Formula Sheet 3 - Web Physics

Constants
J
R = 8.
3145
mK ⋅
23
N = 602 . × 10
A
R
−23
J
kB
= = 138 . × 10
N
A
mK ⋅
−19
e = 1.
602 × 10 C
2
1
9 N⋅
m
k = = 9×
10
2
4πε
C
µ = 4π
× 10
0
8 m
c = 3×
10
s
me
= 911 . × 10
m = 167 . × 10
p
0
−7
T⋅
m
A
−31
−27
kg
kg
Force, Field, Energy
and Potential
r r
F = qE
∆U
= q∆V
b r r
Vab
= ∫ E⋅dl
a
r
E =−∇V
1 Point Charge
r
E = k q r r
V = k q r
2 ˆ
Distributed Charge
r
E = ∫ k dq
r rˆ
2
V = ∫ k dq
r
Many Sources
r r
E = E
U
T
T
=
∑
i
∑
pairs
i
U
ij
Formula Sheet for Physics 251
Flux and Gauss’s Law Addition of Resistors
ϕ = ∫∫ r r
RT
= R1 + R2
+ ... series
E⋅da
1 1 1
Q
= + + ... parallel
enclosed
ϕclosed
=
RT
R1 R2
ε
Capacitors
C =
Q V
CT
= ∑ Ci
parallel
i
1 1
= ∑ series
i
CT
Ci
2
1 2 Q 1
U = CV = = QV
2 2C
2
Current and Current
Density
I =
† indicates formulas that are specific to parallel-plate capacitors
dq
dt
r
I = J ⋅da
r
J = nqv d
∫∫ r
Ohm’s Law
r r
E = ρJ
V = IR
Resistivity and
Resistance
dL
R = ∫ ρ A
ρ = ρ 1 + α T −T
0
[ ( )]
0 0
Electric Power
P = IV
Real Batteries
V
= E −
Ir
Kirchoff’s rules
V = 0
∑
∑
loop
node
I = 0
RC circuits (charging)
t
−
τ
I = Ie
i
⎛
Q= Qf
⎜1
−e
⎝
τ = RC
t
−
τ
⎞
⎟
⎠
RC circuits (discharging)
I = Ie
Q=
Qe
i
τ = RC
i
t
−
τ
t
−
τ
Magnetic Force and
Torque
r r r
F = qv × B
r r r
dF = Idl × B
r r
τ = µ × B
r r
U =−µ
⋅B
Cyclotron Motion
R =
mv
qB
2πm
T =
qB
qB
ωc
=
m
Flux and Gauss’s Law for
Magnetism
r
ϕ = B⋅da
∫∫ r
ϕclosed = 0
Sources of Magnetic
Fields
r
µ qv × rˆ
0
B =
2
4π
r
r
r µ Idl × rˆ
0
B = ∫
2
4π
r
µ
0I
B = center of loop
2R
Ampere’s Law
r r
∫ B⋅ dl = µ
0
Ienc
+ I
where
I
d
= ε
0
dϕ
dt
E
Faraday’s Law
( )
dϕB
E =−
dt
or
r r d r r
∫ E⋅ dl =− ∫∫ B⋅da
C dt S
Mutual Inducance
M
N 1ϕB1
N ϕB
= =
I2
I1
=−M dI 2
E1
dt
=−M dI 1
E2
dt
Self Inductance
Nϕ
B
L =
I
E =−L dI
dt
1 2
U = LI
2
2 2
d

RL Circuits (charging)
⎛
I = If
⎜1
−e
⎝
VL
= Ve
0
L
τ =
R
t
−
τ
t
−
τ
⎞
⎟
⎠
RL Circuits (discharging)
I = Ie
VL
= Ve
i
L
τ =
R
i
t
−
τ
t
−
τ
AC Circuits (general)
XL
= ωL
1
XC
=
ωC
2 2
Z = R + ( XL
− XC)
XL
− XC
tanϕ
=
R
V0 = I0Z
V0
Vrms
=
2
I0
Irms
=
2
VR
,max
= IR
VL,max
= IXL
VC,max
= IXC
1 2 1 2
P = I0
R=
I0
Zcosϕ
2 2
Formula Sheet for Physics 251
Light
Lateral and Angular
Emax
= cB
Magnification
max
1
c =
m
y s
= ′ =− ′
εµ
y s
0 0
mT
= mm
1 2...
c = λf
r 1 r r
θ
S = E×
B
M = ′
µ
θ
0
25
EmaxB
M =
max
I = Sav
=
f ( cm)
Magnifier
2µ
0
fobjective
Sav
M =− Telescope
prad
=
feyepiece
c
Propagation of Light Cameras
c
f
n =
f - number =
v
D
λ0
λ =
Eyeglasses
n
θr
= θ
power = 1 diopters
a
f
nasinθa = nbsinθb
−1⎛
n ⎞
b
θc
= sin ⎜ ⎟ (TIR)
⎝ na
⎠
2
I = I0
cos ϕ
Focal Lengths
R
f =
2
1 ⎛ 1 1 ⎞
= ( n −1) ⎜ − ⎟
f ⎝ R1 R2⎠
Image Location
1 1 1
= +
f s s ′
AC Circuits (resonance)
ϕ = 0
Z = R
1
ω =
LC
X = X
L
C
† indicates formulas that are specific to parallel-plate capacitors