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Dag Gillberg [J]
[J]==[N
[N m]
m]dA
[F]
[F]==[C/V]
[C/V]dA
[Hz]
[Hz]==[s[s 11]] [C]
[V]
[⌦]
[C] =
= [A
[A s]s]
[V] =
= [J/C]
[J/C]
[⌦] =
= [V/A]
[V/A]
= dxdy (cartesian)
Shell
Theorem:
A shell
of
uniform
charge
22
22
[Wb]
=
[T
m
[H]
=
[T
m
/A]
[T]
=
[N/(A
m)]
[Wb]
=
[T
m
[H]
=
[T
m
/A]
[T]
=
[N/(A
= rdrd✓ eV
(polar)
19
Stuff
inattracts
red is not
explained
JJ
eV =
= 1.602
1.602⇥
⇥10
10 19
1)
or repels
an external charge as
a circle:
s10=
[m]
99 2⇡R
33
GG  giga
M
k
kilo
=
10
giga == 10
M  mega
mega =
= 10
1066
k
kilo
=
10
shell’s
charge
were
at its centre;1212
on
the
formula
sheet
2
2 66
33
99
m
 milli
10
µµ  micro
=
nn  nano
pp  pico
mChapters
milli ==A
10= ⇡R2123
micro[m
= 10
10
nano =
= 10
10
pico =
= 10
2) exerts no net electrostatic force on a
2
2
sphere:
A = 4⇡R
[m ]
interior.
Mathematical
Conservation
Mathematical Formulae
Formulae
and Identities
Identities
Conservation of
of E
Emech
in isolated
isolated system
system (Wext
4
3 and
3
mech in
ext = 0):
⇡R
[m
re: (1(1++x)x)nn ==V
=
22
3+n(n
1
+
nx
+
1)x
E
00 1 q2 
1
+
nx
n(n
1)x
/2!+
Emech
= K
K+
+ U
U=
=kq
mech =
• Coulomb’s law 33 /2!+
Coulomb’s
Law:
F
=
[N]
RR ff
2
r
~
~
n(n
n(n 1)(n
1)(n 2)x
2)x /3!
/3!+
+pp...
... (if
(if x
x <
< 1)
1)
W
ss
[J]
Wfforce
= ii F
F ·· d~
d~
[J]
orce =
~
~
tionax
(uncorrelated
errors)
b± bb 4ac
4ac
b±
Electric
Field:theorem:
E = FK
/q==0W
ax22++•bx
bx+
+
c
=
0
has
roots
x
=
c
=
0
has
roots
x
=
WorkKE
theorem:
K
Wnet
WorkKE
net[N/C or V/m
2a
2a
Charge
is
quantized
and
conserved
q
@f
~~ ~
sin(↵±±2 ))=
=sin
sin↵↵cos
cos ±
±cos
cos↵↵sin
sin
sin(↵
~ = E
~
Power:
P
=
dW/dt
=
F
·
~
v
[W]
Power:
P
=
dW/dt
=
F
·
~
v
[W]
point
charge:

E
=
k
and
E
+ ( @y
)
+
...
2
y
r to move a mass
cos(↵•±± ))=
=cos
cos↵↵cos
cos ⌥
⌥sin
sin↵↵sin
sin
cos(↵
!
A
force
is
conservative
if
the
work
!
A
force
is
conservative
if
the
work
Conductors vs nonconductors, ground
~ = /(2
1 issheet:
E
Di↵erential
length:
ds =
= dx
dx (cartesian)
(cartesian)
Di↵erential
length:
ds
or nonconducting
charge between
between two
two points
points
is path
path independent.
or
charge
cz + ...
ds =
= rd✓
rd✓ (polar)
(polar)
ds
~ = /(✏0 )
2
2
2
•
Electric
Field
and
Potential
Electric
Field
and
Potential
Electric
fields
conducting
1
sheet:

