Physics Formulary, Notas de estudo de Engenharia Elétrica

Baixe Physics Formulary e outras Notas de estudo em PDF para Engenharia Elétrica, somente na Docsity!

Physics Formulary

By ir. J.C.A. Wevers

©c 1995, 2001 J.C.A. Wevers Version: November 13, 2001

Dear reader,

This document contains a 108 page LATEX file which contains a lot equations in physics. It is written at advanced undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and often needs to look up equations.

This, and a Dutch version of this file, can be obtained from the author, Johan Wevers ([email protected]).

It can also be obtained on the WWW. See http://www.xs4all.nl/˜johanw/index.html, where also a Postscript version is available.

If you find any errors or have any comments, please let me know. I am always open for suggestions and possible corrections to the physics formulary.

This document is Copyright 1995, 1998 by J.C.A. Wevers. All rights are reserved. Permission to use, copy and distribute this unmodified document by any means and for any purpose except profit purposes is hereby granted. Reproducing this document by any means, included, but not limited to, printing, copying existing prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause, unless upon explicit prior written permission of the author.

This document is provided by the author “as is”, with all its faults. Any express or implied warranties, in- cluding, but not limited to, any implied warranties of merchantability, accuracy, or fitness for any particular purpose, are disclaimed. If you use the information in this document, in any way, you do so at your own risk.

The Physics Formulary is made with teTEX and LATEX version 2.09. It can be possible that your LATEX version has problems compiling the file. The most probable source of problems would be the use of large bezier curves and/or emTEX specials in pictures. If you prefer the notation in which vectors are typefaced in boldface, uncomment the redefinition of the \vec command in the TEX file and recompile the file.

