Electromagnetic Fields : Maxwell's, Poisson's & Laplace equations

Maxwell’s Equations in Time-Varying Field

• Maxwell's equations in point form
  1.  (Faraday’s Law)
     
  2. (Modified Ampere’s circuital Law)
     
     
  3. (Gauss’s Law)
     
     
  4. (Non-existence monopole or Gauss’s law of magnetostatics)
     
     
     Maxwell's equations in integral form
     
     
     
     
     

     Key Points

     • For static fields (or non-time varying fields)
       
       The basic equations of electromagnetism are the four Maxwell Equations and the Lorentz force law. In principle, these, together with Newton’s second law of motion are enough to completely determine the motion of an assembly of charges given the initial positions and velocities of all the charges. Maxwell’s equations are, 
       
       
       
       
       Here,  are the electric and magnetic fields respectively. The sources for the fields are the volume charge density ρ and the current density. The two parameters in these equations are the permittivity of free space  and the permeability of free space . The vector, differential-operator in these equations is defined as
     • 
       
       

       these equations can be used to determine the fields if the charge and current densities are known. Once the fields are known, the force felt by a given charge q moving with velocity  is given by the Lorentz force law,

       

       In principle, Equations (1)-(5) constitute all electromagnetic. However, in practice, the charge and current densities associated with matter are too complicated to specify and so a phenomenological means of dealing with the matter is often introduced.

     Maxwell Equation for Time-Varying Field : 
     
     

             Maxwell Equation for Time-Varying Field in Free Space

      

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