Constants:
Electric Potential (static case!):
9
≡
ke
o
=
8.85 × 10
µo
≡
4π × 10
2
=
1/µo o
c
2
1/4πo = 8.98755 × 10 N · m · C
−12
−7
2
C /N · m
−2
∆V = VB − VA =
2
Vpoint = ke
T · m/A
C
−31
kg
−27
kg
e
=
1.60218 × 10
me−
=
9.10938 × 10
mp+
=
1.67262 × 10
NA
=
6.022 × 10
23
g
=
9.81 m/s
B
~ · d~l
E
A
dq
r
continuous
dV
~ = −∇V
~
→E
dx
q1 q2
= ke
= V1 q2 = V2 q1
r12
X ke qi qj
= sum over unique pairs =
rij
pairs ij
Ex = −
Upair of point charges
things/mol
Usystem
2
Ufield =
1
2
V = ke
2
0
=
ax + bx + c =⇒ x =
~
Fcentr
=
−
−b ±
√
b2
− 4ac
2
o E dVol =
Basic Equations:
2
q
r
V = ke
−19
∆U
=−
q
dq
r
1
2
ρV dVol
continuous
dV
~ = −∇V
~
→E
dx
~ x| cos θ = −qEx ∆x
∆P E = q∆V = −q|E||∆~
Ex = −
2a
mv 2
r̂ Centripetal
r
↑ constant E field
Electric Force & Field:
Other:
q1 q2
~
~1
F12 = ke 2 r̂ = q2 E
r
q
1
~1 = ~
E
F12 /q2 = ke 2 r̂
r
X qi
ρr̂
dq
~ = ke
E
r̂ = ke
dVol
r̂
→
ke
i
2
2
ri
r
r2
i
“
~2 − E
~1
E
”
· n̂ = 4πke σ
Fsheet
sheet of charge with σ
σ
(E1 + E2 )
=
2
Current & Resistance:
Capacitors:
dQ
uniform J
~
~ −
= nqAvd
J · dA
−−−−
→I =
dt
X
I
uniform J
J =
nk qk vk −
−−−−
→J =
= nqvd
A
k
I =
S
Qcapacitor
=
Cparallel plate
=
C∆V
Ceq, par
=
0 A
d
1
Q2
2
Q∆V =
= C (∆V )
2
2C
C1 + C2 + C3 + · · ·
1/Ceq, series
=
1/C1 + 1/C2 + 1/C3 + · · ·
Cwith dielectric
=
κCwithout
I
=
dQ/dt = CdV /dt
Ucapacitor
=
~
~ =−d
J · dA
ρ dVol
dt V
S
%l
R=
ρ = 1/σ
A
qτ ~
~
vd =
E τ = scattering time
m
m
ρ = 1/σ =
nq 2 τ
κair = 1
R = V /I
RC circuits
E = %J
QC (t)
=
h
i
−t/τ
Q0 1 − e
−t/τ
QC (t)
=
Q0 e
Q(t)
=
C∆V (t)
τ
=
RC
Ohm
orJ = σE
P = dU/dt = I∆V
charging
Req = R1 + R2 + . . .
discharging
Ohm
power
series
1/Req = 1/R1 + 1/R2 + . . .
