Electric charge
Charge and charges
A properties of matter that determines its electromagnetic interaction.
Fundamental particles are truly point-like1 : e.g. electron
known at least since the time of the Greeks.
measured in unit of Coulomb (C).
classical electromagnetism is really inadequate for subatomic treatment of
charges.
Come in two opposite flavors.
By point charge we basically mean a small charged object whose charge
distribution is negligible.
pair-creation and matter-antimatter imbalance question.
Electric charge is quantized.
A “point charge” could consists of large number of charged particles.
Is proton a point charge? Is a neutron neutral?
Any object can be treated as a point-charge far enough away
Point-like is the simplest approximation for any charge density distribution.
fundamental unit charge: e = 1.6 × 10−19 C.
leptons, mesons, and hadrons have integer charge in unit of e.
quarks have charge ±e/3 or ±2e/3, but never free.
current research: fractional charged collective states.
point-like neutral charge objects: dipole, quadrupole, octupole, etc. . .
Some extended objects still have negligible size along one or two
dimensions.
Electric charge is conserved.
The total charge of an isolated system do not change.
Responsible for all the everyday non-gravity forces.
line charge characterized by charge per length λ.
planar charge characterized by charge per area σ.
electromagnetism is 1,000,000,000,000,000,000,000,000,000,000,000,000
times stronger than gravity.
the delicate cancellation of the two charge flavors.
1 as
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Continuous charge density
far as experiments show
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The electric force
Magnitude
Systems studied by classical (i.e. not quantum) electrodynamics don’t
really have point-like charge distribution.
F =k
Real material consists of huge amount of charges over sizable volume.
Electrons in molecule: quantum motion blur out its pointy-ness.
Continuous approximation is perfectly valid.
Generally all the charge in the system can be represented by a continuous
function of position ρ(~r).
q1 q2
2
r12
Applies for two point charges q1 and q2 with distance r12 .
Also a 1/r 2 force like Newton’s law of universal gravity. Coincident?
charge can be positive or negative, mass-energy can only be positive.
amount of charge in a differential volume dV .
k is big, G is tiny — hence the bigness of EM force.
dq = ρ(~r)dV
Direction
amount of charge in a some volume of space D.
same flavors repel, opposite flavors attract.
Z
ρ(~r)dV
Q=
D
Coulomb’s force is a central force: force between charged particles only
depends on the displacement vector between them.
can be adapted to represent point charge, line charge, surface charge
too. . . with mathematical caveat.
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There is no repulsive gravitational force.
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Coulomb’s Law
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Many charges
Principle of superposition
Coulomb’s Law for electrostatic forces
Force on a charge exerted by multiple other charges is simply the vector sum
of the force by each charge.
~1 = F
~ 21 + F
~ 31 + F
~ 41 + · · ·
F
~ 12 = k q1 q2 r̂12
F
2
r12
~ 12 force exerted by charge 1 on charge 2.
F
2
k a proportionality constant: k ≈ 9 × 109 N·m
.
C2
q1 , q2 electric charges. let the sign take care of themselves.
~r12 displacement vector from 1 to 2. ~r12 = r12 r̂12 .
~ 21 = −F
~ 12 because r̂21 = −r̂12
F
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For charge (distribution), discrete sum → integral.
ZZZ
k q ρ(~r 0 )
~ =
F
%̂ dV 0
with
%
~ = ~r − ~r 0
%2
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The electric field
Compute electric field
The electric force on a charge is proportional to the charge. Factor it out:
~
Point charge Q at R:
~
~ r) = kQ %̂ = kQ ~r − R .
E(~
2 %2
~ ~ ~r − R
~r − R
~ = F/q
~
E
~ is due to all other charges in the universe (or
where the electric field E
your system).
Why field? Why the extra abstraction?
EHrÓL
·=rÓ-R
~ to ~r.
%
~ is the displacement vector from R
Works. So at least as good as the force description.
Do not depends on test charge.
Modularized the problem: separate effects of charge from effects on charge.
The field have energy and momentum themselves.
