PH2200 Formula Sheet - Physics

PH2200 Formula Sheet
Electric Fields
! kqq e 1 2 1
F ˆ
12
= r k
2
e
=
r
4πε
o
!
! Fe
E = lim
qo
→0
qo
! kq e
E = rˆ
2
r
! " kq
E = E = rˆ
∑
∑
e i
i
2 i
i i ri
! " kdq e
E = ∫dE = ∫ rˆ
2
r
Gauss's Law
! !
Φ = E⋅dA
# ∫
E
! ! qin
E⋅ dA=
ε
σ
E =
ε
°
∫
°
at conductor surface
Electric Potential
B ! !
∆ U =−q E⋅ds
kq e
V()
r =
r
kq e i
V = ∑
r
x
i
12
o
i
A
B
∆U
! !
∆ VA→B
= =− E⋅ds
q
∫
kdq e
V = ∫
r
kqq e 1 2
U =
r
E
dV
=−
dx
∫
o
Capacitance and Dielectrics
Q
C = ∆ V
ε A
C = °
parallel-plate capacitor
d
A
C 1
= cylindrical capacitor
l ⎛b
⎞
2ke
ln⎜
⎟
⎝a
⎠
ab
C =
spherical capacitor
k ( b − a )
e
C = C + C + C + ... parallel
eq
1
2
1 2 3
1 1 1 1
= + + + ... series
C C C C
eq
1 2 3
2
Q
1 1
U = =
2Q∆ V =
2C( ∆V)
2C
2
u = ε E
E
C = κC
°
o
Current and Resistance
∆Q
Iav
= ∆ t
dq
I =
dt
Iav
= nqvd
A
! !
J = nqvd
! !
J = σ E
∆V
R =
I
1
ρ = σ
ρl
R =
A
ρ = ρ [1 + α( T −T
)]
o
P = I∆V
2 ( ∆V
)
P = I R=
R
Direct Current Circuits
2
R = R + R + R + ... series
eq
1 2 3
1 1 1 1
= + + + ... parallel
R R R R
eq
∑Iin
∑
=
closed loop
o
1 2 3
∑
I
out
∆ V = 0
2
qt = CE
−e
−t/
RC
( ) (1 ) charging
It
E
= e
R
qt = Qe
−t/
RC
( ) charging
−t/
RC
( ) discharging
Q
=− e
RC
−t/
RC
( ) discharging
It
Magnetic Fields
! ! !
F = qv×
B
B
FB
= q vBsinθ
! ! !
FB
= I L×
B
! ! !
dFB
= Ids × B
! !
µ = NIA
! ! !
τ = µ × B
! !
UB
=−µ
⋅B
! ! ! !
F = qE + qv×
B
mv qB
r = , ω =
qB m
Sources of the Magnetic Field
!
!
µ
o Ids × rˆ
dB =
2
4π
r
µ
oI
B = long straight wire
2π
a
FB
µ
oII
1 2
=
l 2π
a
! !
B⋅ ds = µ I
# ∫
µ
oNI
B =
2π
r
N
B= µ
o I = µ
onI
l
! !
Φ
B
= ∫ B⋅dA
! !
B⋅ dA=
0
# ∫
Faraday's Law
dΦ
E =−N
dt
E =−Blv
o
B
dΦ
∫ E ! !
# ⋅ ds =−
dt
B
toroid
solenoid

Inductance
dΦ
E
dt
NΦB
L =
I
2
µ
oN A
L =
l
E
I = −e
R
E −Rt / L
I = e
R
2
U = LI
B
L
=− N =−
1
2
dI
L
dt
−Rt / L
(1 ) rising current
2 12 1 21
12 21
I1 I2
solenoid
decaying current
2
B
uB
=
2µ
°
N Φ N Φ
M = = M = = M
dI
dI
E =− M and E =−M
dt
dt
Q= Qmax
cos( ωt+
φ)
1
ω =
LC
U = UC
+ UL
2 2
Qmax
2 LImax
2
= cos ωt+
sin ωt
2C
2
1 2
2 12 1 21
Alternating Current Circuits
Irms
= 0.707I
∆Vrms
0.707V
X = ωL
L
1 1 2 2
max
max
1
X C
=
ωC
2 2
Z = R + ( XL
− XC)
−1
⎛ XL
− XC
⎞
φ = tan ⎜ ⎟
⎝ R ⎠
Pav = Irms∆Vrms
cosφ
2
Pav
= IrmsR
∆Vrms
Irms
=
R + ( X − X )
1
ωo
=
LC
I ∆ V = I ∆V
2 2
L C
Electromagnetic Waves
# ∫
S
# ∫
S
! ! Q
E⋅ dA=
εo
! !
B⋅ dA=
0
! ! dΦ
B
# ∫ E⋅ ds =−
dt
! !
dΦ
# ∫ B⋅ ds = µ
oI
+ ε
oµ
o
dt
1
c =
εµ
o o
fλ
= c
Emax
E
= = c
Bmax
B
! !
! E×
B
S =
µ
°
E B
= = =
max max
I Sav
cuav
2µ
o
2
1 2 B
uB = uE = ε
2 oE
=
2µ
o
B
u = ε E =
av
1
2
o
2
2 max
max
2µ
o
U S
p= P=
c c
2U
2S
p = P =
c c
E
complete absorption
complete reflection
The Nature of Light and
the Laws of Geometric Optics
θ'
1
= θ1
n1sinθ1 = n2sinθ2
c
n =
v
λ
λn
=
n
n
θ = >
2
sin
c
(for n1 n2)
n1
Geometric Optics
h'
M =
h
1 1 2 1
+ = =
p q R f
n n n − n
+ =
p q R
1 2 2 1
1 ⎛ 1 1 ⎞
= ( n −1)
⎜ − ⎟
f ⎝ R1 R2
⎠
1 1 1
+ =
p q f
Physical Constants
9 2 2
ke
= 8.988× 10 N⋅m / C
−12 2 2
εo
= 8.854× 10 C / N⋅m
−19
e= 1.602 × 10 C
−31
melectron
= 9.109 × 10 kg
−27
mproton
= 1.672×
10 kg
−7
µ
°
= 4π
× 10 T⋅m/
A
1
8
c= = 2.998×
10 m/
s
εµ
° °
Useful Geometry
Circle
2
Area = π r
Circumference = 2π
r
Sphere
Surface area = 4π
r
4 3
Volume = π
3
r
Cylinder
Lateral surface
area = 2π
rL
2
Volume = π rL
2