E
) +Di↵erential
(b
)
+
(c
)
+
...
y
Di↵erential
area: z
dA =
= dxdy
dxdy (cartesian)
(cartesian)
area:
dA
Shell Theorem:
Theorem: A
A shell
shell of
of uniform
uniform charge:
charge:
Shell
dA
=
rdrd✓
(polar)
Electric
dipole:
dA
=
rdrd✓
(polar)
.
1) attracts
attracts or
or repels
repels an
an external
external charge
charge as if all of the
1)
• Due to charged particle
Circumference
of
a
circle:
s
=
2⇡R
[m]
Circumference
of
a
circle:
s
=
2⇡R
[m]
m y 2
p z 2
twocharge
opposite
shell’s
charge
were at
at charges
its centre;
centre; (±q) separated
shell’s
were
its
x 2
22
22
) Area
+ ( ofofyaa(point
) + ( charge)
)
+
...
Area
circle:
A
=
⇡R
[m
circle:
A
=
⇡R
[m
z
2) exerts
exerts no
no net
net electrostatic
electrostatic force
force on
on a charge in its
2)
dipole
moment:
~
p

=
qd
directed fro
Surface area
area of
of aa sphere:
sphere: A
A=
= 4⇡R
4⇡R22
[m22]]
Surface
[m
interior.
interior.
⇡R33
[m33]]
Volume
sphere:
= 4343⇡R
[m
Volume
aa sphere:
VV =
• ofof
Dipoles
~
~
kq q
q 
kq
~
⌧
=
p
~
⇥
E
U
=
p
~
·
E
uations
Coulomb’s
Law:
F
=
[N]
Coulomb’s Law: F = Hrr
[N]
qenc
~~ =
~~/q
Error
Propagation (uncorrelated
(uncorrelated
errors)
Error
Propagation
errors)
dv
~
~
Electric
Field: E
E
==
F
/q00 E
[N/C
or=
Electric
Field:
F
[N/C
or
V/m]
q
Gauss’
Law:
·
d
A
cceleration:
a
=
=
constant
•q @f
Charged
objects:
point, sheet,
e
✏0
@f
@f dt
qq
@f
2
2
2
2
~
~
~
~
~
point
charge:

E
=
k
and
E
=
=
(
)
+
(
)
+
...
point
charge:

E
=
k
and
E
=

Eˆ
r
=
(
)
+
(
)
+
...
ff
@x xx
@y yy
rr
@x
@y
vcz
=
vsphere,
Electric
Potential:1
= U/q
wire,
disk,
f+
i + at shell, cylinder
~~ =
nonconducting
1V
sheet:
E
E
=0 /(2✏0[V]
nonconducting
sheet:
0)
ax+
+
by
+
+
...
IfIf ff ==ax
by
+
cz
...
p
p
2
22
~~ =
2
2
2
22 +
v
=
v
+(c(c2a
s......
Uconducting
is the electrostatic
potential
energ
conducting
1 sheet:
sheet: E
E
= /(✏
/(✏00)
1
= (a
(a xx)) f+
+(b
(b yy))i +
+
ff =
zz)) +
• Electric
m pp flux and Gauss’1law
m
Electric dipole:
dipole:
Electric
Axnnyq
yq
...
IfIf ff ==Ax
zz ...
sf =mmsi + viptp + 2 at2
potential
di↵erence:
V = by
Vfd; Vi =
two
opposite
charges
(±q)
separated
two
opposite
charges
(±q)
separated
nn
2
2
2
2
2
2
2
=Using
+(( yvy )) +
+(( zz )) +
+...
...
ff (( xx ))symmetries!
+
ff =
q
•
dipole charge:
moment: ~
~
==
qd directed
directed from q to +q
dipole
moment:
ppV
 =
qd
acceleration: ~a = r
point
k
r
~
~
~
= p~p~ ⇥
⇥E
E
U @V
= p~p~ · E
~⌧~⌧ =
U
=
Kinematics Equations
Equations
Kinematics
~
H
H
E
from
V
:
E
=
s
q
dv
dv
~
~
~~ =
Gauss’
Law:
=
E
·
A
= [email protected]
Gauss’ Law: ee = E · ddA
Uniform linear
linear acceleration:
acceleration:
= dtdt =
= constant
constant
aa =
rgyUniform
✏✏
q1 q2
System
2 charges:
U12 [V]
= V1 q2 = k
= vvii +
+at
at
vvff2=
Electricof
Potential:
= U/q
U/q
[V]
Electric
Potential:
VV =
1
22 11
22
22 22 xx yy zz enc
enc 00 If f = ax +
pby + cz + ...
(a x )2 + (b y )2 + (c
f =
m p
If f = Axn yq
z ...
f =f ( nxx )2 + ( m y 2
y ) Chapters
2425
Kinematics
Equations z) 2 + ... + ( pz z )2 + ... a = dv
dt = constant
vf = vi + at
2
2
v
=
v
+ 2a s
i
f
Electric potential
sf = si + vi t + 12 at2
v2
Uniform circular acceleration:
~
a