Light and heat are waving electromagnetic field.
The causality question led to Einstein theory of relativity.
Q
R
Lorentz’s force formula:
Ór
y
~ = q(E
~ + ~v × B)
~
F
~
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More than one charge: PoS again
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Electric field of highly symmetric distribution
Multiple discrete charges Qi at ~ri :
X kQi
X kQi ~r − ~ri
~ r) =
E(~
2
2 %̂i =
%
r − ~ri | |~r − ~ri |
i
i
i |~
Field near a point-like charge distribution falls like
∼ r −2 .
Continuous charge distribution ρ(~r 0 ):
Z
Z
Z
kdQ
kρ(~r 0 )dV 0
kρ(~r 0 ) ~r − ~r 0
~ r) =
E(~
dV 0
%̂ =
%̂(~r 0 ) =
2
2
2
%
%
|~r − ~r 0 | |~r − ~r 0 |
·=rÓ-rÓ¢
dQ=ΡHrÓ¢ LdV
Ór¢
EHrÓL
Field near a line-like charge distribution falls like
∼ r −1 .
Ór
Field near a plane-like charge distribution do not
change with distance or falls like ∼ r 0 .
z
x
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Charged object in electric field
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Flux of a vector field
Point charge experience a force in the direction of electric field:
Representation of a flat area by a vector ~a = aâ
~ = qE
~
F
A dipole is a neutral closely bounded pairs of positive and negative
charges of the same magnitude. ~p = q ~d. ~d is from the negative end to a
positive end.
~p = 0
~
~
F
~τ = ~p × E
U = −~p · E
magnitude a: area
direction â: perpendicular to area
Flux of a constant vector field ~f through a flat area is the inner product of
the two vectors:
Φ = ~f · ~a = V a cos θfa
a
Most material belong to one of two classes:
Insulator charges are tightly bound, not free to move.
dielectric: have intrinsic and induced dipole momentum.
polarizability: induced dipole momentum versus applied
field.
Θfa
f
Conductor certain charges are free to move around the inside of the
material.
~
electric field leads to electric current: (~J = σ E).
electric field inside is zero in electrostatic equilibrium.
All charges are in the surface.
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Divergence theorem
Gauss’s Law
Z
~ · d~a = 4πkQinside ∂D
E
ΦE ≡
Flux of a vector field ~f (~r) over an arbitrarily shaped area ∂D: break into
arbitrarily small and flat pieces and add (integrate):
∂D
=
0 ≡
Z
Φ=
~f · d~a
Qinside ∂D
0
1
= 8.85 × 10−12 C2 /N · m2 .
4πk
0 vaccum permittivity. Related to
speed of light.
4π solid angle of a closed volume.
Relates the electric field on any surface to the charge in that surface.
Divergence theorem: flux of a vector field through a closed surface equal
the volume integral of the divergence of the field over the interior of that
volume.
I
ZZZ
~f · d~a =
~ · ~f dV
∇
∂D
D
Direction of d~a defines inside/outside
Geometric interpretation of Coulomb’s Law
1/r d−1 law related to having d spacial dimensions.
Universally true, but useful for finding electric field only in rare cases
with sufficient symmetry.
Making smart choice of Gaussian surface: respect the symmetry.
~
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Work and potential energy
~
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Two equations
Z
WA,
~ B
~ =
~
B
~ r) · d~r
F(~
~
A
The work it takes to move a charge q (from far away) to distance d of a
stationary charge Q:
Z d
Qq
Qq
W (d) =
−k 2 dr = k
r
d
∞
Electrostatics
Important observations:
It takes positive energy to assemble two like charges and negative energy
to assemble two unlike charges.
Electrostatic force is a conservative force:
I
~ =0
~ r) · d~r = 0
~ ×F
∇
or
F(~
~ = ρ(~r)
~ ·E
∇
0
~ =0
~ ×E
∇
Electric field obey Gauss’s Law
Electric (static) field is curl-free/conservative.
~ and B,
~ not the path by which they
W only depends on the end points A
connect.
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