Johan Wevers

• Physical Constants Contents I
• 1 Mechanics
  • 1.1 Point-kinetics in a fixed coordinate system
    • 1.1.1 Definitions
    • 1.1.2 Polar coordinates
  • 1.2 Relative motion
  • 1.3 Point-dynamics in a fixed coordinate system
    • 1.3.1 Force, (angular)momentum and energy
    • 1.3.2 Conservative force fields
    • 1.3.3 Gravitation
    • 1.3.4 Orbital equations
    • 1.3.5 The virial theorem
  • 1.4 Point dynamics in a moving coordinate system
    • 1.4.1 Apparent forces
    • 1.4.2 Tensor notation
  • 1.5 Dynamics of masspoint collections
    • 1.5.1 The centre of mass
    • 1.5.2 Collisions
  • 1.6 Dynamics of rigid bodies
    • 1.6.1 Moment of Inertia
    • 1.6.2 Principal axes
    • 1.6.3 Time dependence
  • 1.7 Variational Calculus, Hamilton and Lagrange mechanics
    • 1.7.1 Variational Calculus
    • 1.7.2 Hamilton mechanics
    • 1.7.3 Motion around an equilibrium, linearization
    • 1.7.4 Phase space, Liouville’s equation
    • 1.7.5 Generating functions
• 2 Electricity & Magnetism
  • 2.1 The Maxwell equations
  • 2.2 Force and potential
  • 2.3 Gauge transformations
  • 2.4 Energy of the electromagnetic field
  • 2.5 Electromagnetic waves
    • 2.5.1 Electromagnetic waves in vacuum
    • 2.5.2 Electromagnetic waves in matter
  • 2.6 Multipoles
  • 2.7 Electric currents
  • 2.8 Depolarizing field
  • 2.9 Mixtures of materials
• 3 Relativity II Physics Formulary by ir. J.C.A. Wevers
  • 3.1 Special relativity
    • 3.1.1 The Lorentz transformation
    • 3.1.2 Red and blue shift
    • 3.1.3 The stress-energy tensor and the field tensor
  • 3.2 General relativity
    • 3.2.1 Riemannian geometry, the Einstein tensor
    • 3.2.2 The line element
    • 3.2.3 Planetary orbits and the perihelion shift
    • 3.2.4 The trajectory of a photon
    • 3.2.5 Gravitational waves
    • 3.2.6 Cosmology
• 4 Oscillations
  • 4.1 Harmonic oscillations
  • 4.2 Mechanic oscillations
  • 4.3 Electric oscillations
  • 4.4 Waves in long conductors
  • 4.5 Coupled conductors and transformers
  • 4.6 Pendulums
• 5 Waves
  • 5.1 The wave equation
  • 5.2 Solutions of the wave equation
    • 5.2.1 Plane waves
    • 5.2.2 Spherical waves
    • 5.2.3 Cylindrical waves
    • 5.2.4 The general solution in one dimension
  • 5.3 The stationary phase method
  • 5.4 Green functions for the initial-value problem
  • 5.5 Waveguides and resonating cavities
  • 5.6 Non-linear wave equations
• 6 Optics
  • 6.1 The bending of light
  • 6.2 Paraxial geometrical optics
    • 6.2.1 Lenses
    • 6.2.2 Mirrors
    • 6.2.3 Principal planes
    • 6.2.4 Magnification
  • 6.3 Matrix methods
  • 6.4 Aberrations
  • 6.5 Reflection and transmission
  • 6.6 Polarization
  • 6.7 Prisms and dispersion
  • 6.8 Diffraction
  • 6.9 Special optical effects
  • 6.10 The Fabry-Perot interferometer
• 7 Statistical physics
  • 7.1 Degrees of freedom
  • 7.2 The energy distribution function
  • 7.3 Pressure on a wall
  • 7.4 The equation of state
  • 7.5 Collisions between molecules
  • 7.6 Interaction between molecules Physics Formulary by ir. J.C.A. Wevers III
• 8 Thermodynamics
  • 8.1 Mathematical introduction
  • 8.2 Definitions
  • 8.3 Thermal heat capacity
  • 8.4 The laws of thermodynamics
  • 8.5 State functions and Maxwell relations
  • 8.6 Processes
  • 8.7 Maximal work
  • 8.8 Phase transitions
  • 8.9 Thermodynamic potential
  • 8.10 Ideal mixtures
  • 8.11 Conditions for equilibrium
  • 8.12 Statistical basis for thermodynamics
  • 8.13 Application to other systems
• 9 Transport phenomena
  • 9.1 Mathematical introduction
  • 9.2 Conservation laws
  • 9.3 Bernoulli’s equations
  • 9.4 Characterising of flows by dimensionless numbers
  • 9.5 Tube flows
  • 9.6 Potential theory
  • 9.7 Boundary layers
    • 9.7.1 Flow boundary layers
    • 9.7.2 Temperature boundary layers
  • 9.8 Heat conductance
  • 9.9 Turbulence
  • 9.10 Self organization
• 10 Quantum physics
  • 10.1 Introduction to quantum physics
    • 10.1.1 Black body radiation
    • 10.1.2 The Compton effect
    • 10.1.3 Electron diffraction
  • 10.2 Wave functions
  • 10.3 Operators in quantum physics
  • 10.4 The uncertainty principle
  • 10.5 The Schr¨odinger equation
  • 10.6 Parity
  • 10.7 The tunnel effect
  • 10.8 The harmonic oscillator
  • 10.9 Angular momentum
  • 10.10 Spin
  • 10.11 The Dirac formalism
  • 10.12 Atomic physics
    • 10.12.1 Solutions
    • 10.12.2 Eigenvalue equations
    • 10.12.3 Spin-orbit interaction
