Physics 308: Equations and Principles, Spring 2000
Daniel W. Koon, St. Lawrence University

Chapters are from Purcell's Electricity and Magnetism
Chapter 1:
Coulomb:
Electric field:
Potential energy:
Flux & Gauss: F =
Charge distributions:
      spherical
      line charge
      surface charge
Energy density:
NOTE: For CGS units, set k=1. For MKS, 4p k=1/eo.

You should memorize:
      The MKS, CGS units of charge, force, energy, etc.
      The expressions for force, electric field for collections of discrete charges
You should be able to:
      Calculate force, e-field, potential energy, and energy density for collections of point charges
      Apply Gauss' Law to calculate E for simple symmetries (point, plane, line, spherically symmetric charge density)
      Integrate spherically symmetric charge densities to get E or stored energy
      Identify the appropriate Gaussian surfaces for simple symmetries

Chapter 2:
Nabla ("Del" operator):
Gradient:
Divergence:
Curl:
Laplacian:
Gauss' Law (differential form):
Curl of :
Electric potential
'The circle':




Energy density: u=r f

You should memorize:
      The equations for div, grad, curl, Laplacian
      All six equations from "the wheel"
      The equations for F, E, PE, and V
      What the divergence and curl of the electric field equal
You should know:
      The similarities and differences between F, E, PE, and V
You should know how to:
      Convert between E, r, and F, including being able to do the appropriate path, polar, and spherical integrations
      Apply Gauss' and Stokes' Theorems, using the appropriate integrations


Chapter 3:
Perfect conductors: E = 0 inside
f=constant on surface
E|| = 0 at surface
E^ = 4p[ k]s at surface
Boundary value problems:
      Uniqueness
      Method of images
      Conformal mapping
      Spherical or cylindrical harmonics
      Relaxation, overrelaxation
Capacitance: C=Q/V
      of isolated sphere:
      of isolated disc:
      of parallel plates:
      of parallel capacitors:
      of series capacitors:
Energy storage: U=QV=CV2=
You should memorize:
      The definition of capacitance and its units
      The expressions for capacitance of series and parallel capacitors
      Any one of the three expressions for the energy stored in a capacitor
You should know:
      The qualitative properties of a Faraday cage, the rules about an electric field in a conductor
You should be able to:
      Solve boundary value problems (BVPs) using the method of images, relaxation
      Calculate the capacitance of simple symmetric systems, including parallel plates, isolated sphere, coaxial cable, and nested spheres.

Chapter 4:
Current:
Current density:
Continuity:
Ohm: V = IR
             
Conductance: s = nem =
Resistivity: r = = 1/ s
Kirchhoff: ,      
Equivalent resistances: Rs = R1 + R2 + ...
     
Power dissipation R: P = VI = I2R =V2/R
Thevenin: e Th = Vopen
     
RTh = e Th/Ishort
Charging RC circuit: Q=Q0(1-e-t/RC)
Discharging RC circuit: Q=Q0e-t/RC
You should memorize:
      The continuity equation (Eq. 9)
      Ohm's Law in microscopic form (Both eqs.)
      The formulae for series and parallel resistors
      Both of Kirchhoff's Laws
      The time constant of an RC circuit
You should know:
      What causes electrical resistance
      How resistivity varies with temperature for metals and for semiconductors, and why
You should be able to:
      Simplify complex resistor networks
      Solve complex networks of resistors and batteries
      Apply Thvenin's theorem
      Convert between resistance and resistivity for a wire
      Devise the equations for Q, I, VR, and VC in an RC circuit

Chapter 5:
Lorentz force:
Charge is relativistically invariant.
in different frames: E'|| = E||

Accelerating charges ® electromagnetic waves
is a consequence of relativity.
You should know:
      That charge is both conserved and invariant in all inertial reference frames
      How magnetic fields, electrostatics, and special relativity are interrelated
You should be able to:
      Apply the equations for a relativistic electric field (Sections 5 & 6)
      Apply the equation for the perpendicular component of 'magnetic' force (Eq. 24)

Chapter 6:
Magnetic field: and
Biot & Savart:
Ampere:
Vector potential:

Field due to . . .
      long wire:
      single coil: on axis
                  at center
      solenoid:
Hall effect: , RH=
You should memorize:
      CGS and MKS expressions for force on a charge
      The curl of the magnetic field
      Ampere's law
You should know:
      the right-hand rules for field and force
You should be able to:
      Calculate the total B-field due to more than one long wire, using the appropriate right-hand rules
      Calculate the B-field near a cylindrically symmetric current density
You should keep for future reference:
      The formulae for B near a long wire, coil, solenoid

Comparison of electrostatic and magnetostatic fields:

