184_notes:eflux_curved

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184_notes:eflux_curved [2018/07/24 15:15] – curdemma184_notes:eflux_curved [2021/06/01 15:29] (current) – schram45
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 Section 21.3 from Matter and Interaction (4th edition) Section 21.3 from Matter and Interaction (4th edition)
  
-[[184_notes:q_enc|Next Page: Enclosed Charge]]+/*[[184_notes:q_enc|Next Page: Enclosed Charge]]
  
-[[184_notes:e_flux|Previous Page: Electric Flux and Area Vectors]]+[[184_notes:e_flux|Previous Page: Electric Flux and Area Vectors]]*/
  
 ===== Electric Flux through Curved Surfaces ===== ===== Electric Flux through Curved Surfaces =====
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 ==== Curved Surfaces ==== ==== Curved Surfaces ====
 === Area Vectors === === Area Vectors ===
-[{{184_notes:electricflux4.jpg?350|Area vectors and Efield vectors are parallel  }}]+[{{184_notes:electricflux4.jpg?350|Area vectors along a curved sphere.  }}]
  
 Before we said that for a flat surface, the area vector is given by the magnitude of the area times the vector that points perpendicular to the area. This makes sense for a flat surface, where the area vector will point in the same direction for all points on the surface. However, for a curved surface, this no longer makes sense. It becomes impossible to use a single vector to describe the surface. Before we said that for a flat surface, the area vector is given by the magnitude of the area times the vector that points perpendicular to the area. This makes sense for a flat surface, where the area vector will point in the same direction for all points on the surface. However, for a curved surface, this no longer makes sense. It becomes impossible to use a single vector to describe the surface.
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   - constant electric fields along the surface.   - constant electric fields along the surface.
  
-[{{  184_notes:electricflux5.jpg?350|Along this surface, the electric field vectors from the point charge are not constant and the Efield vectors are not parallel to the area vectors. A plane is not a good choice of gaussian surface for a point charge.}}]+[{{  184_notes:electricflux5.jpg?350|Along this surface, the electric field vectors from the point charge are not constant and the Efield vectors are not parallel to the area vectors. A plane is not a good choice of Gaussian surface for a point charge.}}]
  
 As an example, let's look at the electric field from a positive point charge, which points radially away from the point charge (shown by the red arrows). If we imagine a rectangular plane above the point charge, then the electric field vectors that would point through that plane would both change direction and change in magnitude all over the surface. Since the electric field vectors have different directions all over the plane but the dA vectors all point up for the plane (shown by the green arrows), this means we cannot easily simplify the dot product in the electric flux equation. On top of that, the electric field is not constant along the plane since the magnitudes of the vectors change. In this case, the integral to find the flux would be extremely difficult and would need to account for the changing angle of the vectors and the changing size of the field (aka we do not want to do that). As an example, let's look at the electric field from a positive point charge, which points radially away from the point charge (shown by the red arrows). If we imagine a rectangular plane above the point charge, then the electric field vectors that would point through that plane would both change direction and change in magnitude all over the surface. Since the electric field vectors have different directions all over the plane but the dA vectors all point up for the plane (shown by the green arrows), this means we cannot easily simplify the dot product in the electric flux equation. On top of that, the electric field is not constant along the plane since the magnitudes of the vectors change. In this case, the integral to find the flux would be extremely difficult and would need to account for the changing angle of the vectors and the changing size of the field (aka we do not want to do that).
  
-[{{184_notes:electricflux6.jpg?350|A spherical gaussian surface around a point charge is a good choice. Efield vectors are constant around the surface and they are parallel to area vectors  }}]+[{{184_notes:electricflux6.jpg?350|A spherical Gaussian surface around a point charge is a good choice. Efield vectors are constant around the surface and they are parallel to area vectors  }}]
  
 Instead, imagine a spherical shell or bubble around the point charge. In this case, the $d\vec{A}$'s would point radially outward from the bubble at every point along the surface (again shown by the green vectors). Likewise, for every dA, the electric field points radially away from the point charge (shown by red arrows). This means that the $d\vec{A}$ is always perpendicular to $\vec{E}$ for a given location, so we can simplify the dot product to a simple multiplication.  Instead, imagine a spherical shell or bubble around the point charge. In this case, the $d\vec{A}$'s would point radially outward from the bubble at every point along the surface (again shown by the green vectors). Likewise, for every dA, the electric field points radially away from the point charge (shown by red arrows). This means that the $d\vec{A}$ is always perpendicular to $\vec{E}$ for a given location, so we can simplify the dot product to a simple multiplication. 
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 ====Examples==== ====Examples====
-[[:184_notes:examples:Week5_flux_two_radii|Flux through Two Spherical Shells]] +  * [[:184_notes:examples:Week5_flux_two_radii|Flux through Two Spherical Shells]] 
- +    * Video Example: Flux through Two Spherical Shells 
-[[:184_notes:examples:Week5_flux_dipole|Flux from a Dipole]]+  * [[:184_notes:examples:Week5_flux_dipole|Flux from a Dipole]] 
 +{{youtube>9KCJqEg_8pI?large}}
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