    • 10.12.4 Selection rules
  • 10.13 Interaction with electromagnetic fields
  • 10.14 Perturbation theory
    • 10.14.1 Time-independent perturbation theory
    • 10.14.2 Time-dependent perturbation theory
  • 10.15 N-particle systems IV Physics Formulary by ir. J.C.A. Wevers
    • 10.15.1 General
    • 10.15.2 Molecules
  • 10.16 Quantum statistics
• 11 Plasma physics
  • 11.1 Introduction
  • 11.2 Transport
  • 11.3 Elastic collisions
    • 11.3.1 General
    • 11.3.2 The Coulomb interaction
    • 11.3.3 The induced dipole interaction
    • 11.3.4 The centre of mass system
    • 11.3.5 Scattering of light
  • 11.4 Thermodynamic equilibrium and reversibility
  • 11.5 Inelastic collisions
    • 11.5.1 Types of collisions
    • 11.5.2 Cross sections
  • 11.6 Radiation
  • 11.7 The Boltzmann transport equation
  • 11.8 Collision-radiative models
  • 11.9 Waves in plasma’s
• 12 Solid state physics
  • 12.1 Crystal structure
  • 12.2 Crystal binding
  • 12.3 Crystal vibrations
    • 12.3.1 A lattice with one type of atoms
    • 12.3.2 A lattice with two types of atoms
    • 12.3.3 Phonons
    • 12.3.4 Thermal heat capacity
  • 12.4 Magnetic field in the solid state
    • 12.4.1 Dielectrics
    • 12.4.2 Paramagnetism
    • 12.4.3 Ferromagnetism
  • 12.5 Free electron Fermi gas
    • 12.5.1 Thermal heat capacity
    • 12.5.2 Electric conductance
    • 12.5.3 The Hall-effect
    • 12.5.4 Thermal heat conductivity
  • 12.6 Energy bands
  • 12.7 Semiconductors
  • 12.8 Superconductivity
    • 12.8.1 Description
    • 12.8.2 The Josephson effect
    • 12.8.3 Flux quantisation in a superconducting ring
    • 12.8.4 Macroscopic quantum interference
    • 12.8.5 The London equation
    • 12.8.6 The BCS model
• 13 Theory of groups Physics Formulary by ir. J.C.A. Wevers V
  • 13.1 Introduction
    • 13.1.1 Definition of a group
    • 13.1.2 The Cayley table
    • 13.1.3 Conjugated elements, subgroups and classes
    • 13.1.4 Isomorfism and homomorfism; representations
    • 13.1.5 Reducible and irreducible representations
  • 13.2 The fundamental orthogonality theorem
    • 13.2.1 Schur’s lemma
    • 13.2.2 The fundamental orthogonality theorem
    • 13.2.3 Character
  • 13.3 The relation with quantum mechanics
    • 13.3.1 Representations, energy levels and degeneracy
    • 13.3.2 Breaking of degeneracy by a perturbation
    • 13.3.3 The construction of a base function
    • 13.3.4 The direct product of representations
    • 13.3.5 Clebsch-Gordan coefficients
    • 13.3.6 Symmetric transformations of operators, irreducible tensor operators
    • 13.3.7 The Wigner-Eckart theorem
  • 13.4 Continuous groups
    • 13.4.1 The 3-dimensional translation group
    • 13.4.2 The 3-dimensional rotation group
    • 13.4.3 Properties of continuous groups
  • 13.5 The group SO(3)
  • 13.6 Applications to quantum mechanics
    • 13.6.1 Vectormodel for the addition of angular momentum
    • 13.6.2 Irreducible tensor operators, matrixelements and selection rules
  • 13.7 Applications to particle physics
• 14 Nuclear physics
  • 14.1 Nuclear forces
  • 14.2 The shape of the nucleus
  • 14.3 Radioactive decay
  • 14.4 Scattering and nuclear reactions
    • 14.4.1 Kinetic model
    • 14.4.2 Quantum mechanical model for n-p scattering
    • 14.4.3 Conservation of energy and momentum in nuclear reactions
  • 14.5 Radiation dosimetry
• 15 Quantum field theory & Particle physics
  • 15.1 Creation and annihilation operators
  • 15.2 Classical and quantum fields
  • 15.3 The interaction picture
  • 15.4 Real scalar field in the interaction picture
  • 15.5 Charged spin-0 particles, conservation of charge
  • 15.6 Field functions for spin- 12 particles
  • 15.7 Quantization of spin- 12 fields
  • 15.8 Quantization of the electromagnetic field
  • 15.9 Interacting fields and the S-matrix
  • 15.10 Divergences and renormalization
  • 15.11 Classification of elementary particles
  • 15.12 P and CP-violation
  • 15.13 The standard model
    • 15.13.1 The electroweak theory
    • 15.13.2 Spontaneous symmetry breaking: the Higgs mechanism
    • 15.13.3 Quantumchromodynamics VI Physics Formulary by ir. J.C.A. Wevers
  • 15.14 Path integrals
  • 15.15 Unification and quantum gravity
• 16 Astrophysics
  • 16.1 Determination of distances
  • 16.2 Brightness and magnitudes
  • 16.3 Radiation and stellar atmospheres
  • 16.4 Composition and evolution of stars
  • 16.5 Energy production in stars
• The ∇ -operator
• The SI units

