| Both sides previous revision Previous revision Next revision | Previous revision Next revisionBoth sides next revision |
| 184_notes:gauss_motive [2017/09/21 02:37] – [Motivation for Gauss's Law] pwirving | 184_notes:gauss_motive [2018/11/07 19:57] – dmcpadden |
|---|
| | [[184_notes:e_flux|Next Page: Electric Flux and Area Vectors]] |
| | |
| ===== Motivation for Gauss's Law ===== | ===== Motivation for Gauss's Law ===== |
| | Last week, we learned about Ampere's Law, which was a cool shortcut for finding magnetic fields in highly symmetric situations. This week we will talk about a similar short for finding electric fields in symmetric situations, called Gauss's Law. While Gauss's Law has many similar features to Ampere's Law, there are a couple of key differences. First, we will be talking about enclosing a charge, rather than a current (since we are returning to a discussion about electric fields). Second, rather than talking about the magnetic field around an imagined loop, we will be talking about the electric field around an imagined area. The notes this week will step through each of the pieces of Gauss's Law, starting with introducing the general concept of Gauss's law before we talk about the mathematics and the advantages and disadvantages to using it. |
| 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. | |
| |
| {{youtube>_oktQiCNfEw?large}} | {{youtube>_oktQiCNfEw?large}} |
| ==== Conceptualizing Gauss's Law around a Point Charge ==== | ==== Conceptualizing Gauss's Law around a Point Charge ==== |
| {{ 184_notes:smallsphere.jpg?300}} | {{ 184_notes:smallsphere.jpg?300}} |
| | |
| | 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 always 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**. |
| |
| First, let's go back to our example of the point charge. [[184_notes:pc_efield#Electric_Field_Vectors|The electric field points away from a positive charge]]. If we imagine a spherical bubble (like a very thin shell) around the point charge, we could think about the strength of the electric field that is on the surface of our imagined bubble (shown in the figure to the right). | First, let's go back to our example of the point charge. [[184_notes:pc_efield#Electric_Field_Vectors|The electric field points away from a positive charge]]. If we imagine a spherical bubble (like a very thin shell) around the point charge, we could think about the strength of the electric field that is on the surface of our imagined bubble (shown in the figure to the right). |
| - 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 $A=4\pi r^2$. | - 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 by the surface area of the imaginary sphere is a constant; this product is called [[ | 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 [[ |
| 184_notes:e_flux|electric flux]]. | 184_notes:e_flux|electric flux]]. |
| |