Download Lecture Notes 01: Introduction/Overview, Coulomb's Law, Electric Field, Principle of Superposition

UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
LECTURE NOTES 1
Introduction:
•
In this course, we will study/investigate the nature of the ELECTROMAGNETIC INTERACTION
(at {very} low energies, i.e. E ~ 0 GeV, {1 GeV = 109 electron volts = 1.602×10−10 Joules}).
The electromagnetic interaction is ONE of FOUR known FORCES (or INTERACTIONS) of
Nature:
1) Electromagnetic Force – binds electrons & nuclei together to form atoms
- binds atoms together to form molecules, liquids, solids. . . .
gases
2) Strong Force – binds protons & neutrons together to form nuclei
3) Weak Force – responsible for radioactivity (e.g. β decay) (weak force important @ high
energies)
4) Gravity – binds matter together to form stars, planets, solar systems, galaxies, etc.
•
•
At the MICROSCOPIC (i.e. QUANTUM) LEVEL (elementary particle physics) the forces of
nature are mediated by the exchange of a “force-carrying” particle e.g. between two “charged”
particles:
mediating force
carrier
• charge B
•charge A
Quantum
Field Theory Force
QED
1) EM
QCD
QWD
QGD
Mass
of force
mediator
Range
of force
mediator
≡ 0.000
∞
2) STRONG
Force
Mediator
single
PHOTON
octet of
Force
Type
attractive &
repulsive
attractive &
3) WEAK
GLUONs
W±, Zo
repulsive
attractive &
repulsive
≡ 0.000
Mw ≈ 80.4 GeV/c2
Mz ≈ 91.2 GeV/c2
attractive
only
≡ 0.000
4) GRAVITY single
GRAVITON
Intrinsic
spin of
force mediator
Charge
associated
w/ force
1
±e
r, g, b
~ 1fm
1
r , g,b
~ 1fm
1
± gW
∞
2
MASS, m
(unquantized)
At high energies, QED & QWD unify to become a single force, known as the ELECTROWEAK FORCE
= Planck’s constant divided by 2π = h/2π = 1.0546 x 10-34 Joule – seconds
mproton= 0.93 GeV/c2 = 1.67262158×10−27 kg
1 fm = 1 femto-meter = 1 Fermi = 10-15meters
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
1
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Pattern of Masses for Fundamental, Spin-½ Matter Particles - Fermions
u
c
t
have electric charge + 2/3
d
s
b
have electric charge −1/3
“doublets” of quarks:
fractional
electric charge
charge!!!
Each quark comes in 3 strong (“color”) charges: red, green, blue
“doublet” of anti-quarks:
2
u
c
t
q = -2/3
d
s
b
q = +1/3
with 3 anti-color charges:
red, green, blue
(i.e. anti-red, anti-green, anti-blue)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Questions:
Why are there 3 generations of quarks & leptons? Internal Quantum #? Why not just one? Are there
more? (seemingly not…)
What physics is responsible for the observed pattern of quark/lepton masses?
Why are there four forces of nature? Why not just one? Are there more forces?
Note that ALL 4 fundamental forces of nature have both electric & magnetic fields!!!
“Magnetic” field arises from motion of “electric” charge in space – relativity & space-time involved
here!
FORCE
“ELECTRIC” FIELD
“MAGNETIC” FIELD
EM
STRONG
WEAK
GRAVITY
EM – electric
chromo–electric
weak–electric
gravito–electric
EM – magnetic
chromo–magnetic
weak–magnetic
gravito–magnetic
Nordvedt Effect
e.g. affects motion of
moon’s orbit around
earth (very small)
no motion/movement
Electric Field – time-averaged field (macroscopic) present for static charges exchanging virtual
quanta associated w/given force
Magnetic Field – time averaged field (macroscopic) arises/associated w/moving charges – motional
effect
Magnetic field arises from motion of charge
Any/all/each of
4 fundamental
forces of nature
any/all/each of
4 fundamental
forces of nature
Magnetic field results from charge + space-time structure of our universe!!