Chapter 1

Mechanics

1.1 Point-kinetics in a fixed coordinate system

1.1.1 Definitions

The position ~r, the velocity ~v and the acceleration ~a are defined by: ~r = (x, y, z), ~v = ( ˙x, y,˙ z˙), ~a = (¨x, y,¨ ¨z). The following holds:

s(t) = s 0 +

|~v(t)|dt ; ~r(t) = ~r 0 +

~v(t)dt ; ~v(t) = ~v 0 +

~a(t)dt

When the acceleration is constant this gives: v(t) = v 0 + at and s(t) = s 0 + v 0 t + 12 at^2. For the unit vectors in a direction ⊥ to the orbit ~et and parallel to it ~en holds:

~et =

~v |~v|

d~r ds ~e˙t = v ρ ~en ; ~en =

~e˙t | ~e˙t|

For the curvature k and the radius of curvature ρ holds:

~k = d~et ds

d^2 ~r ds^2

dϕ ds

∣ ;^ ρ^ =^

|k|

1.1.2 Polar coordinates

Polar coordinates are defined by: x = r cos(θ), y = r sin(θ). So, for the unit coordinate vectors holds: ~e^ ˙r = θ~˙eθ , ~e˙θ = − θ~˙er

The velocity and the acceleration are derived from: ~r = r~er , ~v = ˙r~er + r θ~˙eθ, ~a = (¨r − r θ˙^2 )~er + (2 ˙r θ˙ + r θ¨)~eθ.

1.2 Relative motion

For the motion of a point D w.r.t. a point Q holds: ~rD = ~rQ +

~ω × ~vQ ω^2

with QD =~ ~rD − ~rQ and ω = θ˙.

Further holds: α = θ¨. ′^ means that the quantity is defined in a moving system of coordinates. In a moving system holds: ~v = ~vQ + ~v ′^ + ~ω × ~r ′^ and ~a = ~aQ + ~a ′^ + ~α × ~r ′^ + 2~ω × ~v − ~ω × (~ω × ~r ′) with |~ω × (~ω × ~r ′)| = ω^2 ~r ′ n

1.3 Point-dynamics in a fixed coordinate system

1.3.1 Force, (angular)momentum and energy

Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where the mo- mentum is given by ~p = m~v:

F^ ~ (~r, ~v, t) = d~p dt

d(m~v ) dt

= m

d~v dt

• ~v

dm dt

m=const = m~a

Chapter 1: Mechanics 3

Newton’s 3rd law is given by: F~action = − F~reaction.

For the power P holds: P = W˙ = F~ · ~v. For the total energy W , the kinetic energy T and the potential energy U holds: W = T + U ; T˙ = − U˙ with T = 12 mv^2.

The kick S~ is given by: S~ = ∆~p =

F dt^ ~

The work A, delivered by a force, is A =

∫^2

1

F^ ~ · d~s =

∫^2

1

F cos(α)ds

The torque ~τ is related to the angular momentum ~L: ~τ = L~˙ = ~r × F~ ; and ~L = ~r × ~p = m~v × ~r, |~L| = mr^2 ω. The following equation is valid:

τ = −

∂U

∂θ

Hence, the conditions for a mechanical equilibrium are:

Fi = 0 and

~τi = 0.

