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22                                                                 Chapter 1

        such images are clear for simple fields and charges distributions they are much messier in more
        complicated cases. 2D-  and 3D-diagrams,  where the gradients of color variations reflect
        electrical field strength, as shown in Figure 1.6.4 , are much more informative. In Figure 1.6.4a
                                               6
        areas of highest (bright red) and lowest (dark green) strength, are clearly visible and lines of
        force can be added if their structure is not too complicated. Quite often, the images like 1.6.4b
                                                         could be part of a contemporary
                                                         art exhibition!

                                                         In conclusion, let us evaluate the
                                                         integral in (1.20) along the line of
                                                         force. Since  = const. along the
                               a)                   b)   integration curve L

            Figure 1.6.4 Electric field strength around electric      = Δ (Δ )⁄         (1.22)
                                                                     
                              dipole                     If so,  measuring the energy
                                                         acquitted by the sensor #1 we can
        define the electrical field strength as the energy of electric field required to move the point-like
        charge of 1C at the distance of 1m along the line of force.

        1.6.4   Gauss’s Law for Electric Fields (Axiom #2) and Coulomb’s Law
        Gauss’s law expresses the total flux of electric fields at any moment of time through the closed
        surface area  of any shape (see Figure 1.6.5a) and can be written as

                                     ∯ (, ) ∘  =  ()            (1.23)
                                                         ⁄
                                                    0
                                                       Here   ()  is the total charge
                                                              
                                                       within some volume  V  at any
                                                       current  moment of time.  Some
                                                       charges can stay at rest or move in
                                                       any direction inside V, can leave V
                                                       or arrive from outside V, as shown
                              a)                   b)   in Figure 1.6.5a.  They can be
                                                       positive  or   negative.  The
                 Figure 1.6.5 Volume V with charges    coefficient          called  the
                                                                     0
                                                       permittivity of free space or
        vacuum is required by the chosen SI unit set as we have demonstrated above in Example #3 of
        Section 1.3.3.
        Assuming that V is the sphere of radius r and holds a single at rest point-like charge  () = 
                                                                            
        in the sphere center, as shown in  Figure 1.6.5b, we can derive Coulomb’s law combining
        Gauss’s law and Lorentz’s force equation. Using the fact that the geometrical structure in Figure
        1.6.5b has the complete spherical symmetry, we can suggest that the vector of electrical field is
        constant on the sphere surface and can be pulled out of the integral. Then



        6  Public Domain Image, source: http://sciencewise.blogspot.com/2008/01/exploring-electrostatics.html
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