At microscopic level, EM force mediated by (virtual) photons
− two electrically charged particles “know” about each other by exchanging virtual photons.
Virtual photon
• charge e
•charge e
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
3
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Planck’s constant
⎛
⎞
Virtual photons carry linear momentum, pγv ⎜ = h ⎟
λ
γv ⎠
⎝
but have zero total energy:
DeBroglie wavelength
c = speed of light = 3 × 108 m/sec
Eγ2v = pγ2v c 2 + mγ2v c 4 = 0
Real Photons
(e.g. visible light):
Eγ2R = pγ2R c 2 mγ R = 0
If Eγ v = 0, then pγ2v c 2 = − mγ2v c 4
pγ R = h
λ γ , Eγ = hfγ > 0
R
R
R
i.e. pγ = ±imγ v c 2
i= -1 complex!
v
If Eγ v = 0 then: Eγ v = hfγ v = 0 ⇒ fγ v = 0 virtual photons have zero frequency, but have non-zero DeBroglie
wavelength, λγ V > 0 !
FORCE: F = ma (Newton’s 2nd Law)
d mγ v vγ v
Δp
dp
=
=
F =
Δt
dt
dt
(
)
= mγ t
dvγ r
dt
= mγ t a
electric charges emit & absorb virtual photons (lots of them!!!)
− each such photon carries with it momentum, Pγ v
− depending on sign of momentum (emitted/absorbed), a net force will result, acting on each charged particle
+
1
e1+
Opposite charges – attract:
e2+
•e1
Like charges – repulsive: Fe+
•
Fe+
2
−
•
•e2
Fe+ Fe−
1
2
n.b. Your own body can sense virtual photons!!!
o Get your comb out, comb your hair several times - charges up comb via static electricity
o Bring comb near to e.g. hair on your forearm & feel the pull on forearm hairs from electric
charge on comb (works best in winter/dry conditions).
4
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
ELECTROSTATIC FIELDS IN A VACUUM
COULOMB’S LAW
It has been experimentally observed (Charles Augustin Coulomb, 1785) that the net, time-averaged
force (i.e. summed over many, many virtual photons) between two stationary point charges Qa & Qb:
1) Acts along the line joining the two point charges, Qa & Qb (i.e. radial force!)
2) Is linearly proportional to the product of the two point charges, Qa * Qb
(n.b. Force is charge-signed!)
– Net force is repulsive if Qa is same sign as Qb.
– Net force is attractive if Qa is opposite sign as Qb.
3) Is inversely proportional to the square of the separation distance, rab ≡ rb − ra = rab between the
two point charges.
The net force exerted by point charge Qa ON point charged Qb is given by:
F ab = K
Qa Qb
rab2
rˆab (SI UNITS – Newtons)
constant of
unit vector
proportionality (points from Qa at ra to Qb at rb )
rab ≡ rb − ra = rab
rˆab ≡
rab rab
=
=
rab rab
unit vector pointing from point A to point B.
Fab is a radial force, one which points from (to) point A to (from) point B, depending on sign of the
charge product (QaQb)
Qa Qb < 0 is attractive force (F < 0)
Qa Qb > 0 is repulsive force (F > 0)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
5
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
The NET force exerted by point charge Qb ON point charge Qa:
QQ
Fba = K b 2 a rˆba
rba
Fba is radial force, point from (to) point B to (from) point A, depending on sign of charge product
(QaQb)
Qa Qb < 0 is attractive force (F < 0)
Qb Qa > 0 is repulsive force (F > 0)
rba ≡ ra − rb = rba
Fab = K
Qa Qb
r
2
ab
rˆba ≡
rba rba
=
= − rˆab
rba rba
Fba = K
rˆab
Qb Qa
rba2
Now r ab = r ba, since rab ≡ rb − ra = rab
rˆba
and rba = ra − rb = rba
but note that rba = − rab
and/or r ba = − r ab since
( rb − ra ) = − ( ra − rb ) =
rab = − rba
thus, we see that: Fab = − Fba
This is Newton’s 1st Law: For every action, there is equal and opposite reaction.