The force of friction is usually proportional to the force perpendicular to the surface, except when the motion starts, when a threshold has to be overcome: Ffric = f · Fnorm · ~et.

1.3.2 Conservative force fields

A conservative force can be written as the gradient of a potential: F~cons = −∇~U. From this follows that ∇ × F~ = ~ 0. For such a force field also holds:

∮ F^ ~ · d~s = 0 ⇒ U = U 0 −

∫^ r^1

r 0

F^ ~ · d~s

So the work delivered by a conservative force field depends not on the trajectory covered but only on the starting and ending points of the motion.

1.3.3 Gravitation

The Newtonian law of gravitation is (in GRT one also uses κ instead of G):

F^ ~g = −G m^1 m^2 r^2

~er

The gravitational potential is then given by V = −Gm/r. From Gauss law it then follows: ∇^2 V = 4πG%.

1.3.4 Orbital equations

If V = V (r) one can derive from the equations of Lagrange for φ the conservation of angular momentum:

∂L ∂φ

∂V

∂φ

d dt

(mr^2 φ) = 0 ⇒ Lz = mr^2 φ = constant

For the radial position as a function of time can be found that:

( dr dt

2(W − V )

m

L^2

m^2 r^2

The angular equation is then:

φ − φ 0 =

∫^ r

0

[

mr^2 L

2(W − V )

m

L^2

m^2 r^2

]− 1

dr r−^2 field = arccos

1 r −^

1 r 0 1 r 0 +^ km/L

2 z

If F = F (r): L =constant, if F is conservative: W =constant, if F~ ⊥ ~v then ∆T = 0 and U = 0.

Chapter 1: Mechanics 5

1.4.2 Tensor notation

Transformation of the Newtonian equations of motion to xα^ = xα(x) gives:

dxα dt

∂xα ∂ x¯β

dx¯β dt

The chain rule gives:

d dt

dxα dt

d^2 xα dt^2

d dt

∂xα ∂ x¯β

dx¯β dt

∂xα ∂ x¯β

d^2 x¯β dt^2

dx¯β dt

d dt

∂xα ∂ ¯xβ

so: d dt

∂xα ∂ ¯xβ^

∂ x¯γ

∂xα ∂ ¯xβ

dx¯γ dt

∂^2 xα ∂ ¯xβ^ ∂ ¯xγ

dx¯γ dt

This leads to: d^2 xα dt^2

∂xα ∂ x¯β

d^2 x¯β dt^2

∂^2 xα ∂ ¯xβ^ ∂ ¯xγ

dx¯γ dt

dx¯β dt

Hence the Newtonian equation of motion

m

d^2 xα dt^2

= F α

will be transformed into:

m

d^2 xα dt^2

• Γαβγ

dxβ dt

dxγ dt

= F α

The apparent forces are taken from he origin to the effect side in the way Γαβγ

dxβ dt

dxγ dt

1.5 Dynamics of masspoint collections

1.5.1 The centre of mass

The velocity w.r.t. the centre of mass R~ is given by ~v −

R. The coordinates of the centre of mass are given by:

~rm =

mi~ri ∑ mi

In a 2-particle system, the coordinates of the centre of mass are given by:

R^ ~ = m^1 ~r^1 +^ m^2 ~r^2 m 1 + m 2

With ~r = ~r 1 − ~r 2 , the kinetic energy becomes: T = 12 Mtot R˙^2 + 12 μ r˙^2 , with the reduced mass μ given by: 1 μ

m 1

m 2 The motion within and outside the centre of mass can be separated:

L~^ ˙

outside =^ ~τoutside ;^

~L˙

inside =^ ~τinside

~p = m~vm ; F~ext = m~am ; F~ 12 = μ~u

1.5.2 Collisions

With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds: ~p = m~vm is constant, and T = 12 m~v (^) m^2 is constant. The changes in the relative velocities can be derived from: S~ = ∆~p =

μ(~vaft − ~vbefore). Further holds ∆L~C = CB~ × S~, ~p ‖ S~ =constant and ~L w.r.t. B is constant.

