Formula Sheet - Web Physics

Constants
J
R = 8.
3145
m⋅
K
23
N = 602 . × 10
A
R
−23
J
kB
= = 138 . × 10
NA
m⋅
K
−19
e = 1.
602 × 10 C
2
1
9 N⋅
m
k = = 9×
10
2
4πε
C
0
µ = 4π
× 10
0
c = 3×
10
8
−7
m
s
T⋅
m
A
Thermal Expansion
∆L = αL0∆T
β = 3α
Heat
Q=
mc∆T
Q=±
mL
Q=
nCV
∆T
Q=
nC ∆T
P
Heat Current
H = k A L T − T
H
H = AeσT
4
Quantities
Work
Vf
W = ∫ PdV
Vi
( )
Isochoric Processes
W = 0
Q=
nCV
∆T
* ∆U = nC ∆T
V
C
Formula Sheet for Physics 251
mtot
= nM
Heat Engines
M = NAm
W = QH
+ QC
N = NAn
W
e =
QH
Ideal gases
TC
* PV = nrT
ec
= 1 −
TH
* PV = NkT
* v
3RT
M
* indicates formulas that are specific to ideal gases
† indicates formulas that are specific to parallel-plate capacitors
RMS
P
=
3
* CV
= R(monatomic)
2
5
* CV
= R(diatomic)
2
* C = C + R
V
Isobaric Processes
W = P∆V
Q=
nCP∆T
* ∆U = nC ∆T
Isothermal Processes
⎛ Vf
⎞
* W = nRT ln⎜
⎟
⎝ V ⎠
⎛ Vf
⎞
* Q = nRT ln⎜
⎟
⎝ Vi
⎠
* ∆U
= 0
Adiabatic Processes
* W =−nCV
∆T
Q = 0
* ∆U
= nCV
∆T
γ γ
* PV
1 l
= PV
2 2
γ −1
γ −1
* TV
1 l
= TV
2 2
CP
γ =
C
V
V
Cyclic Processes
W = Q
∆U
= 0
i
Refrigerators
W = Q + Q
H
C
K
Q C
=
W
TC
Kc
=
T − T
H
Force, Field, Energy
and Potential
r r
F = qE
∆U
= q∆V
b r r
Vab
= ∫ E⋅dl
a
r
E =−∇V
C
1 Point Charge
r
E = k q r r
V = k q r
2 ˆ
2 Point Charges
r
F k qq 1 2
= rˆ
2
r
U = k qq r
1 2
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
“Elementary” E Fields
r
E = k Q r ˆ sphere
2
r
r λ
E =
r r ˆ line
2πε0
r σ
E = nˆ
plane
2ε0
r σ conducting
E = nˆ
ε surface
0
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
A
† C = ε0
d
V σ
† E = =
d ε
Flux and Gauss’s Law
ϕ = ∫∫ r r
E⋅da
Qenclosed
ϕclosed
=
ε
Current and Current
Density
I =
dq
dt
r
I = J ⋅da
r
J = nqv d
∫∫ r
0
0

Ohm’s Law
r r
E = ρJ
V = IR
Resistivity and Resistance
dL
R = ∫ ρ A
ρ = ρ 1 + α T −T
[ ( )]
0 0
Uniform Currents only
V
E =
L
L
R = ρ
A
I
J =
A
Electric Power
P = IV
2
P = I R
P =
V 2
R
Real Batteries
V = E − Ir
Addition of Resistors
RT
= R1 + R2
+ ... series
1 1 1
= + + ... parallel
R R R
T
1 2
Kirchoff’s rules
V = 0
∑
∑
loop
node
I = 0
RC circuits (charging)
t
−
τ
I = Ie
i
⎛
Q= Qf
⎜1
−e
⎝
t
−
τ
⎞
⎟
⎠
Formula Sheet for Physics 251
RC circuits (discharging) Ampere’s Law
t
r r
−
τ
I = Ie
∫ B⋅ dl = µ
0
Ienc
+ I
i
t
where
−
τ
Q=
Qe
i
dϕ
E
Id
= ε0
τ = RC
dt
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 = ∞straight wire
2πr
µ
0I
B = center of loop
2R
B = µ
0nI∞solenoid
µ
0NI
toroidal
B =
2πr
solenoid
τ = RC
* indicates formulas that are specific to ideal gases
† indicates formulas that are specific to parallel-plate capacitors
Faraday’s Law
( )
dϕ
B
E =−
dt
or
r r r r
E⋅ dl = B⋅da
∫
C
∫∫
Mutual Inducance
M
N 1ϕB1
N ϕ
= =
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
S
d
2 B2
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
I = I0
cosωt
V = V0
cos( ωt
+ ϕ)
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
AC Circuits (resonance)
ϕ = 0
Z = R
1
ω =
LC
X = X
L
C

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
nb
θc
= (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 ′
* indicates formulas that are specific to ideal gases
† indicates formulas that are specific to parallel-plate capacitors