Electrostatics
Electrostatic Energy
Electric charge create electric field
~ = kq Q r̂ = kq Q ~r dE
~ = kq dQ r̂ = kq Q ~r
E
r2
r3
r2
r3
Electric charge in electric field experience force, electric dipole experience
no net force, but a torque
~ = qE
~ ~τ = ~p × E
~
F
Potential energy difference;
kq Q
kq dQ
ϕ(~r) =
dϕ =
∆PE = q [ϕ(b) − ϕ(a)]
r
r
Energy it takes to assemble system of charges
Move one charge in at a time in any order.
First charge is free.
U = U12 + U13 + U23 + U14 + U24 + U34 + · · ·
Direction of dipole points from negative charge to positive charge.
Alternative: sum over energy of each pair (don’t double count).
Gauss’s Law: the electric flux on some closed surface depends only on
how much charge it encloses.
Z
~ · dA
~ = Qins ∇
~ = ρ
~ ·E
ΦE = E
0
0
U =
What is electric field everywhere of sphere, ball, plane, line, cylinder?
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Current, resistor, capacitor
Alternative 3: For each charge, sum of energy with all other in place; ÷2:
1 X X kq qj
U =
qi
2
dij
i
Energy is stored in electric field:
0
uE = E 2
2
LMZ (UCSD)
uE dV
Reivew
Q=CV
U =
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1
1
1
=
+
+ ···
Rpl
R1
R2
Rpl
V =L
R1 R2
=
R1 + R2
U =
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L I2
2
Time constant: τC = RC
Charging: I = I0 e −t/τC , VC = V0 (1 − e −t/τC )
Discharging: I = I0 e −t/τC , VC = V0 e −t/τC
Capacitor network:
Inductor behaves like break at start, short after long time
Cpl = C1 + C2 + · · ·
Time constant: τL = L/R
Charging: I = I0 (1 − e −t/τL ), VL = V0 (1 − e −t/τL )
Dischargeing I = I0 e −t/τL ), VL = V0 e −t/τL
Series capacitor: same charge, voltage add
Parallel capacitor: same voltage, charges add
Reivew
dI
dt
Capacitor behaves like short at start, break after long time.
Series resistor: same current, voltage add
Parallel resistor: same voltage, current adds
1
1
1
=
+
+ ···
Csr
C1
C2
QV
C V2
Q2
=
=
2
2
2C
Inductor stores energy in magnetic field
Resistor network:
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Static magnetic field
LMZ (UCSD)
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Magnetic induction
Moving charge/current creates magnetic field:
~
~ = µ0 Id l × r̂
dB
4π r 2
Ampère’s law allows one to find magnetic field due to symmetric current
distribution:
I
~ · d~I = µ0 Ienc ∇
~ = µ0~Jenc
~ ×B
B
Z
f =
qB
2πm
No magnetic monopole. North and south poles always come in pair:
~ = 0.
~ ·B
∇
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~ · dA
~ ΦB = B A cos θ
B
(if uniform field)
Inductance is magnetic flux to current ratio:
ΦB,2
ΦB,1
M21 =
L=
I1
I1
A steady current in an inductor makes a magnetic field, stores energy:
Z
1 2
uB =
B
UB = uB dV
2µ0
Charged particle undergoes circular cyclotron motion in uniform field:
mv
qB
If magnetic flux is increasing[decreasing], magnetic field of induced current
is in the opposite[same] direction.
ΦB =
~ = q~v × B
~ F
~ = I~l × B
~ ~τ = m
~
~ ×B
F
r=
Changing magnetic field/flux induces electric field/EMF.
I
~
~ · d~l = − dΦB ∇
~ = ∂B
~ ×E
E= E
dt
∂t
The − sign denotes that direction of induce current opposite change in
flux.
Magnetic flux:
What the magnetic field of a wire, cyliner, solenoid, toroid?
Moving charge/current experience a force in magnetic field. Magnetic
dipole feels a torque.
LMZ (UCSD)
Z
UE =
Capacitor stores energy in electric field.
Ohm’s law: relates potential difference/electric field to current/current
density.
~J = σ E
~ V =IR P=IV
LMZ (UCSD)
j6=i
RL and RC circuit
Current is motion of charge (charge per time). Distributed over some
area: current density (charge per time per length2 ).
Rsr = R1 + R2 + · · ·
dij
i,j<i
Electrostatic field is conservative, we can have something called potential:
Z ~r
~ x) · d~x E
~ = −∇ϕ(~
~ r)
ϕ=−
E(~
LMZ (UCSD)
X k q qi qj
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LMZ (UCSD)
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AC Circuit
Source geometry and field
Impedance is complex:
ZR = R
ZC =
1
iωC
Geometry
ZL = iωL
Dipole
Can be treated as resistance in the following ways:
Allows use of resistance addition formulae for series and parallel
connection:
Zserie = Z1 + Z2 + · · ·
Point, Sphere
1
1
1
=
+
+·
Zparallel
Z1
Z2
Line, Cylindrical (outside)
The magnitude of the impedance related peak voltage and peak current
like Ohm’s law:
|Z | =
p
(Re Z )2 + (Im Z )2
Plane, sheet, slab (outside)
Vpeak = |Z |Ipeak
The phase of the impedance is how much in phase voltage is ahead of
current.
φZ = arctan
LMZ (UCSD)
Im Z
Re Z
V (t) = Vpeak sin(ωt)
I (t) = Ipeak sin(ωt − φZ )
1 E-field
2 B-field
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Electric field
qd
20 (x 2 + d 2 )3/2
q
4π0 r 2
λ
2π0 r
σ
20
Mangetic field
1
µ0 (I a 2 )
2(z 2 + a 2 )3/2
2
D.N.E.
µ0 I
2πr
µ0 Js
2
Inside shell
∼0
∼0
Inside solid
∼r
∼r
along the x-axis of a dipole consists of charge ±q posited at y = ±d/2.
along the center axis, distance z away from center
LMZ (UCSD)
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