SI units for electric charge Q: Coulombs (C)
Fundamental unit of electric charge, Qe = 1.602 x 10−19 Coulombs
Question: What is the physics that dictates (specifies/determines) the value of e??
i.e. Why is e = 1.602 x 10−19 Coulombs?
What is K? K =
6
1
4πε o
in SI units
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Coulombs 2
Newton -m 2
⎧ Farads ⎫
⎨=
⎬
⎩ meter ⎭
(Farad is SI unit of
capacitance)
Question: If free space is truly empty, how can it have any measurable physical properties associated
with it???
ε o = electric permittivity of free space = 8.8542 x 10−12
Answer: Free space is NOT empty!!! It is “filled” with virtual particle-anti-particle pairs!!
(e.g. e+-e−, μ+-μ−, q − q , W+W−, etc. pairs) existing for short time(s), as allowed by the
Heisenberg Uncertainty Principle – can “violate” energy (momentum) conservation only for time
interval Δt ≤ / ΔE (and over a distance of Δx ≤ / Δpx ).
ε0 is the macroscopic, time-averaged (over many many such virtual pairs) electric permittivity of
(quantum) vacuum - the physical vacuum behaves like a dielectric medium!!!
Thus: Fab =
1 Qa Qb
4πε o rab2
rˆab
Fab
ẑ
QA r ab
QB
A
B
ra
rb
ϑ•
r ab = rb − ra = rab
ŷ
x̂
Factor of 4π = “flux factor” for solid angle associated with flux of virtual photons emitted by point
charge!!! Virtual photons “emitted” from QA are emitted into 4π steradians @ point A:
γv
γv
γv
γv
QA
Force decreases as
1
r2
γv
Just like/analogous to real
γv
Photons emitted from e.g.
100 watt light bulb - Intensity
decreases as 1 2 from light source.
r
•
γv
A
γv
γv
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Note the similarity between Coulomb’s Law and Newton’s Law of Gravity:
⎛ 1 ⎞ Qa Qb
MaMb
rˆ
FC = ⎜
⎟ 2 rˆ ↔ FG = GN
r2
⎝ 4πε o ⎠ r
Newton’s constant, GN = 6.673 x 10−11m3kg -1s -2
Or can define:
Define:
1
1
1
GE ≡
GN ≡
ε og ≡
g
4πε 0
4 ε0
4π GN
FC = GE
Qa Qb
r
2
rˆ
⎛ 1 ⎞ MaMb
then FG = ⎜
rˆ
g ⎟
2
⎝ 4πε o ⎠ r
Coulomb’s
Constant
Coulomb’s Law
FC =
1 Qa Qb
4πε o
r2
rˆ
“nothing”
Note that if dielectric properties of free space (vacuum) were different than they are, then Coulomb’s
Law, i.e. the force between electrically charged particles would be different. Consider a universe in
which we could change the EM properties of the vacuum at will:
im ( ε o → 0 ) : FC → ∞ !! “strong” electromagnetism
im ( ε o → ∞ ) : FC → 0 !! “weak” electromagnetism
(assuming this doesn’t also affect value of fundamental electric charge, e)
Note further/we shall see that: c = speed of light = 1
ε o μo
= 3 x 108 m / sec
μo = magnetic permeability of free space = 4π x 10−7 Newtons/Ampere
1 Ampere of electric current = 1 Coulomb/sec (I = dQ/dt)
Thus:
im ( ε o → 0 ) ⇒ c → ∞⎫⎪
⎬ If μo is unchanged
im ( ε o → ∞ ) ⇒ c → 0 ⎪⎭
8
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
THE ELECTRIC FIELD E (Vector Quantity!!)
