Physics Formulae/Electromagnetism Formulae

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Lead Article: Tables of Physics Formulae


This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Electromagnetism.


Laws of Electromagnetism[edit]

Maxwell's Equations[edit]

Below is the set in the differential and integral forms, each form is found to be equivalant by use of vector calculus. There are many ways to formulate the laws using scalar/vector potentails, tensors, geometric algebra, and numerous variations using different field vectors for the electric and magnetic fields.

Name Differential form Integral form
Gauss's law
Gauss's law for Magnetism
Maxwell–Faraday Law
(Faraday's law of induction)
Maxwell-Ampère Circuital law
(Ampere's Law with Maxwell's correction)
Lorentz Electromagnetic

Force Law


The Field Vectors


Central to electromagnetism are the electric and magnetic field vectors. Often for free space (vacumm) only the familiar E and B fields need to be used; but for matter extra field vectors D, P, H, and M must be used to account for the electric and magnetic dipole incluences throughout the media (see below for mathematical definitions).


The electric field vectors are related by:
The magnetic field vectors are related by:


Interpretation of the Field Vectors


Intuitivley;

— the E and B (electric and magnetic flux densities) fields are the easiest to interpret; field strength is propotional to the amount of flux though cross sections of surface area, i.e. strength as a cross-section density.

— the P and M (electric polarization and magnetization respectivley) fields are related to the net polarization of the dipole moments thoughout the medium, i.e. how well they respond to an external field, and how the orientation of the dipoles can retain (or not) the field they set up in response to the external field.

— the D and H (electric displacement and magnetic intensity field) fields are the least clear to understand physically; they are introduced for convenient thoretical simplifications, but one could imagine they relate to the strength of the field along the flux lines, strength as a linear density along flux lines.


Hypothical Magnetic Monopoles


— As far as is known, there are no magnetic monopoles in nature, though some theories predict they could exist.

— The approach to introduce monopoles in equations is to define a magnetic pole strength, magnetic charge, or monopole charge (all synonomous), treating poles analogously to the electric charges.

— One pole would be north N (numerically positive by convention), the other south S (numerically negative). There are two units which can be used from the SI system for pole strength.

— Pole srength can be quantified into densities, currents and current densities, as electric charge is in the previous table, exactly in the same way.

Maxwell's Equations would become one of the columns in the table below, at least theoretically. Subscripts e are electric charge quantities; subscripts m are magnetic charge quantities.


Name Weber (Wb) Convention Ampere meter (A m) Convention
Gauss's Law
Gauss's Law for magnetism
Faraday's Law of induction
Ampère's Law
Lorentz force equation


They are consistent if no magnetic monopoles, since monopole quantities are then zero and the equations reduce to the original form of Maxwell's equations.

Pre-Maxwell Laws[edit]

These laws are not fundamental anymore, since they can be derived from Maxwell's Equations. Coulomb's Law can be found from Gauss' Law (electrostatic form) and the Biot-Savart Law can be deduced from Ampere's Law (magnetostatic form). Lenz' Law and Faraday's Law can be incorperated into the Maxwell-Faraday equation. Nonetheless they are still very effective for simple calculations, especially for highly symmetrical problems.


Coulomb's Law

For a non uniform charge distribution, this becomes:

Biot-Savart Law
Lenz's law Induced current always opposes its cause.

Electric Quantities[edit]

Electric Charge and Current


Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Elementary Charge Quantum C = A s [I][T]
Quantized Electric Charge q C = A s [I][T]
Electric Charge (any amount) C = A s [I][T]
Electric charge density of dimension n

( = n-space)

n = 1 for linear mass density,

n = 2 for surface mass density,

n = 3 for volume mass density,

etc

linear charge density ,

surface charge density ,

volume charge density ,


no general symbol for

any dimension


n-space charge density:

special cases are:

C m-n [I][T][L]-n
Total descrete charge C [I][T]
Total continuum charge

n-space charge density

special cases are:


C [I][T]
Capacitance F = C V-1
Electric Current A [I]
Current Density A m-2 [I][L]-2
Displacement current A [I]
Charge Carrier Drift Speed m s-1 [L][T]-1


