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--- 184_notes:gauss_motive [2017/07/08 20:14] – [Conceptualizing Gauss's Law around a Point Charge] caballero
+++ 184_notes:gauss_motive [2018/07/24 14:47] – [Conceptualizing Gauss's Law around a Point Charge] curdemma
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+[[184_notes:e_flux|Next Page: Electric Flux and Area Vectors]]
 ===== Motivation for Gauss's Law =====
+Last week we learned about Ampere's Law for finding magnetic fields. This week we will look into Gauss's Law in order to be better able to find Electric Fields.
-We have been analyzing the kinds of electric fields that are produced by charges. We started by finding the [[184_notes:pc_efield|electric field of a point charge]] and then used the electric field from a point charge to build up the [[184_notes:linecharge|electric field from a line of charge]]. The method of building a field from a point charge can also be used for a 2D sheet of charge or a 3D volume of charge and will //always// work; however, the mathematical calculation needed to determine the field can become much more complicated (you have to use either a double or triple integral). In some cases, we can use the symmetric nature of the situation to apply Gauss's Law as a shortcut for finding the electric field, even though there are trade offs for using this shortcut (it is alway true, but only useful in highly symmetric situations). In the end, both of these methods can tell us the same thing, even though the mathematics we will use is different: **charges create electric fields**. These notes will introduce the general concept of Gauss's law before we talk about the mathematics and the advantages and disadvantages to using it.
+Back at the beginning of the semester, we talked about how to find the electric field from [[184_notes:pc_efield|point charges]], [[184_notes:line_fields|lines of charge]], and [[184_notes:dist_charges|distributions of charge]] (i.e., cylinders, spheres, or planes of charge). The method of building a field from a point charge can also be used for a 2D sheet of charge or a 3D volume of charge and will //always// work; however, the mathematical calculation needed to determine the field can become much more complicated (you have to use either a double or triple integral). In some of these cases, Gauss's Law works as a shortcut for finding the electric field from complicated charge distributions. However, there are trade offs for using this shortcut -- it is alway true, but only useful in highly symmetric situations. In the end, both of these methods are built around the same idea, even though the mathematics we will use is different: **charges create electric fields**. These notes will introduce the general concept of Gauss's law before we talk about the mathematics and the advantages and disadvantages to using it.
+{{youtube>_oktQiCNfEw?large}}
 ==== Conceptualizing Gauss's Law around a Point Charge ====
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   - The area of the imagined bubble or imagined Gaussian surface increases. The increase in area is proportional to $r^2$ because the [[https://en.wikipedia.org/wiki/Sphere#Surface_area|surface area of a sphere]] is $A=4\pi r^2$.
-This suggests that the electric field at the surface of the imaginary sphere multiplied the surface area of the imaginary sphere is a constant. The idea of electric field at the surface of an area is called **electric flux**, which we will talk about [[184_notes:e_flux|more in the next page of notes]]. If you draw the Gaussian surface with a larger radius, the electric field will be smaller, so the electric flux is constant. If you draw the Gaussian surface with a smaller radius, the electric field will be greater, so the electric flux is constant.
+This suggests that //the electric field at the surface of the imaginary sphere multiplied by the surface area of the imaginary sphere is a constant//; this product is called [[
+_notes:e_flux|electric flux]].
+If we are considering a closed surface, like a bubble that encapsulates a charge, we are dealing with the total electric flux. If you draw the Gaussian surface with a larger radius, the electric field will be smaller, but the total electric flux is constant. If you draw the Gaussian surface with a smaller radius, the electric field will be greater, but the total electric flux is constant.
 {{  184_notes:bigsphere.jpg?300}}
-Gauss's Law is built around this idea that the electric flux is constant through the imagined bubble (no matter where we draw the surface of the bubble) and says that electric flux must be related to the amount of charge **//inside//** the bubble (the point charge in this case). //__Note that the Gaussian surface is completely imaginary__// - we are not physically placing some bubble around the charge. As we will talk about later, the choice of where to draw the surface and what shape to draw it is up to you; however, certain choices will lead to math that is virtually impossible, so we will want to be smart in our choice of surface.
+**Gauss's Law is built around this idea that the total electric flux is constant through the imagined bubble (no matter where we draw the surface of the bubble)** and that electric flux is related to the amount of charge **//inside//** the bubble (the point charge in this case). It turns out this relationship is always true - for any shape of charge (line, sphere, blob, etc.) and for any closed Gaussian surface, the electric flux on that surface is directly related to the charge inside the surface.
-It turns out this relationship is always true - for any shape of charge (line, sphere, blob, etc.) and for any closed Gaussian surface, the electric flux on that surface is directly related to the charge **inside** the surface. We will spend use the next few pages of notes talking about the mathematical formalism for Gauss's Law, but this is the general conceptual idea behind the math.
+//__Note that the Gaussian surface is completely imaginary__// - we are not physically placing some surface around the charge. As we will talk about later, the choice of where to draw the surface and what shape to draw it is up to you; however, certain choices will lead to math that is virtually impossible, so we will want to be careful in our choice of surface. We will spend the next few pages of notes talking about the mathematical formalism for Gauss's Law, but this is the general conceptual idea behind the math.