(Also known as the Electric Field Intensity)
We’ve introduced/discussed the net/time averaged force, F e.g. of Qa acting on Qb:
1 Qa Qb
rˆab
Fab =
4πε o rab2
We now introduce the concept of a net/time averaged electrostatic field,
Ea , due to Qa, at a (separation) distance, rb − ra from Qa (i.e. at Qb), which is defined in terms of the
ratio of the net/time averaged force Fab ( rb ) to the strength of the test charge Qb used as a probe:
Ea ( rb ) ≡ Fab ( rb ) Qb
Fab ( rb ) = Qb Ea ( rb )
or:
Fab
ẑ
Qb • B
Point A is known as source point
rab
Qa • A
Qa is known as source charge
ra
electrostatic force &
electrostatic field evaluated
rb
at point B = “field point”
•
ŷ
O
Qb is known as test charge
x̂
ra points from the local origin, O to point A where the source charge Qa is located.
rb points from the local origin, O to point B where the test charge Qb is located.
rb points from the local origin, O to point B where the electric field (net/time averaged)
due to Qa is to be evaluated (i.e. by experimentally measuring Fab , and knowing (apriori) Qa and Qb).
Fab ( rb ) = Qb Ea ( rb ) =
Then:
1 Qa Qb
4πε o r
Ea ( rb ) =
2
ab
1 Qa
4πε o r
2
ab
rˆab =
rˆab =
1
Qa Qb
( rb − ra )
Qa
( rb − ra )
4πε o rb − ra 3
1
4πε o rb − ra 3
Very often, we will be considering situations in electrostatics where we use one charge, QT to TEST
for the presence/existence of “source” charge(s) qs.
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
9
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
We want to know e.g. the electric field due to qs, a separation distance, r from it:
vector r ≡ ( r − r ′ ) with magnitude: r = r − r ′
Source Point, S ( @ r ′ )
ẑ
qs
n.b.
r
r′
r
Origin, O
ŷ
Field Point, P (@
r)
QT
FT ( r ) = Force on test charge QT (at field point r ),
a separation distance r from source charge
qs (located at source point, r ′ ):
FT ( r ) =
1 QT qs
4πε o r
2
rˆ =
1
QT qs
4πε o r − r ′ 3
( r − r ′)
x̂
primed quantities (e.g. r ′ ) always refer to source (charge) distribution.
unprimed quantities (e.g. r ) refer to field/observation point.
E (r ) = Electrostatic field ( @ point r ) due to source charge qs a distance r = r − r ′ away from qs:
⎛ 1 ⎞ qs
⎛ 1 ⎞ qs ( r − r ′ ) ⎛ 1 ⎞ qs ( r − r ′ ) ⎛ 1 ⎞ qs
E (r ) = ⎜
r − r′)
=⎜
=⎜
⎟ 2 rˆ = ⎜
⎟ 2
⎟
⎟
2
3 (
⎝ 4πε o ⎠ r
⎝ 4πε o ⎠ r r − r ′ ⎝ 4πε o ⎠ r − r ′ r − r ′ ⎝ 4πε o ⎠ r − r ′
cumbersome notation, but very explicit!!!
FT ( r ) = QT E ( r )
Obviously, SI Units of E ( r ) are Newtons / C (also ≡ volts m)
Units of E = force per unit charge (N/C)
from dimensional analysis
10
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
A Detail:
⎛ F (r ) ⎞
A more rigorous definition of electric field intensity, E ( r ) is given by: E ( r ) ≡ im ⎜
⎟⎟
QT → 0 ⎜ Q
⎝ T ⎠
We really do need this limiting process – experimentally/in real life, the presence of a finite-singed test
charge QT necessarily perturbs the source charge distribution that one is attempting to measure!! This
is especially true for spatially-extended source charge distributions. As the test charge is made smaller
and smaller, the perturbing effect on the original/unperturbed source charge distribution is made
smaller and smaller. In the limit QT → 0, the true source charge distribution is obtained.
THIS IS VERY IMPORTANT TO KEEP THIS IN MIND!!! IT IS NOT A TRIVIAL POINT!!!
