Physics Formulae/Electromagnetism Formulae

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


— 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,


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  



(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
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  
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