184_notes:examples:week14_b_field_capacitor

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184_notes:examples:week14_b_field_capacitor [2017/11/27 02:34] – [Solution] tallpaul184_notes:examples:week14_b_field_capacitor [2017/11/27 03:12] – tallpaul
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   * We represent the situation with the following visual:   * We represent the situation with the following visual:
  
-{{ 184_notes:14_capacitor_side_view.png?400 |Plane in which we wish to find B-field}}+{{ 184_notes:14_capacitor_side_view.png?300 |Plane in which we wish to find B-field}}
 ====Solution==== ====Solution====
 We wish to find the magnetic field in the plane we've shown in the representations. Due to the circular symmetry of the problem, we choose a circular loop in which to situate our integral $\int \vec{B}\bullet\text{d}\vec{l}$. We also choose for the loop to be the perimeter of a flat surface, so that the entire thing lies in the plane of interest, and there is no enclosed current (so $I_{enc} = 0$). We show the drawn loop below, split into two cases on the radius of the loop. We wish to find the magnetic field in the plane we've shown in the representations. Due to the circular symmetry of the problem, we choose a circular loop in which to situate our integral $\int \vec{B}\bullet\text{d}\vec{l}$. We also choose for the loop to be the perimeter of a flat surface, so that the entire thing lies in the plane of interest, and there is no enclosed current (so $I_{enc} = 0$). We show the drawn loop below, split into two cases on the radius of the loop.
  
-{{ 184_notes:14_capacitor_loops.png?500 |Circular Loops}}+{{ 184_notes:14_capacitor_loops.png?600 |Circular Loops}}
  
 Below, we also draw the direction of the magnetic field along the loops. We know the magnetic field is directed along our circular loop -- if it pointed in or out a little bit, we may be able to conceive of the closed surface with magnetic flux through it, which would imply the existence of a magnetic monopole. This cannot be the case! We also know that the field is directed counterclockwise, due to the increasing electric field into the page. This comes from an extension of Lenz's Law, upon which discussion is not needed for this course. Below, we also draw the direction of the magnetic field along the loops. We know the magnetic field is directed along our circular loop -- if it pointed in or out a little bit, we may be able to conceive of the closed surface with magnetic flux through it, which would imply the existence of a magnetic monopole. This cannot be the case! We also know that the field is directed counterclockwise, due to the increasing electric field into the page. This comes from an extension of Lenz's Law, upon which discussion is not needed for this course.
  
-{{ 184_notes:14_capacitor_b_field_loops.png?500 |Circular Loops, with B-field shown}}+{{ 184_notes:14_capacitor_b_field_loops.png?600 |Circular Loops, with B-field shown}}
  
 We are pretty well set up to simplify our calculation of the integral in the representations, since the B-field is parallel to the loop's perimeter. Below, we show the integral calculation, where the magnetic field at a radius $r$ is displayed as $B(r)$. We are pretty well set up to simplify our calculation of the integral in the representations, since the B-field is parallel to the loop's perimeter. Below, we show the integral calculation, where the magnetic field at a radius $r$ is displayed as $B(r)$.
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 Notice, the distance between the plates has no effect on the magnetic field calculation. Also, the amount of the charge on the plates at a given time does not matter -- we only care about how fast the charge is changing (the current!). Also, it is interesting that outside the plates, the magnetic field is the same as it would be for a long wire. This would be just as if the capacitor were not there, and the wire were connected. Below, we show a graph of the magnetic field strength as a function of the distance from the center of the capacitor. Notice, the distance between the plates has no effect on the magnetic field calculation. Also, the amount of the charge on the plates at a given time does not matter -- we only care about how fast the charge is changing (the current!). Also, it is interesting that outside the plates, the magnetic field is the same as it would be for a long wire. This would be just as if the capacitor were not there, and the wire were connected. Below, we show a graph of the magnetic field strength as a function of the distance from the center of the capacitor.
  
-{{ 184_notes:14_capacitor_b_field_graph.png?500 |B-Field Strength, Graphed}}+{{ 184_notes:14_capacitor_b_field_graph.png?400 |B-Field Strength, Graphed}}
  
 We have enough information to find the maximum B-field, which is at the edge of the plates: We have enough information to find the maximum B-field, which is at the edge of the plates:
 $$B_{\text{max}} = \frac{\mu_0 I}{2\pi R} = \frac{4\pi \cdot 10^{-7} \text{Tm/A} \cdot 3\text{ A}}{2\pi \cdot 10 \text{ m}} = 60 \text{ nT}$$ $$B_{\text{max}} = \frac{\mu_0 I}{2\pi R} = \frac{4\pi \cdot 10^{-7} \text{Tm/A} \cdot 3\text{ A}}{2\pi \cdot 10 \text{ m}} = 60 \text{ nT}$$
  • 184_notes/examples/week14_b_field_capacitor.txt
  • Last modified: 2021/07/22 13:51
  • by schram45