Usually, we might think of e.g. QT = 1 e and e.g. qs = 1019 e, thus qs >> QT, and thus perturbing effects
are negligible (in this case).
We have shown that: E ( r ) ≡
F (r )
1 qs
1 qs QT
=
rˆ and thus: F ( r ) =
rˆ = QT E ( r )
2
4πε o r
4πε o r 2
QT
If F ( r ) is a radial force ⎫⎪
⎬ for point source charge, qs
then E ( r ) is also radial ⎪⎭
Convention: direction of electric field lines for qs = +e and qs = −e
•
•
qs = −e
inward
qs = +e
outward
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
11
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
ELECTRIC FIELD LINES
Associated with Two Point Charges
Equal but opposite charges
Figure 2.13
Equal charges
Figure 2.14
12
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
THE PRINCIPLE of LINEAR SUPERPOSITION
-VERY IMPORTANT⎛ F (r ) ⎞
Assuming we are always in im ⎜
⎟⎟ (i.e. QT << qs) regime, then suppose we have N discrete point
QT → 0 ⎜ Q
⎝ T ⎠
source charges: q1 , q2 , q3 , q4 … qN
What is the (total or net) force, FToT ( r ) due to all of the N source charges?
N
Vectorially, we know that FToT ( r ) = F1 ( r ) + F2 ( r ) + F3 ( r ) + … FN ( r ) = ∑ Fi ( r ) . More explicitly:
i =1
N
FTOT ( r ) = F1 ( r ) + F2 ( r ) + F3 ( r ) + … EN ( r ) = ∑ Fi ( r )
i =1
⎧ Q ⎫⎧ q
⎫ Q
q
q
q
= ⎨ T ⎬ ⎨ 12 rˆ1 + 22 rˆ2 + 32 rˆ3 + … N2 rˆN ⎬ = T
r2
r3
rN
⎩ 4πε o ⎭ ⎩ r1
⎭ 4πε o
where: rˆ ≡ ( r − ri ) = ri
N
qi
rˆ
∑
2 i
i =1 ri
What is (total or net) electric field intensity, ETOT ( r ) due to all of the N source charges?
We know that:
FTOT ( r ) = QT ETOT ( r ) or: ETOT ( r ) ≡ FTOT ( r ) QT
N
ETOT ( r ) = E1 ( r ) + E2 ( r ) + E3 ( r ) + … EN ( r ) = ∑ Ei ( r )
∴
i =1
⎧ 1 ⎫ ⎧ q1
q3
qN ⎫
q2
1
=⎨
⎬ ⎨ 2 rˆ1 + 2 rˆ2 + 2 rˆ3 + … 2 rˆN ⎬ =
r2
r3
rN
⎩ 4πε o ⎭ ⎩ r1
⎭ 4πε o
N
qi
i =1
i
∑r
2
rˆi
We can extend the use of the principle of linear superposition to mathematically describe the net/total
force + net/total electric field intensity at the field point, r for arbitrary continuous charge
distributions:
Q
1 ⎛ 1 ⎞
⎛ 1 ⎞
FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ dqs and ETOT ( r ) =
⎜ rˆ ⎟ dqs
4πε o ⎝ r ⎠
4πε o ∫ ⎝ r 2 ⎠
where:
r = ( r − r ′ ) , r = r − r ′ , rˆ = r r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
13
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Then for volume, surface & line charge source distributions:
A.)
VOLUME CHARGE DISTRIBUTIONS: Volume Charge Density, ρ ( r ′ ) :
(e.g. inside cylinders, spheres, boxes, etc.)
Q
⎛ 1 ⎞
dqs = ρ ( r ′ ) dτ ′ : FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ ρ ( r ′ ) dτ ′
4πε o v ⎝ r ⎠
Coulombs/m3
B.)
