1 of 10 Gauss s Law Gauss s Law is one of the 4 fundamental laws of electricity and magnetism called Maxwell s Equations Gauss s law relates charges and electric fields in a subtle and powerful way but before we can write down Gauss s Law we need to introduce a new concept the electric flux through a surface E surface with area A Consider an imaginary surface which cuts across some E field lines We say that there is some electric flux through this surface To make the notion of flux precise we must first define a surface vector K Definition surface vector A A A n associated with a flat surface of area A Magnitude of vector A area A of surface A Direction of vector A direction perpendicular normal to area A surface direction of unit normal n Notice that there is an ambiguity in the direction n Every flat surface has two perpendicular directions A smaller area shorter A The electric flux through a surface A is defined as K K E A E A cos A E for E constant surface flat The flux has a the following geometrical interpretation flux the number of electric field lines crossing the surface Think of the E field lines as rain flowing threw an open window of area A The flux is a measure of the amount of rain flowing through the window To get a big flux you need a large E a large A and you need the area perpendicular to the E field vector which means the area vector A is parallel to E In the rain analogy you need the window to be facing the rain direction A E E A 0 cos 1 max PHYS1120 Lecture Notes Dubson 9 10 2009 90o cos 0 0 University of Colorado at Boulder 2 of 10 A cos is the projection of the area A onto the plane perpendicular to E It is the area which faces the rain Only the area facing the rain gives flux E A E area A cos K K The formula E A is a special case formula it only works if the surface is flat and the E field is constant If the E field varies with position and or the surface is not flat we need a more general definition of flux K K E da surface integral of E E E da To understand a surface integral do this in your imagination break the total surface up K into many little segments labeled with an index i The surface vector of segment i is da i If the segment is very very tiny it is effectively flat and the E field is constant over that K K tiny surface so we can use our special case formula E A K K The flux through segment i is therefore i E i da i Ei is the field at the segment i K K K K The total flux is the sum E i da i E da i In the limit that the segments become infinitesimal there are an infinite number of segments and the sum becomes an integral K K In general computing surface integral E da can be extremely messy So why do we care about this thing called the electric flux The electric flux is related to charge by Gauss s Law PHYS1120 Lecture Notes Dubson 9 10 2009 University of Colorado at Boulder 3 of 10 the 1st of 4 Maxwell s Equations Gauss s Law K K q enclosed E da v 0 In words the electric flux through any closed surface S is a constant 1 0 times the total charge inside S K K E v da closed surface integral closed da A surface is closed if it has no edges like a sphere For a closed surface the direction of da is always the outward normal 1 4 0 The constant 0 is related to k by k Fcoul k q1 q 2 r2 da 1 q1 q 2 4 0 r 2 0 8 85 10 12 SI units Gauss Law can be derived from Coulomb s Law if the charges are stationary but Gauss s Law is more general than Coulomb s Law Coulomb s Law is only true if the charges are stationary Gauss s Law is always true whether or not the charges are moving It is easy to show that Gauss s Law is consistent with Coulomb s Law From Coulomb s kQ 1 Q Law the E field of a point charge is E 2 We get the same result by r 4 0 r 2 applying Gauss s Law r K K v E da S v E da E v da since E is parallel to da on S since E is constant on S EA imaginary spherical surface S radius r Solving for E we have E E 4 r2 Q 0 says Mr Gauss 1 Q Done 4 0 r 2 PHYS1120 Lecture Notes Dubson 9 10 2009 University of Colorado at Boulder 4 of 10 When viewed in terms of field lines Gauss s Law is almost obvious after a while Recall that flux is proportional to the number of field lines passing through the surface Notice also that flux can be positive or negative depending on the angle between the Efield vector and the area vector Where the field lines exit a closed surface the flux there is positive where the field lines enter a closed surface the flux there is negative So the total flux through a closed surface is proportional to field lines exiting field lines entering da da E exiting S E entering S If a closed surface S encloses no charges then the number of lines entering must equal the number of lines exiting since there are no charges inside for the field lines to stop or start on imaginary surface S E K K E v da 0 S S So only charges inside the surface can contribute to the flux through the surface Positive charges inside produce positive flux negative charges produce negative flux The net flux is due only to the net charge inside K K q E v da enclosed 0 Using Gauss s Law to solve for the E field Gauss s Law is always true it s a LAW But it is not always useful Only in situations K K with very high symmetry is it easy to compute the flux integral v E da In these few cases of high symmetry we can use Gauss s Law to compute the E field PHYS1120 Lecture Notes Dubson 9 10 2009 University of Colorado at Boulder 5 of 10 Example of Spherical Symmetry Compute E field everywhere inside a uniformly charged spherical shell By symmetry E must be radial along a radius so E E r We choose an imaginary surface S concentric with and inside the charged sphere S Q Since the E field is radial and the surface vector K K da on S is also radial we have E da E da The dot …
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