Electric Fields


Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Electric Field, Field Strength,

Flux Density, Potential Gradient

N C-1 = V m-1 [M][L][T]-3[I]-1
Electric Flux N m2 C-1 [M][L]3[T]-3[I]-1
Electric Permittivity F m-1
Dielectric constant,

Relative Permittivity

F m-1
Electric Displacement Field C m-2 [I][T][L]-2
Electric Displacement Flux C [I][T]
Electric Dipole Moment vector

is the charge separation

directed from -ve to +ve charge

C m [I][T][L]
Electric Polarization C m-2 [I][T][L]-2
Absolute Electric Potential

relative to point

Theoretical:

Practical:

(Earth's radius)

V = J C-1
Electric Potential Difference
Electric Potential Energy J [M][L]2[T]2

Magnetic Quantities[edit]

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Magnetic Field, Field Strength,

Flux Density, Induction Field

T = N A-1 m-1
Magnetic Flux Wb = T m-2
Magnetic Permeability H m-1
Relative Permeability H m-1
Magnetic Field Intensity,

(also confusingly the field strength)

Magnetic Dipole Moment vector

N is the number of turns of conductor

A m2 [I][L]2
Magnetization
Self Inductance Two equivalent definitions are in fact possible:

H = Wb A-1
Mutual Inductance Again two equivalent definitions are in fact possible:

X,Y subscripts refer to two conductors mutually inducing

voltage/ linking magnetic flux though each other

H = Wb A-1


Electric Fields[edit]

Electrostatic Fields[edit]

Common corolaries from Couloumb's and Gauss' Law (in turn corolaries of Maxwell's Equations) for uniform charge distributions are summarized in the table below.


Uniform Electric Field accelerating a charged mass
Point Charge
At a point in a local

array of Point Charges

Electric Dipole

Line of a Charge
Charged Ring
Charged Conducting Surface
Charged Insulating Surface
Charged Disk
Outside Spherical Shell r>=R
Inside Spherical Shell r<R
Uniform Charge r<=R
Electric Dipole Potential Energy

in a uniform Electric Eield

Torque on an Electric Dipole

in a uniform Electric Eield

Electric Field Energy Density

Linear media (constant throughout)

For non-uniform fields and electric dipole moments, the electrostatic torque and potential energy are:



Electric Potential and Electric Field


Electrostatic Potentials[edit]

Point Charge
Pair of Point Charges


Electrostatic Capacitances[edit]

Parallel Plates
Cylinder
Sphere
Isolated Sphere
Capacitors Connected in Parallel
Capacitors Connected in Series
Capacitor Potential Energy

Magnetic Fields[edit]

Magnetic Forces[edit]

Force on a Moving Charge


Force on a Current-Carrying Conductor


Magnetostatic Fields[edit]

Common corolaries from the Biot-Savart Law and Ampere's Law (again corolaries of Maxwell's Equations) for steady (constant) current-carrying configerations are summarized in the table below.


For these types of current configerations, the magnetic field is easily evaluated using the Biot-Savart Law, containing the vector , which is also the direction of the magnetic field at the point evaluated.

For conveinence in the results below, let be a unit binormal vector to and , so that



then also the unit vector for the direction of the magnetic field at the point evaluated.

Hall Effect
Circulating Charged Particle
Infinite Line of Current
Magnetic Field of a Ray
Center of a Circular Arc
Infinitley Long Solenoid
Toroidal Inductors and Transformers
Current Carrying Coil
Magnetic Dipole Potential Energy

in a uniform Magnetic Eield

Torque on a Magnetic Dipole

in a uniform Magnetic Eield

For non-uniform fields and magnetic moments, the magnetic potential energy and torque are:


Magnetic Energy[edit]

Magnetic Energy for Linear Media

( constant at all points in meduim)

Magnetic Energy Density for Linear

Media ( constant at all points in meduim)

EM Induction[edit]

Self Induction of emf
Mutual Induction
transformation of voltage

Induced Magnetic Field

inside a circular capacitor

Induced Magnetic Field

outside a circular capacitor

External Links[edit]

Maxwell's Equations

Electric Field

Magnetic Field

Electric Charge

Magnetic Monopoles