flux of an Electric Field

flux of an Electric Field

To define the flux of an electric field consider  which shows an arbitrary
(asymmetric) Gaussian surface immersed in a nonuniform electric field let  us divide the surface into small squares of area each square being small enough to permit us to neglect any curvature and to consider the individual square to be flat We represent each such element of area with an area vector whose magnitude is the area  Each vector  is perpendicular to the Gaussian surface and directed away from the interior of the surface Because the squares have been taken to be arbitrarily small the electric field E may be taken as constant over any given square The vectors j A and E for each square then make some angle with each other enlarged view of three squares (I. 2. and 3) on the  Gaussian surface and the angle  for each A provisional definition for the flux of the electric field for the Gaussian surface This instructs us to visit each square on the Gaussian surface to evaluate the scalar product E for the two vectors E and  that we find there and to sum the results algebraically (that is with signs included) for all the squares that make up the surface The sign or a zero resulting from each scalar product determines whether the flux through its square is positive negative or zero Squares like I in which E points inward make a negative contribution to the sum of Squares like 2 in which E lies in the surface make zero contribution Squares like 3 in which E points outward make a positive contribution The exact definition of the flux of the electric field through a closed surface is found by allowing the area of the squares shown in to become smaller and smaller approaching a differential limit  The area vectors then approach a differential limit  The sum of  then becomes an integral and we have for  the definition of electric flux.1

The circle on the integral sign indicates that the integration is to be taken over the entire (closed) surface The flux of the electric field newton-square-meter per coulomb (N . m”/C). We can interpret in the following way First recall that we can the density of elector field lines passing through an area as a proportional measure of an electric field there Specific all the magnitude  is proportional to the number of electric field lines per unit area Thus the scalar product of is proportional to the number of electric field lines passing through area Then because the integration in is carried out over a Gaussian surface which is closed, we see that The electric flux through a Gaussian surface is proportional to the net number of electric field lines passing through that surface.

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