ETOT ( r ) =
1
4πε o
⎛ 1
∫ ⎜⎝ r
2
v
⎞
⎠
rˆ ⎟ ρ ( r ′ ) dτ ′
SURFACE CHARGE DISTRIBUTIONS: Surface Charge Density, σ ( r ′ ) :
(e.q. on surfaces of cylinders, spheres, boxes, etc.)
Q
⎛ 1 ⎞
dqs = σ ( r ′ ) da′: FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ σ ( r ′ ) da′
4πε o S ⎝ r ⎠
Coulombs/m2
ETOT ( r ) =
where:
14
1
4πε o
⎛ 1
∫ ⎜⎝ r
S
r = ( r − r′) ,
2
⎞
⎠
rˆ ⎟ σ ( r ′ ) da′
r = r − r ′ , rˆ = r r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
C).
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
LINE CHARGE DISTRIBUTIONS: Linear Charge Density, λ ( r ′ ) :
(e.q. wire)
dqs = λ ( r ′ ) d ′ : FTOT ( r ) =
Coulombs/m
where:
ETOT ( r ) =
QT ⎛ 1 ⎞
⎜ rˆ ⎟ λ ( r ′ ) d ′
4πε o C∫ ⎝ r 2 ⎠
1
4πε 0
r = ( r − r′) ,
⎛ 1
∫ ⎜⎝ r
C
2
⎞
rˆ ⎟ λ ( r ′ ) d ′
⎠
r = r − r ′ , rˆ = r r
Thus, a complete description of all possible charge distributions, consisting of discrete and continuous
charge distributions:
FTOT ( r ) =
QT
4πε o
⎧⎪ N qi
⎛ ρ ( r′) ⎞
⎛ σ ( r′) ⎞
⎛ λ ( r ′) ⎞
⎨∑ 2 rˆi + ∫ ⎜ 2 rˆ ⎟ dτ ′ + ∫ ⎜ 2 rˆ ⎟ da′ + ∫ ⎜ 2 rˆ ⎟ d
r
r
r
⎪⎩ i =1 r i
⎠
⎠
⎠
V⎝
S⎝
C⎝
⎫⎪
′⎬
⎪⎭
ETOT ( r ) =
⎛ ρ ( r′) ⎞
⎛ σ ( r ′) ⎞
⎛ λ ( r′) ⎞
1 ⎪⎧ N qi
⎨∑ 2 rˆi + ∫ ⎜ 2 rˆ ⎟ dτ ′ + ∫ ⎜ 2 rˆ ⎟ da′ + ∫ ⎜ 2 rˆ ⎟ d
r
r
r
4πε o ⎩⎪ i =1 r i
⎠
⎠
⎠
V⎝
S⎝
C⎝
⎪⎫
′⎬
⎭⎪
Please Note:
For all integrals (above), when integrals over dτ ′ , da′, and/or d ′ are carried out, FTOT ( r ) and thus
ETOT ( r ) have NO r ′ (i.e. source-position) dependence - it has been integrated over/integrated out!!!
FTOT ( r ) and ETOT ( r ) ≡ FTOT ( r ) QT are functions of the field point variable r ONLY
i.e. they are not functions of r ′ (source point{s}) !!!
PLEASE work/grind through example 2.1 Griffiths p. 62-63) on your own to better learn/understand
this!
