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CU-Boulder PHYS 1120 - Gauss's Law

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1 of 11 PHYS1120 Lecture Notes, Dubson, 1/26/2017 ©University of Colorado at Boulder E surface with area A A area A A smaller area, shorter A A E 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. 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. Definition: surface vector A = ˆA An, associated with a flat surface of area A. Magnitude of vector A = area A of surface. Direction of vector A = direction perpendicular (normal) to surface = direction of unit normal ˆn. Notice that there is an ambiguity in the direction ˆn. Every flat surface has two perpendicular directions. The electric flux  through a surface A is defined as E A EAcos     (for E = constant, surface flat ) The flux  has the following geometrical interpretation: | flux |  the number of electric field lines crossing the surface. Think of the E-field lines as rain flowing through 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.) E A  = 0, cos =1,  max E A = 90o, cos = 0 ,  = 02 of 11 PHYS1120 Lecture Notes, Dubson, 1/26/2017 ©University of Colorado at Boulder (A cos is the projection of the area A onto the plane perpendicular to E. The plane perpendicular to E is the area which "faces the rain". Only the area facing the rain contributes to the flux. Let’s consider this flux business in a little more detail. In the diagram below, we have a constant electric field E, passing through surface 1, represented by vector A, tilted at angle . [We use bold font A for vectors.] This tilted surface 1 has area | A | = A = LW. The projection of this surface 1 onto the plane perpendicular to the E-field is surface 2, which has an area that we call A (for area of surface perpendicular to direction of E). The area of this plane, this surface 2, is A = L W = LcosW = A cos. So we have A = A cos. Now we are going to show that the flux EA through a surface A is proportional to the number of field lines passing through the surface. Recall that, from the definition of a field line diagram, the magnitude of the E-field is proportional to the density of the lines: | E| E (# field lines)/A N/ A   (N is the number of field lines through the area A). So, we haveN E A. Now, we showed above that A Acos, so we have N E A E Acos E A     . Done! The number N of field lines through a surface is proportional to the fluxEA. E E A area = A cos E E A Surface 2: area = A = A cos W L L = L cos Surface 1: area = A = LW3 of 11 PHYS1120 Lecture Notes, Dubson, 1/26/2017 ©University of Colorado at Boulder We can now see that, since the same number of E-field lines pass through both surfaces 1 and 2, they must have the same magnitude flux. The math shows the same thing: For surface 1, 1EAcos  . For surface 2, 2EA EAcos   . Now, the formula EA 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: E da "surface integral of "   E To understand a surface integral, do this: in your imagination, break the total surface up into many little segments, labeled with an index i. The surface vector of segment i is ida. If the segment is very, very tiny, it is effectively flat and the E-field is constant over that tiny surface, so we can use our special case formula EA   . The flux through segment i is therefore i i iE da  . (Ei is the field at the segment i) The total flux is the sum: iiiE da E da     (In the limit that the segments become infinitesimal, there are an infinite number of segments and the sum becomes an integral.) 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. E E da 4 of 11 PHYS1120 Lecture Notes, Dubson, 1/26/2017 ©University of Colorado at Boulder imaginary spherical surface S, radius r r + Gauss's Law (the 1st of 4 Maxwell's Equations) enclosed0qE da In words, the electric flux through any closed surface S is a constant (1/0) times the total charge inside S. E da surface integralclosed 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. The constant 0 is related to k by 01k4. 1 2 1 2coul220kq q q q1Fr 4 r 1208.85 10   (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 Law, the E-field of a point charge is220kQ 1 QEr 4 r . We get the same result by applying Gauss's Law: (since E is parallel to da on S) (since E is constant on S) (says Mr. Gauss) Solving for E, we have 201QE4r . Done. "closed" da da  S20E da EdaE daEAQE 4 r  5 of 11 PHYS1120 Lecture Notes, Dubson, 1/26/2017 ©University of Colorado at Boulder 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 E-field

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