Download Coulomb`s law:

208 Formulae Sheet
r
Coulomb’s law: F =
r
q1q2
q
. Electric field created by a charge q: E =
2
4πε 0 r
4πε 0 r 2
1
Permittivity of free space:
1
4πε 0
= 9 × 109 Newton ⋅ meter 2 / coulomb 2 = 9 ⋅ 109 N ⋅ m 2 / C 2
r
r
∫ E ⋅ dS =
S
Qenclosed
ε0
Gauss’s law (electric flux through a closed surface):
Surface area of a sphere of radius R is S = 4πR 2
r r Qenclosed
E
.
∫ ⋅ dS =
ε0
S
r |σ |
A jump of the electric field over a charged surface: δ | E |=
ε0
σ
(when an axes is directed from left to right !)
ε0
r
q
. Unit: 1volt=J/C
Electric potential of a point charge q: V (r ) − V (∞ ) =
4πε 0 r
Erhs − Elhs =
ε0 = 8.85×10-12 C/(Vm)
r
r2
Definition of the electric potential difference:
r r
r
r
Edr
∫ = −[V (r2 ) − V ( r1 )]
r
r1
Conservation of energy for a charge Q: K + QV ( r ) = const .
Energy of an electron in electric potential=1volt (electron volt): 1eV=1.6×10-19 J
(J=1Joule).
Q
A
=ε
Capacitance: C=Q/V;
Parallel-plate capacitor: C =
ΔV
d
Ra Rb
1
1 ⎛ 1
1 ⎞
Spherical capacitor:
=
−
⎜
⎟ ; C = 4πε
Rb − Ra
C 4πε ⎝ Rinner Router ⎠
Unit: 1F ≡ 1 farad = 1coulomb / volt
1 pF = 10−12 F 1μ F = 10−6 F
ε0 = 8.85×10-12 F/m
Capacitors in parallel: Ctot = C1 + C2 + C3 + ...
Capacitor as energy storage U =
ε =K ε0
Q 2 CV 2
=
2C
2
1
1
1
1
=
+
+
+ ...
Ctot C1 C2 C3
r
r
E 2 (r)
u=energy density u( r ) = ε
2
Capacitors in series:
+++++++++++++++++++++++++++++++++++++++++++++++++
ΔQ dQ
=
Definition of current I (t ) = lim
. Unit: 1A=1ampere= C/s.
Δt →0 Δt
dt
Current I= qnvS ( q=charge, n=density, v=velocity, S the cross-section area)
Ohm’s law: V/I=R; V=RI; V/R=I
Unit: 1Ω=V/A=Vs/C
Length
L
=ρ .
Resistor with a constant cross section: R = ρ
cross − sec tion ' s area
S
Resistivity ρ is measured in [Ωm].
1
1
1
1
= +
+ + ...
Rtot R1 R2 R3
ρ ⇔ 1/ ε
Resistors in series Rtot = R1 + R2 + R3 +… Resistors in parallel
Similarity between resistance and capacitance: R ⇔ 1/ C
Power output (energy loss rate): P = IV = RI 2 = V 2 / R . Unit: [J/s]
q( t )
;
Discharging capacitor: q(t ) = Qinitial exp( −t / RC ) I = dq / dt = −
RC
negative I implies that the charge flows out from the plate, i.e., it is discharging
Q
Charging capacitor q(t ) = Q final [1 − exp( −t / RC )] I (t ) = final exp( −t / RC )
RC
Kirchhoff’s rules: sum of the directed currents in each of the junctions is zero;
sum of the voltage drops and rises along each of the closed loops is zero.
++++++++++++++++++++++++++++++++++++++++++++++
r
r r
Force acting on a charge q moving in the magnetic field F = qv × B
r
r
Force acting on an element dl of a current-carrying conductor: F = Id l × B
Cyclotron frequency: f = ω = qB
2π 2π m
ur
ur ur
Dipoles. Electric dipole moment of a pair charges ±q separated by d : p = qd ;
ur
ur
Magnetic dipole
moment
of
a
small
area
surrounded
by
a
current
I:
μ
=
I
(
d
S)
r ur ur r ur ur
Torque [Nm]: τ = p × E ; τ = μ × B .
ur ur
ur ur
Energy of a dipole in a field: U = − p ⋅ E ; U = − μ ⋅ B
r
r μ0 q( v × rr )
Magnetic field created by a moving charge q (Biot-Savart law): B =
4π
r3
r
r μ0 I ( d l × rr )
Magnetic field created by an element dl carrying current I: dB =
r3
4π
Units for magnetic field 1T (tesla ) = 1N / C ⋅ m / s = 1N / A ⋅ m
Permeability
μ0 = 4π × 10−7 T ⋅ m / A = 4π × 10−7 N ⋅ s 2 / C 2 = 4π × 10−7 N / A2
Magnetic field created by a straight wire carrying current I : | B |=
μ0 I
2π r
Steady-state version of Ampere’s law (current enclosed by a path):
∫
ur r
Bdl = μ0Ienclosed
contour
Magnetic field created by a solenoid: B = μ0nI , n=N/l is number of turns per unit length.