ACTIVE “LEARNING BY DOING”
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
15
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
EXAMPLE 2.1 p. 62 Griffiths
- very explicit detailed derivationFind the electric field intensity, E(r) a distance z above mid-point of a straight line segment of length
2L, which carries a uniform line charge λ (Coulombs/meter) (n.b. QTOT = 2λ L )
λ ( r ′)
1
Here, E ( r = zzˆ ) =
rˆd ′
4πε o C∫ r 2
Notice the symmetry of this problem – contribution to net E ( r ) @ field point, P from infinitesimal
line charges d λ associated with infinitesimal line segments, dL located at ± x such that x̂
components of net electric field @ field point, P cancel each other:
cos θ =
sin θ =
z
r
x
r
=
=
z
x2 + z 2
x
x2 + z 2
dENET ( r = zzˆ ) = dE + ( r = zzˆ ) + dE − ( r = zzˆ )
⎧⎪⎛ 1
= ⎨⎜
⎪⎩⎝ 4πε o
⎫⎪
⎞ ⎛ λ dL ⎞
+
⎟ ⎜ 2 ⎟ ⎡⎣( − sin θ xˆ ) + ( cos θ zˆ ) ⎤⎦ ⎬ ← dE ( r = zzˆ )
⎪⎭
⎠⎝ r ⎠
⎧⎪⎛ 1
+ ⎨⎜
⎪⎩⎝ 4πε o
⎫⎪
⎞ ⎛ λ dL ⎞
−
⎟ ⎜ 2 ⎟ ⎡⎣( + sin θ xˆ ) + ( cos θ zˆ ) ⎤⎦ ⎬ ← dE ( r = zzˆ )
⎪⎭
⎠⎝ r ⎠
⎛ 1 ⎞ ⎛ λ dL ⎞
=2 ⎜
⎟ ⎜ 2 ⎟ cos θ zˆ
4
πε
⎠
o ⎠⎝ r
⎝
16
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 1
Prof. Steven Errede
Now only need to integrate this expression over x from 0 ≤ x ≤ L :
{
}
L
L ⎛
1
+
−
ˆ
ˆ
=
+
=
=
dE
r
zz
dE
r
zz
(
)
(
)
∫0
∫0
∫0 2 ⎜⎝ 4πε o
0
⎤⎡
L⎡
⎤
⎛ 1 ⎞
1
z
⎢
⎥⎢
= 2⎜
λ
⎟ ∫0
⎥ dx zˆ
2
2
⎢⎣ ( x + z ) ⎥⎦ ⎣ x 2 + z 2 ⎦
⎝ 4πε o ⎠
ENET ( r = zzˆ ) = ∫ dE ( r = zzˆ ) =
L
L
⎞ ⎛ λ dL ⎞
⎟ ⎜ 2 ⎟ cos θ zˆ
⎠⎝ r ⎠
⎛ 2λ z ⎞ L
1
=⎜
dx zˆ
⎟ ∫0 2
2 3/ 2
4
πε
o ⎠
⎝
(x + z )
⎤
⎛ 2λ z ⎞ ⎡
x
=⎜
⎟⎢ 2 2
⎥
2
⎝ 4πε o ⎠ ⎣ z x + z ⎦
L
0
⎛ 2λ ⎞ ⎡ z
zˆ = ⎜
⎟⎢ 2
⎝ 4πε o ⎠ ⎣ z
⎤
⎞
⎛ 2λ L ⎞ ⎛
1
⎟ zˆ
⎟⎜
⎥ zˆ = ⎜
2
2
L +z ⎦
⎝ 4πε o ⎠ ⎝ z z + L ⎠
L
2
2
2
⎛ ε⎞
⎛L⎞
If z >> L; then (Taylor Series Expansion) 1 + ε ≈ ⎜1 + ⎟ ≈ 1 for ε = ⎜ ⎟
1
⎝ 2⎠
⎝z⎠
Q
2λ L
ENET ( r = zzˆ ) ≈
zˆ = TOT 2 zˆ ← same E-field as that due to a point charge, q!
2
4πε o z
4πε o z
If L → ∞ (i.e. infinite straight wire): use the same Taylor series expansion, but for L z :
⎛ ε⎞
⎛z⎞
i.e. 1 + ε ≈ ⎜1 + ⎟ ≈ 1 for ε = ⎜ ⎟
⎝ 2⎠
⎝L⎠
Then: ENET ( r = zzˆ ) ≈
2
1
2λ
λ
zˆ =
zˆ
4πε o z
2πε o z
1
The E-field is actually in the radial ( ρ̂ ) direction for an infinite straight wire – in cylindrical
coordinates:
ENET ( r ) ≈
1
2λ
4πε o ρ
ρˆ =
λ
ρˆ
2πε o ρ
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
17