6 Physics Formulary by ir. J.C.A. Wevers

1.6 Dynamics of rigid bodies

1.6.1 Moment of Inertia

The angular momentum in a moving coordinate system is given by:

L^ ~′^ = I~ω + ~L′ n

where I is the moment of inertia with respect to a central axis, which is given by:

I =

i

mi~ri 2 ; T ′^ = Wrot = 12 ωIij~ei~ej = 12 Iω^2

or, in the continuous case:

I =

m V

r′^2 ndV =

r′^2 ndm

Further holds: Li = Iij^ ωj ; Iii = Ii ; Iij = Iji = −

k

mkx′ ix′ j

Steiner’s theorem is: Iw.r.t.D = Iw.r.t.C + m(DM )^2 if axis C ‖ axis D.

Object I Object I

Cavern cylinder I = mR^2 Massive cylinder I = 12 mR^2 Disc, axis in plane disc through m I = 14 mR^2 Halter I = 12 μR^2

Cavern sphere I = 23 mR^2 Massive sphere I = 25 mR^2 Bar, axis ⊥ through c.o.m. I = 121 ml^2 Bar, axis ⊥ through end I = 13 ml^2

Rectangle, axis ⊥ plane thr. c.o.m. I = 121 m(a^2 + b^2 ) Rectangle, axis ‖ b thr. m I = ma^2

1.6.2 Principal axes

Each rigid body has (at least) 3 principal axes which stand ⊥ to each other. For a principal axis holds:

∂I ∂ωx

∂I

∂ωy

∂I

∂ωz

= 0 so L′ n = 0

The following holds: ω˙k = −aijk ωiωj with aijk =

Ii − Ij Ik if I 1 ≤ I 2 ≤ I 3.

1.6.3 Time dependence

For torque of force ~τ holds:

~τ ′^ = I θ¨ ;

d′′^ ~L′ dt

= ~τ ′^ − ~ω × ~L′

The torque T~ is defined by: T~ = F~ × d~.

1.7 Variational Calculus, Hamilton and Lagrange mechanics

1.7.1 Variational Calculus

Starting with:

δ

∫^ b

a

L(q, q, t˙ )dt = 0 with δ(a) = δ(b) = 0 and δ

du dx

d dx

(δu)

8 Physics Formulary by ir. J.C.A. Wevers

If the equation of continuity, ∂t% + ∇ · (%~v ) = 0 holds, this can be written as:

{%, H} +

∂t

For an arbitrary quantity A holds: dA dt

= {A, H} +

∂A

∂t

Liouville’s theorem can than be written as:

d% dt

= 0 ; or:

pdq = constant

1.7.5 Generating functions

Starting with the coordinate transformation: { Qi = Qi(qi, pi, t) Pi = Pi(qi, pi, t)

one can derive the following Hamilton equations with the new Hamiltonian K:

dQi dt

∂K

∂Pi

dPi dt

∂K

∂Qi

Now, a distinction between 4 cases can be made:

1. If pi q˙i − H = PiQi − K(Pi, Qi, t) −

dF 1 (qi, Qi, t) dt , the coordinates follow from:

pi =

∂F 1

∂qi

; Pi =

∂F 1

∂Qi

; K = H +

dF 1 dt

2. If pi q˙i − H = − P˙iQi − K(Pi, Qi, t) +

dF 2 (qi, Pi, t) dt

, the coordinates follow from:

pi =

∂F 2

∂qi

; Qi =

∂F 2

∂Pi

; K = H +

∂F 2

∂t

3. If − p˙iqi − H = Pi Q˙i − K(Pi, Qi, t) + dF 3 (pi, Qi, t) dt

, the coordinates follow from:

qi = −

∂F 3

∂pi

; Pi = −

∂F 3

∂Qi

; K = H +

∂F 3

∂t

4. If − p˙iqi − H = −PiQi − K(Pi, Qi, t) +

dF 4 (pi, Pi, t) dt

, the coordinates follow from:

qi = −

∂F 4

∂pi

; Qi =

∂F 4

∂pi

; K = H +

∂F 4

∂t

The functions F 1 , F 2 , F 3 and F 4 are called generating functions.