=
• Equipotential
surfaces
r
Uniform linear acceleration: • • • Work and Energy
Electric
potential energy
Springblock: F = k s,
Usp = 12 k( s)2
> 0 ) energy
transferred
to object by
force
• WFrom
system
of
charged
particles
K = K f Ki
K = 12 mv 2
Ug = mgh
Ignoring dissipative energy losses:
Calculating
Esys = K + U = Wext or Ef = Ei + Wext
U = Uf Uifield
= Wffrom
[J]
• Electric
orce
potential
If external agent does work against force: U = Wext • ~ = /(2✏0 )
nonconducting 1 sheet: E
~ = /(✏0 )
conducting 1 sheet: E
Electric dipole:
two opposite charges (±q) separated by d;
dipole moment: ~
p = qd directed from q to +q
~
~
~⌧ = p~ ⇥ E
U = p~ · E
H
~ · dA
~ = qenc
Gauss’ Law: e = E
✏0
Electric Potential: V = U/q0 [V]
U is the electrostatic potential energy
Rf
~ · d~s
potential di↵erence: V = Vf Vi =
E
i
point charge: V = k rq
~ from V : Es = @V
E
@s
System of 2 charges: U12 = V1 q2 = k q1rq2
System of several charges: U = U12 + U13 + U23 + ... Capacitance: C = q/ VC
[F]
✏0 A
parallelplate capacitor:
C
=
d
P
Potential from electric field
in series: 1/Ceq = P i (1/Ci )
in parallel: Ceq = i Ci
Capacitance
UC = 12 C( VC )2
uE = 12 ✏0 E 2
• Depend only on geometry!
Capacitor with dielectrics: ✏0 ! ✏ = ✏0
dielectric constant: = ✏/✏0
• Ceq in parallel and series
Current: i = dq/dt [A]
• Energy stored in capacitor
Current density: J~ [A/m2 ]
(stored in the electric field of capacitor)
~ = i/A in direction of E
~
J
J = ne evd
• Capacitor with dielectric
ne is # of conduction electrons/m3
vd is drift speed of electrons
Resistance: R = VP
[⌦]
R /i Chapters 2627
• • Current & current density
• Resistance and resistivity
• Ohm’s law & power
Circuits
• Kirchhoff’s laws
• loop = voltage
• junction = current
• The RC circuit
• charging
• discharging
• time constant Capacitance: C = q/ VC
[F]
✏0 A
parallelplate capacitor:
C
=
d
P
in series: 1/Ceq = P i (1/Ci )
in parallel: Ceq = i Ci
UC = 12 C( VC )2
uE = 12 ✏0 E 2
Capacitor with dielectrics: ✏0 ! ✏ = ✏0
dielectric constant: = ✏/✏0
Current: i = dq/dt [A]
Current density: J~ [A/m2 ]
~ = i/A in direction of E
~
J
J = ne evd
ne is # of conduction electrons/m3
vd is drift speed of electrons
Resistance: R = VP
[⌦]
R /i
in series: Req = i RP
i
in parallel: 1/Req = i (1/Ri )
Ohm’s Law ) R independent of VR
R = ⇢L/A
resistivity: ⇢ [⌦m]
conductivity: = ⇢ 1 [⌦ 1 m 1 ]
current density: J = E
Power: P = iV (general case) [W]
if Ohm’s law holds: P = i2 R = V 2 /R
emf: E = dW
[V]
dQ
Kirchho↵ ’s Rules:P
voltage (loop):
Vi = 0
i P
current (junction):
i ii = 0
RC circuit: ⌧ = RC [s]
discharging: q(t) = qmax e t/⌧
charging: q(t) = qmax (1 e t/⌧ )
