in each of three regions: (i) to the left of both, (ii)

between them, (iii) to the right of both.

022 00

For a Gaussian rectangular box extending equal distances

above and below the plane, the Gauss’s law becomes

[Example 2.6]

(Solution)

In the region (i),

xxxEEE000 22

In the region (ii),x

022 00

In the region (iii),

Inha University 19

2.2.4 The Curl of EFor a point charge q at the origin, the electric field at a distance r from the origin becomes

Let us calculate the line integral of the electric field from a point a to

another point b.

In the spherical coordinates, ˆdsinrˆdrrdrld

When the path is a close case (ra = rb),

When there are many charges, the principle of superposition

states holds (중첩의 원리가 적용) x

.ldE 0

: Stoke’s Theorem

: Curl Theorem

adEldE

21 EEE

02121 EEEEE

: always true for

static charges

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2.3.1 Introduction to Potential

The potential difference between two points a and b is

where is a standard reference point (typically, ).

: Electric Potential (전위)

: The fundamental

theorem of gradients

rldErV

ldEldEldE

ldEldEaVbV

From the fundamental theorem for gradients, Eq. (1.55) in Chapter 1,

aldVaVbV

aTbTldTb

aldEldV

(Integral form)

: Electric Potential(Differential form)

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2.3.2 Comments on Potential

Potential (V) Potential Energy (U) (W)

In the most cases, V(r = ) = 0

rVKldEldEldErVrr

for static charges,Since

E xyzxyz

= 0 = 0

Potential (V) is a scalar quantity.

Potential (V) is independent on the reference coordinates.

aVbVa'Vb'V V'V

Exception) The case of a uniformly charged infinite plane.

??? 22 00

zdzzVz

In real world there is no charged infinite plane.

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2.3.2 Comments on Potential

The total force acting on the test charge Q is Superposition Principle

Unit of potential (V) :

VoltC/JC/mN

“Test” charge

The total electric field acting on the test charge Q is

The total potential acting on the test charge Q is

321 VVVV

rldEEEldErV

: vector sum

: scalar sum

(단위: N)

(단위: N/C)

(단위: Nm/C=Volt)

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qEout 2

'drqldErV

[Example 2.7]

From Gauss’s law, the field inside is

(Solution)

inSin adE

From Gauss’s law, the field outside is

outSout

For a point outside the sphere (r > R), the potential

becomes

For a point inside the sphere (r < R), the potential becomes

'drqldErV

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2.3.3 Poisson’s Equations and Laplace’s Equation

: Poisson’s equation

In region, where there is no charge, so = 0,

From the curl of E relation,

(divergence of E) and 0 E

(curl of E),

02 V : Lapalce’s equation

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2.3.4 The Potential of a Localized Charge Distribution

The electric field from a point charge q at the distance r is

Since V() = 0,

1) A Point Charge

P The linear infinitesimal displacement in the spherical coordinates is

ˆdsinrˆdrrdrld

'drqldEVrV

2) Multiple discrete charges

With the superposition principle

3) A continuous charge distribution

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The Electric Potential & Electric Field of a Localized Charge Distribution

Electric Field

Electric Potential

for the case of V() = 0.

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'd'd'sinR'da 2

[Example 2.8]

Let us set the point P on the z axis. Then,

(Solution)

'cosRzzRr 2222

Then, the potential equation becomes

The infinitesimal surface area on the sphere is

'cosRzzR

'd'sinR

'd'cosRzzR

'd'sinRzV

Let .u'cosRzzR 222 du'd'sinRz 2

Rz'cosRzzR

'd'sin zR

<다른풀이방법은[Example 2.7] 참조>

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[Example 2.8] (continued)

zRzRRz

'cosRzzR

'd'sinRzV

For points outside the sphere, z > R & RzzR 2

For points inside the sphere, z < R & zRzR 2

Since the total charge on the shell is q = 4R2 ,

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2.3.5 Boundary Conditions

[Problem 2.11] case

Relationship among the charge density , electric field and electric potential

Boundary condition at a discontinuity

Field discontinuity at the surface of the conducting sphere

• Let us take a Gaussian surface of very thin rectangular-

type box shape overlapped over the conducting surface.

Gauss’s law for the conducting surface

adE enclosed

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The normal component of the electric field is discontinuous by an amount of /0 at the boundary.

• In the limit of the thickness of the rectangular-box-shaped Gaussian surface close to zero,

AEEadE belowaboveS

(continued -1)

.EE belowabove

• The tangential component of the electric field is always continuous.

.ldE 0

For a very thin rectangular loop covering the surface

.lEElElEldE //

above 0

above EE

.nEEEEEE //

belowbelow

aboveabovebelowabove

where is a unit vector perpendicular to the surface from “below” to “above”n

(2.33)

aldEaVbV

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As the path length shrinks to zero,

• The potential is continuous across any boundary, since

(continued - 2)

• The gradient of V inherits the discontinuity in E, since E = -V,

: the normal derivative of V

V belowabove

belowabove VV (2.34)

(2.36)

(2.37)

abelow

aabovebelowabove ldEldEVV

above EE

nVV belowabove

(2.35)

nEE belowabove

Inha University 32

Next Class

Chapter 2. Electrostatics

2.1 The Electric Field

2.2 Divergence & Curl of Electrostatic Fields

2.2.2 The Divergence of E

2.2.4 The Curl of E

2.4 Work and Energy in Electrostatics

2.5 Conductors

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Appendix

zzzyyyxxxrr ˆ'ˆ'ˆ''

222'''' zzyyxxrr

zzzyyxxz

yzzyyxxy

xzzyyxxx

ˆ '''

ˆ '''ˆ '''1

21-222

21-22221-222

zzzyyxxzz

yzzyyxxyyxzzyyxxxx

ˆ ''''22

1ˆ ''''2

23-222

23-22223-222

23-222'''ˆ 'ˆ 'ˆ ' zzyyxxzzzyyyxxx

Proof of2

32 411

r 1ˆ2

lecture note #2a chapter 2. electrostatics...

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