X
X
Iin =
Iout junction
X
∆V = 0 loop
parallel
closed path
ac Circuits
τ
=
L/R RL circuit
τ
=
RC RC circuit
1
“resistance” of a capacitor for ac
2πf C
XC
=
XL
=
ωcutoff
=
EM Waves:
2πf L “resistance” of an inductor for ac
1
= 2πf
τ
1
c
=
I
=
I
=
~
|E|
~
|B|
»
–»
–»
–
photons
energy
1
time
photon
Area
λf =
energy
Emax Bmax
power (P)
E2
=
=
= max
time · area
2µ0
area
2µ0 c
Magnetism:
Vectors:
~
~
FB = q~
v×B
|~
F| =
~
d~
FB = Id~l × B
~ =
dB
~ =
B
B=
FB
=
l
Fx2 + Fy2 magnitude
»
–
−1 Fy
θ = tan
direction
Fx
I-carrying wire
µo Id~l × r̂
wire
4π
r2
µ0 I
θ̂ loop
2r
µo N I
= µo nI solenoid
l
µo I1 I2
2 wires
2πd
r̂ = ~
r/|~
r|
d~l = dx x̂ + dy ŷ + dz ẑ
let ~
a = ax x̂ + ay ŷ + az ẑ and ~
b = bx x̂ + by ŷ + bz ẑ
~
a·~
b =ax bx + ay by + az bz =
~ with ~
~ I loop
~
τ =~
µ×B
µ = IA
Ufield
1
=
2µo
dipole
2
B dVol
Induction & Maxwell
~ = |E|l
~ motional voltage
∆V = |~
v||B|l
L = N ΦB /I
∆VL = −LdI/dt
2
L = µo N A/l solenoid
“
”
−t/τ
I = (∆V /R) 1 − e
I = (∆V /R) e
−t/τ
τ = L/R
τ = L/R
LR close
LR open
1
2
U = LI
2
M12 = N2 Φ12 /I1 = M21 = N1 Φ21 /I2 = M
mutual
~ · d~l = − dΦB
E
dt
∆V =
∆V = Blv
motional
~ · dA
~ = 4πke qencl = qencl
E
o
ΦE =
~ · dA
~ =0
B
Derived unit
Symbol
equivalent to
newton
joule
watt
coulomb
V
farad
ohm
tesla
electron volt
-
N
J
W
C
W/A = m2 ·kg/·s3 ·A
F
Ω
T
eV
1 T · m/A
1 T · m2
1 N/C
kg·m/s2
kg·m2 /s2 = N·m
J/s=m2 ·kg/s3
A·s
Power
Prefix
Abbreviation
−12
pico
nano
micro
milli
centi
kilo
mega
giga
tera
p
n
µ
m
c
k
M
G
T
10
10−9
10−6
10−3
10−2
103
106
109
1012
∆V2 = −M dI1 /dt − LdI2 /dt
ai bi = |~
a||~
b| cos θ
|~
a×~
b| = |~
a||~
b| sin θ
˛
˛
˛ x̂
ŷ
ẑ ˛˛
˛
˛
˛
~
~
a × b = ˛ax ay
az ˛ = (ay bz − az by ) x̂ + (az bx − ax bz ) ŷ + (ax by − ay bx ) ẑ
˛
˛
˛ bx
by
bz ˛
~ · dA
~
B
~
U = −~
µ×B
n
X
i=1
ΦB =
q
~ · d~l = µo I + 1 dΦE
B
c2 dt
C/V = A2 ·s4 /m2 ·kg
V/A = m2 ·kg/s3 ·A2
Wb/m2 = kg/s2 ·A
1.6 × 10−19 J
1 N/A2
1V · s
1 V/m
Right-hand rule #1
1. Point the fingers of your right hand along the direction of ~
v.
~
2. Point your thumb in the direction of B.
Optics:
3. The magnetic force on a + charge points out from the back of your hand.
hc
λ
speed of light in vacuum
c
=
speed of light in a medium
v
E
=
n
=
λ1
λ2
=
v1
c/n1
n2
=
=
v2
c/n2
n1
n1 sin θ1
=
n2 sin θ2
λf
=
c
M
=
1
f
n1
n2
+
p
q
=
=
q
=
1
f
=
Right-hand rule #2:
Point your right thumb along the wire in the direction of the current. Your fingers curl
around the direction of the magnetic field, which circulates around the wire.
hf =
refraction
Snell’s refraction
h0
q
=−
h
p
1
1
2
+ =
mirror & lens
p
q
R
n2 − n1
spherical refracting
R
n2
−
p flat refracting
n1
„
«»
–
n2 − n 1
1
1
−
lensmaker’s
n1
R1
R2
2