qmax = CE
Magnetic Field: Chapters 2829
• • Magnetic fields
• Force on moving charge • The Hall effect • Charged particle moves
in circle! What radius? • Force on current carrying wire • Torque on current loop Magnetic field due to currents
• Magnetic flux • Ampere’s law • Force between wires • Solenoid & toroid if Ohm’s law holds: P = i R = V /R
emf: E = dW
[V]
dQ
Kirchho↵ ’s Rules:P
voltage (loop):
Vi = 0
i P
current (junction):
i ii = 0
RC circuit: ⌧ = RC [s]
discharging: q(t) = qmax e t/⌧
charging: q(t) = qmax (1 e t/⌧ )
qmax = CE
Magnetic Field:
~ = µ0 i
long straight wire: B
2⇡ d
(direction tangent to circle, RHR)
solenoid: Bsolenoid = µ0 ni where n = N/l
~ = µ0 id~s2⇥ˆr
BiotSavart Law: dB
4⇡ r
H
~ · d~s = µ0 ienc + µ0 ✏0 d e
Ampere’s Law: B
dt
~
Force from B:
~
moving charge: F~on q = q~v ⇥ B
~
wire current: F~wire = i~l ⇥ B
li1 i2
2 parallel wires: Fk wires = µ02⇡d
(k attract, antik repel)
Magnetic dipoles:
loop’s magnetic dipole moment: µ
~ = N iA
(direction from RHR of I)
~ loop = µ0 2~µ3
magnetic field on axis: B
4⇡ z
~
torque on current loop: ~⌧ = µ
~ ⇥B
~
potential energy: U = µ
~ ·B
Magnetic flux through loop:
R
~
~ [Wb]
m = N loop B · dA Chapters 3031
• Faraday’s and Lenz’ laws
• • Capacitance: C = q/ VC
[F]
parallelplate capacitor:
C = ✏0dA
P
in series: 1/Ceq = P i (1/Ci )
in parallel: Ceq = i Ci
UC = 12 C( VC )2
uE = 12 ✏0 E 2
Capacitor with dielectrics: ✏0 ! ✏ = ✏0
dielectric constant: = ✏/✏0
Current: i = dq/dt [A]
Current density: J~ [A/m2 ]
~ = i/A in direction of E
~
J
J = ne evd
ne is # of conduction electrons/m3
vd is drift speed of electrons
Resistance: R = VP
[⌦]
R /i
in series: Req = i RP
i
in parallel: 1/Req = i (1/Ri )
Ohm’s Law ) R independent of VR
R = ⇢L/A
resistivity: ⇢ [⌦m]
conductivity: = ⇢ 1 [⌦ 1 m 1 ]
current density: J = E Inductor and inductance • RL circuit • Energy stored in magnetic field EM oscillations and AC
• LC circuit • Damped oscillations (RLC) • Forced oscillation Power: P = iV (general case) [W]
if Ohm’s law holds: P = i2 R = V 2 /R
emf: E = dW
[V]
dQ
Kirchho↵ ’s Rules:P
voltage (loop):
Vi = 0
i P
current (junction):
i ii = 0
RC circuit: ⌧ = RC [s]
discharging: q(t) = qmax e t/⌧
charging: q(t) = qmax (1 e t/⌧ )
qmax = CE • Three simple circuits • Power in AC circuits, power factor • Transformers Magnetic Field:
~ =
long straight wire: B µ0 i Faraday’s Law: E = d dtm or E =  d dtm 
direction of induced current such that
~ will oppose the charge in m .
induced B
Induced electric field:
H
~ · d~s = d m
E= E
dt Inductance: L [H] = [Wb/A]
solenoid: L = N i m = µ0 n2 `A
di
di
Ecoil = L dt