Faraday’s law (the EMF induced in a closed loop as response to a change of magnetic flux
r r
dΦ
through the loop):
∫ E ⋅ dr = − dt B
Ampere’s law (including “displacement” current created by varying in time electric
ur r
fields):
∫ Bd r = μ0 ( I c + ε 0 ddtΦ E )
Maxwell’s equations: two Gauss’s laws + Faraday’s and Ampere’s laws
d Φ B2
N 2Φ B 2 = M 21 I1
dt
d Φ B1
Emf1 = − N1
N1Φ B1 = M 12 I 2
dt
Mutual Inductance: M mutual = μ0n1n2loverlap Soverlap
Mutual Inductance: Emf 2 = − N 2
Units for flux (weber) and EMF:
M 21 = M 12
M=
ΦB
I
flux [Φ B ] = 1T ⋅ m 2 = 1N ⋅ m ⋅ s / C = 1J ⋅ s / C = 1V ⋅ s
1T ⋅ m 2 = 1Wb 1V = 1Wb / s
Units of the mutual inductance (henry):
1henry = 1H = 1Wb /1A = 1V ⋅ s /1A = 1Ω ⋅ s = 1J / A2
Inductance (self-inductance):
Emf = − N
dΦB
dI
= −L
dt
dt
Another units for permeability: μ0 = 4π × 10−7 H / m
NΦB
N2
= μ0
× Area
Inductance of a toroidal solenoid: L =
I
2π r
Emf
(1 − exp( − Rt / L))
Current growth in an R-L circuit: I =
R
Decay of current in an R-L circuit: I = I (t = 0) exp( − Rt / L)
LI 2 (t )
( dQ / dt ) 2
Magnetic field energy: U (t ) =
=L
2
2
2
B
Density of magnetic field energy uB =
2 μ0
1 2
q 2 (t ) Q 2
d 2q 1
LI (t ) +
=
= const
2
q = 0 , ω = 1/ LC ; 2
Oscillations in a L-C circuit: 2 +
2C
2C
dt
LC
I (t ) = I sin(ωt + ϕ )
++++++++++++++++++++++++++++++++++++++++++++++
2π
2π
λ
Waves (frequency, wave vector, speed):
ω=
k=
v = = ω/k
T
λ
T
ω = vk
Wave propagating along x: y ( x; t ) right / left = A cos(kx m ωt + ϕ ) ϕ = phase
Wave equation:
∂ 2 y (t , x ) ∂ 2 y (t , x )
v
=
∂x 2
∂t 2
2
Set of wave equations in electromagnetism:
∂E y (t , x )
∂E (t , x )
∂B (t , x )
∂B (t , x )
=− z
− z
= ε 0 μ0 y
∂x
∂t
∂x
∂t
∂ 2 E y (t , x )
∂x 2
Speed of light in vacuum and medium; index of refraction n:
1
1
=
≈ (3 × 108 m / s )2
c2 =
−12 2
−7
2
2
ε 0 μ0 (8.85 × 10 C / Nm ) ⋅ (4π × 10 N / A )
v2 =
1
εμ
n = c / v = KK magn ≈ K
v = c/n
= ε 0 μ0
∂ 2 E y (t , x )
∂t 2
Relation between the amplitudes of the electric and magnetic fields in electromagnetic
fields: E=cB.
Radiation power: P=IA
Intensity of radiation far away from the source:
I = P /(4π r 2 )
2
2
Density of energy: u = ε 0 E 2 ( x, t ) ; average
ur ur density of energy u = ε 0 E ( x, t ) = ε 0 E / 2
Poynting vector S, intensity I: uSr = E × B P = uSr ⋅ uAr I = S = EB
μ0
2 μ0
Radiation pressure: Prad = α I / c ; for totally reflecting mirror α=2; for black body α=1.
+++++++++++++++++++++++++++++++++++++++++++++++++++
Angle of reflection: θincident = θ reflected
Snell’s law: nincident sin θincident = nrefracted sin θ refracted
Angle of total internal reflection: sin θ critical =
nrefracted
nincident
r ur
r r ur
filter
Polarizing by a linear filter along the direction n: E incident ⎯⎯⎯
→ n n ⋅ E incident
(
)
Malus’s law (consequence of the relation above): I = I max cos2 φ
ϕ y = ϕz m
Polarizations: circular E y = E z
Bruster’s angle: tgθ polar =
π
2
; elliptical E y ≠ E z
nrefracted
ϕ y = ϕz m
linear : ϕ y = ϕ z .
nincident
Huygens’s and Fermat’s principles.
y'
s'
=−
y
s
1 1 1
+ =
f =| R | / 2
Concave spherical mirror →), focal length:
s s' f
1 1
1
+ =
| f |=| R | / 2
Convex spherical mirror →( :
s s' − | f |
Spherical refractive image na + nb = nb − na m = y ' = − na s '
R
y
nb s
s s'
1 1 1
s'
m=−
Thin lenses (converging lens, f>0; diverging lens, f<0): + =
s s' f
s
⎛ 1
1
1 ⎞
Lens maker’s equation: = (n − 1) ⎜ − ⎟
Images; lateral magnification: m =
f
Double convex/concave lenses:
⎝ R1
R2 ⎠
⎛ 1
1
1 ⎞
= ( n − 1) ⎜
+
⎟
|f |
|
R
|
|
R
2 |⎠
⎝ 1
π
2
;
++++++++++++++++++++++++++++++++++++++++++++++
Integrals:
N
∫ x dx =
∫
∫
R
1 N +1
x
N +1
∫x
∞
0
(x
1
∫x
+y
∫ dτ exp( −τ / τ
0
)
2 3/ 2
=
1
x
a
x2 + a2
) = τ 0 (1 − exp( −t / τ 0 )
Averaging
cos2 (ωt − kx ) = sin 2 (ωt − kx ) =
cos(ωt − kx ) ⋅ sin(ωt − kx ) = 0
cos π / 6 = 3 / 2 sin π / 6 = 1/ 2
cos π / 3 = 1/ 2 sin π / 3 = 3 / 2
1
R N +1
N +1
dx =
1
1
N − 1 R N −1
dx = ln(b / a )
a
t
0
N
R
b
1
2
1
∫x
1
dx = ln x
x
a
dx =
0
1
1
1
dx =
N
N −1
x
− N +1 x
x ∫ dy
N
1
2