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# Differences

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- | ==== Activity: Wrap-up ==== | ||

- | * Discuss which variables are variable and which variables are held constant - Students frequently think of anything represented by a letter as a `variable' and do not realize that for each particular situation certain variables remain constant during integration. For example students frequently do not see that the $R$ representing the radius of the ring is held constant during the integration over all space while the r representing the distance to the origin is varying. Understanding this is something trained physicists do naturally while students frequently don't even consider it. This is an important discussion that helps students understand this particular ring problem and also lays the groundwork for better understanding of integration in a variety of contexts. For more information on this topic, see [[whitepapers:variables:start|Students understanding of variables and constants]]. | ||

+ | ==== Activity: Wrap-up ==== | ||

+ | |||

+ | * Discuss which quantities are variable and which variables are held constant - Students frequently think of anything represented by a letter as a `variable' and do not realize that for each particular situation certain quantities remain constant during integration. For example students frequently do not see that the $R$ representing the radius of the ring is held constant during the integration over all space while the r representing the distance to the origin is varying. Understanding this is something trained physicists do naturally while students frequently don't even consider it. This is an important discussion that helps students understand this particular ring problem and also lays the groundwork for better understanding of integration in a variety of contexts. For more information on this topic, see [[whitepapers:variables:start|Students understanding of variables and constants]]. | ||

+ | * Emphasize that while one may not be able to perform a particular integral, the power series expansion of that integrand can be integrated **term by term**. | ||

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This activity is a part of the [[whitepapers:sequences:emsequence:start|Ring Sequence]], which uses a sequence of activities with similar geometries to help students learn how to solve a hard activity by breaking it up into several steps ([[publications:start#student_theses|A Master's Thesis]] about the Ring Sequence). The other activities in the sequence are: | This activity is a part of the [[whitepapers:sequences:emsequence:start|Ring Sequence]], which uses a sequence of activities with similar geometries to help students learn how to solve a hard activity by breaking it up into several steps ([[publications:start#student_theses|A Master's Thesis]] about the Ring Sequence). The other activities in the sequence are: | ||

- | *Prerequisite Activity: | + | *Preceding Activity: |

- | *[[courses:activities:vfact:vfvpoints|Electrostatic Potential Due to Two Point Charges]] | + | *[[courses:activities:vfact:vfvpoints|Electrostatic Potential Due to Two Point Charges]]: Students use what they learned about finding the distance between two points and apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges. |

*Follow-up Activities: | *Follow-up Activities: | ||

- | *[[courses:activities:vfact:vfering|Electric Field Due to a Ring of Charge]] | + | *[[courses:activities:vfact:vfering|Electric Field Due to a Ring of Charge]]: In this small group activity, students use the definition of the electric field as $\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\int_{ring}{\frac{\lambda(\vec{r'})(\vec{r}-\vec{r'})|d\vec{r'}|}{|\vec{r}-\vec{r'}|^3}} $ in order to work out the electric field due to a ring of charge. |

- | *[[courses:activities:vfact:vfaring|Magnetic Vector Potential Due to a Spinning Ring of Charge]] | + | *[[courses:activities:vfact:vfaring|Magnetic Vector Potential Due to a Spinning Ring of Charge]]: This small group activity is the magnetic analogy to [[courses:activities:vfact:vfvring|Electrostatic Potential Due to a Ring of Charge]] where students solve for the magnetic vector potential due to a spinning ring of charge. The set up for this problem is more complex because students now must think about current densities and the inherent vector nature of the magnetic vector potential. |

- | *[[courses:activities:vfact:vfbring|Magnetic Field Due to a Spinning Ring of Charge]] | + | *[[courses:activities:vfact:vfbring|Magnetic Field Due to a Spinning Ring of Charge]]: Building from their experience with the previous exercises, students in this small group activity use techniques from the previous activities to obtain an expression for the magnetic field due to a spinning ring of charge. |

This activity is part of the sequence of activities addressing [[whitepapers:sequences:scalarfieldseq|Representations of Scalar Fields]] in the context of electrostatic potentials. | This activity is part of the sequence of activities addressing [[whitepapers:sequences:scalarfieldseq|Representations of Scalar Fields]] in the context of electrostatic potentials. | ||

- | *Prerequisite Activities: | + | *Preceding Activities: |

- | *[[courses:activities:vfact:vfptcharge|Electrostatic Potential due to a Point Charge]]: This small whiteboard question typically results in an algebraic expression of one variable: the distance from the origin to the point charge. Discussions which will likely arise include notation of the distance from the origin to the point charge, the constants in the equation, and the dimensions of the equation. The representation used by students is predictably algebraic in form, however, the discussion can include other representations of a one-dimensional electrostatic potential. | + | *[[swbq:emsw:vfswpointpot|Electrostatic Potential due to a Point Charge]]: This small whiteboard question asks students to recall the electrostatic potential due to a point charge which results in discussions likely to include notation of the distance from the origin to a point charge. |

- | *[[courses:activities:vfact:vfdrawquadrupole|Drawing Equipotential Surfaces]]: This small group activity encourages students to work in the plane of four point charges arranged in a square to find level curves of equipotential. Students construct a contour plot of the electrostatic potential in the plane of the four charges and explore the constructed scalar field close to the charges, far from the charges, and at important points in the field. Most students are familiar with the elementary equation of the electrostatic potential but few reconcile the equation with the geometry of a scalar field. This small group activity forces students to explicitly work out the geometry of the potential of a quadrupole, allowing them to realize what's "scalar" about the electrostatic potential. | + | *[[courses:activities:vfact:vfdrawquadrupole|Drawing Equipotential Surfaces]]: This small group activity has students construct a contour plot of the electrostatic potential, level curves of equipotential, in the plane of four point charges. |

- | *[[courses:activities:vfact:vfvisv|Visualizing Electrostatic Potentials]]: Students begin by brainstorming ways in which to represent three-dimensional scalar fields in two-dimensions and then use a Mathematica notebook to explore various representations for a distribution of point charges. This activity allows students to check their solutions to [[courses:activities:vfact:vfdrawquadrupole|Drawing Equipotential Surfaces]] as well as explore other representations. Students recognize that the electrostatic potential is a function of three spatial variables which requires an alternative way to represent the potential such as the use of color and plotting equipotential surfaces. | + | *[[courses:activities:vfact:vfvisv|Visualizing Electrostatic Potentials]]: Students begin by brainstorming ways in which to represent three-dimensional scalar fields in two-dimensions and then use a Mathematica notebook to explore various representations for a distribution of point charges. |