Chapter 2

Electricity & Magnetism

2.1 The Maxwell equations

The classical electromagnetic field can be described by the Maxwell equations. Those can be written both as differential and integral equations:

∫∫ © ( D~ · ~n )d^2 A = Qfree,included ∇ · D~ = ρfree ∫∫ © ( B~ · ~n )d^2 A = 0 ∇ · B~ = 0 ∮ E^ ~ · d~s = − dΦ dt

∇ × E~ = −

∂ B~

∮ ∂t H^ ~ · d~s = Ifree,included + dΨ dt

∇ × H~ = J~free +

∂ D~

∂t

For the fluxes holds: Ψ =

( D~ · ~n )d^2 A, Φ =

( B~ · ~n )d^2 A.

The electric displacement D~, polarization P~ and electric field strength E~ depend on each other according to:

D^ ~ = ε 0 E~ + P~ = ε 0 εr E~, P~ = ∑^ ~p 0 /Vol, εr = 1 + χe, with χe = np

2 0 3 ε 0 kT

The magnetic field strength H~, the magnetization M~ and the magnetic flux density B~ depend on each other according to:

B^ ~ = μ 0 ( H~ + M~) = μ 0 μr H~, M~ = ∑^ m/~ Vol, μr = 1 + χm, with χm = μ^0 nm

2 0 3 kT

2.2 Force and potential

The force and the electric field between 2 point charges are given by:

F^ ~ 12 = Q^1 Q^2

4 πε 0 εrr^2

~er ; E~ =

F~

Q

The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field. The origin of this force is a relativistic transformation of the Coulomb force: F~L = Q(~v × B~ ) = l(I~ × B~ ).

The magnetic field in point P which results from an electric current is given by the law of Biot-Savart , also known als the law of Laplace. In here, d~l ‖ I~ and ~r points from d~l to P :

d B~P = μ 0 I 4 πr^2

d~l × ~er

If the current is time-dependent one has to take retardation into account: the substitution I(t) → I(t − r/c) has to be applied.

The potentials are given by: V 12 = −

∫^2

1

E^ ~ · d~s and A~ = 12 B~ × ~r.

Chapter 2: Electricity & Magnetism 11

2.5.2 Electromagnetic waves in matter

The wave equations in matter, with cmat = (εμ)−^1 /^2 the lightspeed in matter, are:

( ∇^2 − εμ

∂^2

∂t^2

μ ρ

∂t

E^ ~ = 0 ,

∇^2 − εμ

∂^2

∂t^2

μ ρ

∂t

B^ ~ = 0

give, after substitution of monochromatic plane waves: E~ = E exp(i(~k ·~r −ωt)) and B~ = B exp(i(~k ·~r −ωt)) the dispersion relation:

k^2 = εμω^2 +

iμω ρ

The first term arises from the displacement current, the second from the conductance current. If k is written in the form k := k′^ + ik′′^ it follows that:

k′^ = ω

1 2 εμ

(ρεω)^2

and k′′^ = ω

1 2 εμ

(ρεω)^2

This results in a damped wave: E~ = E exp(−k′′~n ·~r ) exp(i(k′~n ·~r − ωt)). If the material is a good conductor,

the wave vanishes after approximately one wavelength, k = (1 + i)

μω 2 ρ

2.6 Multipoles

Because

|~r − ~r ′|

r

∑^ ∞

0

r′ r

)l Pl(cos θ) the potential can be written as: V =

Q

4 πε

n

kn rn

For the lowest-order terms this results in:

