Download Capacitance

Equations – Electricity and Magnetism
Circuits
Current: I 
C 0 is the capacitance in a vacuum.
dq
P
 nqvd  
dt
V
 j dA
Ohm’s Law: V  IR 
Inductance
Mutual inductance:
d B
dI
dB
  N
 M
 NA cos 
,
dt
dt
dt


M  B2  B1
I1
I2
d B
dI
 L ,
Self inductance:   
dt
dt
total flux N B
L

current
I
dI
Voltage: V  L
dt
1 2
Energy: E  LI
2
Complex impedance of a resistor: Z R  R
AC voltage: V  V0 eiwt , Vact  Re V 
Width of power resonance curve:  
V
1
1

Complex impedance: Z  0 
I0 C
j C
Impedance in series: ZT  Z1  Z 2
Impedance in parallel:
1
1
1
ZZ


, ZT  1 2
ZT Z1 Z 2
Z1  Z 2
Kirchoff’s Law:
The sum of all voltages around a circuit is 0
A
AC current: I  I 0 ei  t  
Current density: J  nqvd 
Electron gun equation:
I
A
1 2
mv  eV
2
i
,
A
j   E V0 ei t  I 0 ei  t   Z ,
  conductivity
dU
 VI  I 2 R
Power: P 
dt
Charge: Q   I dt
Natural frequency of a circuit:  o 
Quality factor: Q 
R
0
, 
L

1
LC
Capacitance
Q
A
 0
V
d
1
1
1
Combining in series:


Ceq C1 C2
Combining in parallel: Ceq  C1  C2
Capacitance (Parallel plates): C 
dq
d E
 o
 iC
dt
dt
I
dE
Displacement current density: J D  D   o
A
dt
1
Potential Energy: EP   V dq  CV 2
2
U
 qV
Energy density: u 
volume
Dielectric: C   C0
 is the dielectric constant, or relative
permeability.
Displacement current: I D 
R
L
Complex resistance: Z L  i L
Inductance with magnetic material: Lm   L0
Electric
Coulomb’s Law: F 
1
q1q2
 E1 q2
4 0 r 2
Electric field:
F2 
1
qi
E
r̂ 
  V

2 ij
4 0 i  j rij
q2  0
Electric potential:   E  d ,
qj
U
1
V

   E  dl

q0 4 0 j ri
Dipole moment: p  qd   E , where  is the
atomic polarizability
Polarization (number times dipole moment):
P  N p   E0 E ,  E    1
Polarization charge per unit volume     P
1
Energy: U   p  E   D  E dV
2V
Electric torque:   p  E
Energy density:
 E2 E  D
u  0

  0 E0 2 cos 2 kz   t 
2
2
Potential energy:
dW  dU  F  dl  p  E   d
1
Equations – Electricity and Magnetism
Potential difference: E  
Electric flux:
 E  ds
S
Laplace’s Equation (no charge): 2  0

Poisson’s equation:  2  
0
Induced surface charge:  p  P  n̂
  P  F
D   0 E  P   0 E
 D  ds    dV
F
S
V
Magnetism
d B
 
Biot-Savart: B   o   I
 4 
r2
Dipole moment: m  I A
Energy density: u 
Field energy: U 
Torque:   m  B
U
B2

Volume 2 o
LI 2
  P dt
2
Potential energy: U  m  B 
dE    B B
B A
A
E
 
t
 2 A   o j
BH
2
Potential energy: U    m  B
Magntism;  B is magnetic susceptibility.
M

M  N m  B B ; B  0
B
o
Current per unit length of a cylinder; M  is
Overall Magnetism
B0 is magnetism in air
(magnetic cylinder inside solenoid):
B  B0  o M
jB    M
“Magnetic Field” or “Magnetic Intensity”:
B
H
M
0
 H  dl  i
B  0 H
f
1
1  B
 1 for most materials
 1000 in magnetic materials.

Diamagnetism:
B , ve , m 
m   B  field down overall.
Magnitude: ~ 10 5
Paramagnetism
Orbital and spin components of the atom
don’t cancel out. Randomly orientated 
B field applied  alignment  B  . ~ 10 3
Ferromagnetism:
Materials contain “domains” where dipoles
are aligned. Applying a magnetic field aligns
all the domain’s fields. Magnetism remains
after B is removed.
B ~ 1000 .
Hysterisis (history dependant)
Lenz’s Law
“The direction of any magnetic induction effect
is to oppose the cause of the effect”
ElectroMagnetism
Force (plus force on a conductor):
F  q E  v  B  qE   I d  B
E
 0 0 B
Waves in a vacuum (transverse):
2 E
2 B
2 E





