Differences

This shows you the differences between the selected revision and the current version of the page.

courses:order20:vforder20:vfcalculating 2019/05/28 13:46 courses:order20:vforder20:vfcalculating 2019/05/30 08:08 current
Line 7: Line 7:
===== In-class Content ===== ===== In-class Content =====
 +
====Lecture: Electric Potential==== ====Lecture: Electric Potential====
Line 18: Line 19:
Using what you one about the geometry of the situation, one can possibly simplify the numerator. For example: Using what you one about the geometry of the situation, one can possibly simplify the numerator. For example:
$$\text{Coordinate and Geometry Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda s'\ d\phi'}{| s'^2 + s^2 +2ss' \cos(\phi-\phi') + z^2|}$$ $$\text{Coordinate and Geometry Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda s'\ d\phi'}{| s'^2 + s^2 +2ss' \cos(\phi-\phi') + z^2|}$$
 +Emphasize that "primes" (i.e., $s'$, $\phi'$, $z'$, etc.) are used to indicate the location of charge in the charge distribution.
====Lecture: Chop, Calculate, and Add==== ====Lecture: Chop, Calculate, and Add====
Line 23: Line 25:
  * 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.   * 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|}} {{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 (add).+  * 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.   * 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.
-  * 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 change $dQ$ on each length ($dQ = \lambda dx$), and add all of the $dQ$s together in a sum or integral.+{{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.   *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|}}
====Activities==== ====Activities====
Line 32: Line 38:
  * [[..:..:activities:vfact:vfvring|Electrostatic potential due to a ring of charge]] (SGA - 50 min)   * [[..:..:activities:vfact:vfvring|Electrostatic potential due to a ring of charge]] (SGA - 50 min)
-  * [[..:..:activities:vfact:vfvring|Series expansion of potential due to a ring of charge ]] (Extension of previous SGA + 20-30 min) 
  * [[..:..:lecture:vflec:vflines|Lines of Charge]] (Lecture: 30 min)   * [[..:..:lecture:vflec:vflines|Lines of Charge]] (Lecture: 30 min)

Personal Tools