• Monopole: l = 0, k 0 =

ρdV

• Dipole: l = 1, k 1 =

r cos(θ)ρdV

• Quadrupole: l = 2, k 2 = (^12)

i

(3z i^2 − r^2 i )

1. The electric dipole: dipole moment: p~ = Ql~e, where ~e goes from ⊕ to , and F~ = (p~ · ∇) E~ext, and W = −~p · E~out. Electric field: E~ ≈

Q

4 πεr^3

3 ~p · ~r r^2

− ~p

. The torque is: ~τ = ~p × E~out

2. The magnetic dipole: dipole moment: if r 

A: ~μ = I~ × (A~e⊥), F~ = (~μ · ∇) B~out

|μ| =

mv ⊥^2 2 B , W = −~μ × B~out

Magnetic field: B~ =

−μ 4 πr^3

3 μ · ~r r^2

− ~μ

. The moment is: ~τ = ~μ × B~out

2.7 Electric currents

The continuity equation for charge is:

∂ρ ∂t

• ∇ · J~ = 0. The electric current is given by:

I =

dQ dt

( J~ · ~n )d^2 A

For most conductors holds: J~ = E/ρ~ , where ρ is the resistivity.

12 Physics Formulary by ir. J.C.A. Wevers

If the flux enclosed by a conductor changes this results in an induced voltage Vind = −N

dΦ dt

. If the current

flowing through a conductor changes, this results in a self-inductance which opposes the original change:

Vselfind = −L

dI dt

. If a conductor encloses a flux Φ holds: Φ = LI.

The magnetic induction within a coil is approximated by: B =

μN I √ l^2 + 4R^2

where l is the length, R the radius

and N the number of coils. The energy contained within a coil is given by W = 12 LI^2 and L = μN 2 A/l.

The capacity is defined by: C = Q/V. For a capacitor holds: C = ε 0 εrA/d where d is the distance between the plates and A the surface of one plate. The electric field strength between the plates is E = σ/ε 0 = Q/ε 0 A where σ is the surface charge. The accumulated energy is given by W = 12 CV 2. The current through a

capacity is given by I = −C

dV dt

For most PTC resistors holds approximately: R = R 0 (1 + αT ), where R 0 = ρl/A. For a NTC holds: R(T ) = C exp(−B/T ) where B and C depend only on the material.

If a current flows through two different, connecting conductors x and y, the contact area will heat up or cool down, depending on the direction of the current: the Peltier effect. The generated or removed heat is given by: W = ΠxyIt. This effect can be amplified with semiconductors.

The thermic voltage between 2 metals is given by: V = γ(T − T 0 ). For a Cu-Konstantane connection holds: γ ≈ 0. 2 − 0. 7 mV/K.

In an electrical net with only stationary currents, Kirchhoff’s equations apply: for a knot holds:

In = 0, along a closed path holds:

Vn =

InRn = 0.

2.8 Depolarizing field

If a dielectric material is placed in an electric or magnetic field, the field strength within and outside the material will change because the material will be polarized or magnetized. If the medium has an ellipsoidal shape and one of the principal axes is parallel with the external field E~ 0 or B~ 0 then the depolarizing is field homogeneous.

E^ ~dep = E~mat − E~ 0 = − N^

P~

ε 0 H^ ~dep = H~mat − H~ 0 = −N M~

N is a constant depending only on the shape of the object placed in the field, with 0 ≤ N ≤ 1. For a few limiting cases of an ellipsoid holds: a thin plane: N = 1, a long, thin bar: N = 0, a sphere: N = 13.

2.9 Mixtures of materials

The average electric displacement in a material which is inhomogenious on a mesoscopic scale is given by:

〈D〉 = 〈εE〉 = ε∗^ 〈E〉 where ε∗^ = ε 1

φ 2 (1 − x) Φ(ε∗/ε 2 )

where x = ε 1 /ε 2. For a sphere holds: Φ = 1 3 +^

2 3 x. Further holds:^ ( ∑

i

φi εi

≤ ε∗^ ≤

i

φiεi