Electrostatics -- 
Magnetostatics -- 
Source of field Coulomb:  Biot & Savart: 
Force = q Lorentz: 
CGS Units of field statvolt/cm = dyne/esu = esu/cm/cm Gauss = same as units of 
How to calculate: Gauss' Law:  Ampere's Law: 
     (local form) Gauss' Law (local form):  Ampere's Law (local form): 
Potential Since , there is a scalar
      potential, , such that 
Since , there is a vector
      potential, , such that 
"Sandwich" DE = 4p[k]s = 4p[k] Q/A DB=4pJ/c =4pI/lc
Chapter 7
Motional emf: bar: e=
Motional emf: loop:
e=
Faraday:     Flux form:
e =
      Differential form:
Eddy currents, eddy fields, Meissner effect
Flux through a coil:     NBAcosf
Self Inductance: L = -e11/
LR circuit: t = L/R
Energy storage: u =
You should memorize:
      Both forms (flux and differential) of Faraday's law
      The expression for the flux through a coil
      The expression for the emf across an inductor
      The expression for the time constant in an LR circuit
You should know:
      What eddy currents are, where they come from, and some examples of them
You should be able to:
      Calculate all the voltages and currents in an LR circuit
      Compare the energy stored by an inductor to the energy density in its magnetic field
You should keep for future reference:
      The motional emf of a moving bar or loop
      The energy stored in an inductor
      The energy stored in a magnetic field

Chapter 8:
Resonant circuit: XC = XL,
          Z = R
at resonance
Quality factor:   Q = RCw = w L/R
Complex analysis: I = Re(Ioeiwt)
Admittance: Y = I/V
Reactance: X = Im [V/I]
Impedance: Z = V/I
Z   of resistor: R
     
of inductor: iwL
     
of capacitor:
Power dissipation:
      instantaneous: P = VI
     
average, for resistor: P = VrmsIrms
     
average, in general: P = Vrm sIrm scosf

You should memorize:
      The impedance for R, L, and C
You should know:
      The condition for resonance (ZC=ZL)
You should be able to:
      Calculate the impedance of any combination of R, L, and C
      Calculate the power dissipated in any circuit containing R, L, and C
You should keep for future reference:
      The expression for the quality factor, Q
Chapter 9:
Gauss:
Faraday:
Gauss for magnetic field:
Ampere/Maxwell:

Electromagnetic waves in free space: v=c and E0 = B0
energy density: <u> =
Poynting:
               
You should memorize:
      The CGS relation between B0 and E0
      The velocity of an electromagnetic wave
You should be able to:
      Show whether a given expression for E or B, or both, is consistent with Maxwell's equations
      Solve for E(x,y,z,t), given B, or vice versa
You should keep for future reference:
      Maxwell's equations (so far)
      The expressions for energy density and for the magnitude of the Poynting vector (in MKS)

Chapter 10:
Capacitors: C =
Energy density: u =
Multipole expansion:
Point electric dipole:
Electric field due to dipole:
     
     
     
Dipole in -field:
     
     
Polarization:
Displacement vector:
           
           
Boundary conditions: continuous
Maxwell update:

Electromagnetic waves:


Rayleigh scattering: µ w 4
 
Note: e o = 1 for CGS, 4p k = 1/e o for MKS.
You should memorize:
      The definition of the electric dipole
      The equations for energy and torque of a dipole in a field
      The capacitance of a capacitor with a dielectric between its plates
You should know:
      What causes Rayleigh scattering, and its properties
      What Fresnel's equations are, and how they relate to E&M
You should be able to:
      Calculate the dipole moment of a collection of discrete charges
      Calculate U, t, and F of a dipole in a field
You should keep for future reference:
      The definitions of the vectors D, P, and related quantitites
      The updated Maxwell's equations

Chapter 11:
Diamagnets, Paramagnets, Ferromagnets
Absence of magnetic monopoles
Point magnetic dipole:
Magnetic field due to dipole:
     
Dipole in -field:
     
     
Magnetic field ()magnetization ()
Magnetic induction:
     
     
Boundary conditions: perp and par are continuous
Maxwell's equations in materials: m o=1 in CGS
Electromagnetic waves:
You should know:
      What the three major categories of materials are (in terms of magnetic response), and the properties of each
You should be able to:
      Calculate magnetic [dipole] moment
      Calculate U, t, and F of a magnetic dipole in a magnetic field
      Describe magnetism in a material both microscopically and macroscopically
You should keep for future reference:
      The final version of Maxwell's equations (See below)

MAXWELL'S EQUATIONS, final version:
Equation:
CGS
MKS
Electric: divergence
Electric: curl
Magnetic: divergence
Magnetic: curl

COMPARISON BETWEEN:
Polarization and magnetization     /     Electric dipoles and magnetic dipoles     /     Chapter 10 and Chapter 11
Electric
Magnetic
Dipoles
      E-field
     
      E-field
     
      Energy
      Torque
      Force
Internal field
Free charges and currents
       
       

CONVERSIONS:
Charge: 1 esu = 1/3´ 10-9C
Voltage: 1 statvolt = 300V
Capacitance: 1 cm = 1.11pF
Current: 1 esu/s = 1/3 ´ 10-9A
Resistance: 1 s/cm = 9´ 1011W
Inductance: 1 s2/cm = 9´ 1011H
Magnetic field: (B): 1 G = 10-4T