VL = L dt
direction from Lenz’s Law
Uinductor = 12 Li2
uB = 2µ1 0 B 2
L
LR circuit: ⌧ = R
[s]
rising current: i(t) = imax (1 e t/⌧ )
falling current: i(t) = imax e t/⌧
imax = E/R
Oscillatory Motion: x(t) = A cos(!t + 0 )
Frequency: f = 1/T
[Hz]
period: T = 1/f
[s]
angular frequency: ! = 2⇡f
[rad/s]
LC Circuit: q(t) = q0 cos(!t + 0 )
1
2
Analogy )
q springblock system (U = 2 kx )
!= k
m for LC circuits: ! = q 1
LC AC Circuit:
p
Impedance: Z = R2 + (XL XC )2
capacitive reactance: XC = 1/(!C) [⌦]
inductive reactance: XL = !L [⌦]
E(t) = Emax sin(!t)
i(t) = imax sin(!t
)
where imax = Emax /Z
Phase: tan
p = (XL XC )/R
p
Irms = imax / p2
Vrms = Vmax / 2
2
Erms = Emax / 2
Pave = Irms
R Kirchho↵ ’s Rules:P
voltage (loop): i PVi = 0
current (junction): i ii = 0
RC circuit: ⌧ = RC [s]
discharging: q(t) = qmax e t/⌧
charging: q(t) = qmax (1 e t/⌧ ) Chapter 3233
• Magnetic Field:
~ = µ0 i
long straight wire: B
2⇡ d
(direction tangent to circle, RHR)
solenoid: Bsolenoid = µ0 ni where n = N/l
~ = µ0 id~s2⇥ˆr
BiotSavart Law: dB
4⇡ r
H
~
Ampere’s Law: B · d~s = µ0 ienc + µ0 ✏0 ddte
~
Force from B:
~
moving charge: F~on q = q~v ⇥ B
~
wire current: F~wire = i~l ⇥ B
li1 i2
2 parallel wires: Fk wires = µ02⇡d
(k attract, antik repel)
Magnetic dipoles:
loop’s magnetic dipole moment: µ
~ = N iA
(direction from RHR of I)
~ loop = µ0 2~µ3
magnetic field on axis: B
4⇡ z
~
torque on current loop: ~⌧ = µ
~ ⇥B
~
potential energy: U = µ
~ ·B
Magnetic flux through loop:
R
~ · dA
~ [Wb]
=
N
B
m
loop Maxwell’s equations
•
•
• • qmax = CE Gauss’ law for magnetic fields
Induced magnetic fields
Displacement current Electromagnetic waves
•
•
•
• Impedance: Z = R2 + (XL XC )2
capacitive reactance: XC = 1/(!C) [⌦]
inductive reactance: XL = !L [⌦]
E(t) = Emax sin(!t)
i(t) = imax sin(!t
)
where imax = Emax /Z
Phase: tan
p = (XL XC )/R
p
Irms = imax / p2
Vrms = Vmax / 2
2
Erms = Emax / 2
Pave = Irms
R Travelling waves: D(x, t) = A sin(kx !t + 0 )
v= f
k = 2⇡/
! = vk
Electromagnetic waves:
E = Emax sin(kx !t)
B = Bmax sin(kx !t)
p
~ B
~
c = 1/ ✏0 µ0
E = cB
E?
~= 1E
~ ⇥B
~
Poynting vector: S
[W/m2 ]
µ0
2
I = Save = Erms
/(cµ0 )
Optics:
Index of refraction: n = c/v
Snell’s Law: n1 sin ✓1 = n2 sin ✓2 Direction of E and B wave components, and the wave
Direction of EM wave Power and intensity & the Poynting vector
Point source, isotropic, plane wave ...
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Full Document
 Spring '11
 ALANSTEELE
 Law, Thermodynamics, mechanics, Charge, Magnetic Field, Electric charge, Qmax

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