Maxwell's Equations

Maxwell's Equations

There are two different types of field, electric and magnetic. The relationship between these two fields and charges and currents can be summarized in just four statements or equations. These are known as Maxwell's equations. An extremely clear and very detailed description of Maxwell's equations, how you derive them and how you apply them using both basic mathematics and vector calculus is given by Feynman in the early chapters of volume 2 of his famous books The Feynman Lectures on Physics (Ref 3a). In words, Maxwell's equations are :-

1a)  For any closed surface the average normal component of the

            electric field through the surface times the surface area

                        EQUALS

            The sum of the charges enclosed divided by εo

Where εo is just a constant to ensure the units come out correctly. The sign of the charges is also important so if equal number of positive and negative charges are enclosed the sum of these charges is zero. The normal component is the field component at right angles to the surface.

 

If we consider any surface bounded by a loop (similar to considering a drawstring bag as the surface and the perimeter of the bag opening, however wide, as the loop) :-

2a)       Around any loop in space the average tangential component

                  of electric field times the distance around the loop

                                    EQUALS

            The rate of change of the average normal component of

             magnetic field through the surface of the loop times the

                                    surface area.

 

3a)       For any closed surface the average normal component of the

              magnetic field through the surface times the surface area

                                    EQUALS

                                       Zero

This is similar to equation (1a) but for the magnetic field rather than the electric.

 

4a)       Around any loop in space the average tangential component

                    of the magnetic field times the loop length times c2

                                    EQUALS

            The rate of change of the average normal component

             of electric field through the loop times the loop area

                                    MULTIPLIED BY

            The total electric current through the surface divided by εo

Where c2 is the velocity of light squared. This is similar to equation (2a) for the electric field but involves an extra current term.

In using the above definitions a closed surface is the bounding surface of any volume of space ie the outside surface of a sphere or the outside of a cube or in fact the outside of any object you care to imagine. The normal component of a field is just the component at right angles to the surface being considered.

The above equations are all that is required to explain the relationships between the electric and magnetic fields and it is amazing how much complexity can arise from such a seemingly simple set of relationships. Also it is surprising that it is so difficult to find the various field configurations that can be formed which still satisfy all the above. To this end it is much easier to put the expressions into mathematical equations which have a more precise meaning and allow the various terms to be manipulated further. There are a number of different mathematical ways of doing this but the one chosen for this web site is vector calculus. Using this notation some of the phrases used above take on a symbolic form. For example:-

The average normal component of a field (electric E or magnetic B) per unit volume is denoted as the divergence and abbreviated to divE or divB, so the total electric field out of a volume (dV) is divE.dV .

The charge per unit volume is denoted by ρ, so the total charge in a volume (dV) is just ρ.dV . Using these symbols allows equation (1a) to be rewritten as:-

            divE.dV =   

            or divE =                ________________(1b)

The average tangential component of an electric field around any loop is the normal component of curlE times the area of the loop. The usual convention is that the magnitude of this curl is shown by a vector normal to the surface area of the loop. However, it is possible to go around the loop either clockwise or anticlockwise. The convention always adopted is that if you curl your right hand around the loop with your fingers pointing in the direction you have decided is positive, then your thumb points in the direction of the positive normal to the surface. i.e. If you have a circulation in the direction of your fingers, the curl is a vector in the direction of your thumb.

The rate of change of a quantity in calculus is shown as  , hence the rate of change of the average normal component of the magnetic field through the surface of the loop of area (dA) is written  . With the convention adopted for the curl direction this rate of change is negative. Using these two expressions we can

write the second of Maxwell's equations as:-

            curlE dA = -       

            or curlE = -   ________________(2b)

Using similar notation as for equation (1b), equation (3b) becomes:-

            divB = 0 _________________(3b)

Equation (4a) becomes:-

           c2. curlB dA =     +   

      and as I = j dA, where j is the average normal component of current per unit area through the surface. Then:-

           c2. curlB =         __________________(4b) 

The last equation involves a flow of current. Often we know the currents and charges and can use the equations to calculate the fields that they produce. In problems involving the propagation of fields in a vacuum the current and charges are taken to be zero and equation (4b) simplifies to:-

              c2. curlB =       ___________________(5b)

The above equations numbers (1b), (2b), (3b) and (4b) are Maxwell's equations in mathematical form. In a charge free vacuum we can just use equations (1b), (2b), (3b) and (5b) and these enable us to calculate the propagation of any electromagnetic field.

 

 

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