0 0
z 2
zt
t 2
E
B

z
t
Electro-: E  E0 cos kz   t   
B
-Magnetism: B  0 cos kz   t 
c
Speed of Light: c 
1

Relationship between E and B : B 

1
k̂  E
c
2 E
t 2
Poynting Vector (energy flux per second):
1
N
E  B  , N  cU A t ,
0
 2 E   0 0
N 
1 E0 2
2 0 c
2

Equations – Electricity and Magnetism
Radiation pressure:
E2 N
Pabsorbed  o o 
2
c
2N
Preflected   o Eo 2 
c
Reflection (perfect conductor):
E incident  x̂Eo cos kz   t  ,
E reflected   x̂Eo cos kz   t 
Linear polarization:
E z,t   x̂Eox cos kz   t   ŷEox cos kz   t 
Circular polarization:
E z,t   x̂Eox cos kz   t   ŷEox sin kz   t 
Photon momentum: p 
E  hf
E
c
FB
v2
2
  o o v  2
FE
c
Maxwell Equations
1) Gauss's Law (Electric flux):
 qi Q
E   E  ds  i

s
o
o

o
  D  F
2) Gauss’s Law (Magnetic Flux):
B   B  dA  BAcos
E 
B  0
3) Ampere’s (corrected) law

B  d  o I enclosed
loop
 o I c  I d 
d E 

 o  I c   o


dt 
1 E
  B  o J  2
c t
D
  H  jf 
t
4)
d B
d
B
E 
t
In free space,   j  0 .
 E  dl 
EM in Materials
2 B
2 B   o o 2
t
2 E
 2 E   o o 2
t
1  n 2
 o o  2  2  2
v
c
c
c ck
n 
 
v 
  1 for most materials
  D  f
B  0
B
E 
t
E
  B  o o
t
sin  c
Snell’s Law:
 n 
sin  v
Incident:
E1  x̂E0 I ei  t  kz
B1  ŷB0 I ei  t  kz
k1    o o
Transmitted:
E1  x̂E0T ei  t  kz
B1  ŷB0T ei t  kz
k1    o o
Reflected:
E1  x̂E0 Rei  t  kz
B1   ŷB0 Rei  t  kz
E I  E R  ET  E0I  E0R  E0T
(continuous at surface)
H I  H R  HT

B0 I
o

B0 R
o

B0T
o
 E0 I  E0 R  E0T
3
Equations – Electricity and Magnetism
E 2 1 n
Reflection coefficient: R  0 R2  
 1  n 
E0 I
Transmission coefficient:
E0T 2 v E0T 2
4n
E0T 2 v
T

n
T
E0 I 2 c E0 I 2
E0 I 2 c
1  n 2
2
EM and conducting media
E
0 J  0 0
t
j  E
2 E
E
 o
2
z
t
i kz   t 
E  E0 e
k   1  i 
0

 1  i 
2

Skin depth
Distance over which amplitude decreases by e
2
(2.72):  
0
EM and Plasma
Electron density ne , temperature Te
  0, j 0
E
t
Fe  me r  eE (collisionless)
j  ner
0 j   0 0
dj
ne2 E
 ner 
dt
me
 2 E   0 0
0 nee
2
me
Plasma frequency
p 
ne e2
 0 me
   p , waves are attenuated.

c
vp 
vg 
k

 p2
1 2

d
c

dk n   dn
d
Optics
nr
ni
1 1 1
Lensmaker’s Equation:  
u v f
Snell’s Law: n1 sin  i  n2 sin  r
h
v f
Lens Magnification: v 
hu
f
4GM
D
Gravitational Lenses:  
,      ls
2
bc
Ds
Stoke’s Parameters:
I  E0 x 2  E0 y 2
Brewster’s Angle: tan  i 
Q  E0 x 2  E0 y 2
U  2E0 k E0 y cos 
V  2E0 x E0 y sin 
Young’s Double Slit d sin  n
Resolution  

d
(due to diffraction)
Refractive index n 
2 E
e2 E


n
0 e
t 2
me
k 2   0 0 2 
Refractive index in a plasma:
  p2 
 p2
n  1  2   1 
 
2 2

If the pulse is observed at two different
frequencies, then Dne can be measured.
D Dn
t 
v
c
c ck

v 
Units
Flux (Weber): 1Wb  1Tm2  1NmA1
Inductance (Henry):
1H  1Wb A1  1V s A1  1s
Magnetism (Gauss): 1G  10 4 T
Magnetism (Tesla):
1T  1N sC 1 m1  1N A1 m1
Magnetic Dipole Moment  :
1N s 2 C 2  1N A2 Wb A1 m1  1T m A1  1H m1
Volt: 1V  1J C 1
4