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whitepapers:sequences:emsequence:start 2014/08/06 11:18 whitepapers:sequences:emsequence:start 2019/07/22 11:56 current
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Students who are just beginning upper-division courses are being asked to simultaneously learn physics concepts, use mathematical processes in new ways, apply geometric reasoning, and use extended multi-step problem solving. Having students successfully deal with a problem such as finding the magnetic field in all space due to a spinning ring of charge is a significant challenge. If we are to avoid doing the thinking for them and creating a template they can use, then we must create a sequence of learning opportunities that allow them to genuinely develop for themselves the ability to solve a problem like this. Students who are just beginning upper-division courses are being asked to simultaneously learn physics concepts, use mathematical processes in new ways, apply geometric reasoning, and use extended multi-step problem solving. Having students successfully deal with a problem such as finding the magnetic field in all space due to a spinning ring of charge is a significant challenge. If we are to avoid doing the thinking for them and creating a template they can use, then we must create a sequence of learning opportunities that allow them to genuinely develop for themselves the ability to solve a problem like this.
-We created a sequence of five small group activities that help students do deeper thinking while making each of the steps manageable. These five activities take roughly 30 to 60 minutes each and are designed to be used over the course of one or two months in conjunction with other forms of instruction such as lecture, individual homework, computer visualizations, and kinesthetic activities.+We created a sequence of five small group activities that help students do deeper thinking while making each of the steps manageable. These five activities take roughly 30 to 60 minutes each and are designed to be used over the course of one or two months in conjunction with other forms of instruction such as lecture, individual homework, computer visualizations, and kinesthetic activities. Additionally, we have included two lectures to help students reason geometrically about charge distributions and electric potentials.
A more in-depth discussion of the rationale, student thinking, and the way these activities fit together can be found in a discussion of how these activities break the learning into manageable [[.:pieces]] A more in-depth discussion of the rationale, student thinking, and the way these activities fit together can be found in a discussion of how these activities break the learning into manageable [[.:pieces]]
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+====Lecture: Electric Potential====
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+Students may be familiar with the iconic equation for the electric potential (due to a point charge):
+$$\text{Iconic:} \qquad V=\frac{1}{4 \pi \epsilon_0} \frac{Q}{r}$$
+With information about the type of source distribution, one can write or select the appropriate coordinate independent equation for $V$. For example, if the source is a linear (i.e. 1 dimensional):
+$$\text{Coordinate Independent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda | d\vec r' |}{| \vec r - \vec r' |}$$
+Looking at symmetries of the source, one can choose a coordinate system and write the equation for the potential in terms of this coordinate system. Note that this step is often combined with the following step, though one may wish to keep them separate for the sake of careful instruction.
+$$\text{Coordinate Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda |ds'\ \hat s + s'\ d\phi'\ \hat \phi + dz'\ \hat z|}{| s'^2 + s^2 +2ss' \cos(\phi-\phi') + z^2|}$$
+Using what you one about the geometry of the source, one can simplify the expression. For example, if the source is a ring of charge with radius $R$ and charge $Q$ in the $x$-, $y$-plane the integral becomes:
+$$\text{Coordinate and Geometry Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0}\, \frac{Q}{2\pi R} \int_0^{2\pi}\frac{R\ d\phi'}{| R^2 + s^2 +2Rs \cos(\phi-\phi') + z^2|}$$
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+  * To find the area under a curve, one may chop up the x-axis into small pieces (of width $dx$). The area under the curve is then found by calculating the area for each region of $dx$ (which is $f(x) dx$) and then summing up all of those areas. In the limit where $dx$ is small enough, the sum becomes an integral.
+{{courses:order20:vforder20:chop_x.png?300|}}
+  * One could also find the area under a curve by chopping up both the x- and y-axes (chop), calculating the area of each small area under the curve (calculate), and adding all of those together with a double sum or double integral.
+{{courses:order20:vforder20:chop_x_y.png?300|}}
+  * This approach can be used to find the area of a cone, where the 'horizontal' length of each area is $r d\phi$ and the 'vertical' length is $dr$, giving an area of $dA = r d\phi dr$. It is important to make sure that the limits of integration are appropriate so that the integrals range over the whole area of interest.
+{{courses:order20:vforder20:dA_for_cone.png?200|}}
+  * If one wants to calculate something other than length, area, or volume, such as if one sprinkled charge over a thin bar, then chop, calculate, and add still works. Again, chop the bar up into small lengths of $dx$. Then calculate the charge $dQ$ on each length ($dQ = \lambda dx$), and add all of the $dQ$s together in a sum or integral.
+{{courses:order20:vforder20:chop_lambda.png?300|}}
+  *This also works for calculating something (such as charge) over a volume. For a thick cylindrical shell with a charge density $\rho(\vec r)$, chop the shell into small volumes of $d \tau$ (which will be a product of 3 small lengths, e.g. $d \tau = r d\phi\ dr\ dz$), multiply this volume by the charge density at each part of the shell (defined by e.g. $r, \phi,$ and $z$), and add the resulting $dQ$s together.
+{{courses:order20:vforder20:chop_rho.png?200|}}

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