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electrical_engineering_2:the_time-dependent_magnetic_field [2023/04/23 08:51]
ott 		electrical_engineering_2:the_time-dependent_magnetic_field [2023/09/19 23:51] (aktuell)
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-====== 4. time-dependent magnetic Field ======+====== 4 Time-dependent magnetic Field ======
  
 <callout> This chapter is based on the book 'University Physics II' ([[https://creativecommons.org/licenses/by/4.0|CC BY 4.0]], Authors: [[https://openstax.org/details/books/university-physics-volume-2|Open Stax]] ). In detail this is chapter [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)|11. Magnetic Forces and Fields]] </callout> <callout> This chapter is based on the book 'University Physics II' ([[https://creativecommons.org/licenses/by/4.0|CC BY 4.0]], Authors: [[https://openstax.org/details/books/university-physics-volume-2|Open Stax]] ). In detail this is chapter [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)|11. Magnetic Forces and Fields]] </callout>
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 <imgref ImgNr05> depicts a circuit and an arbitrary surface $S$ that it bounds. Notice that $S$ is an open surface: The planar area bounded by the circuit is not part of the surface, so it is not fully enclosing a volume. <imgref ImgNr05> depicts a circuit and an arbitrary surface $S$ that it bounds. Notice that $S$ is an open surface: The planar area bounded by the circuit is not part of the surface, so it is not fully enclosing a volume.
  
-Since the magnetic field is a source-free vortex field, the flux over a closed area is always zero: $\Phi_{\rm m} = \oiint_{A} \vec{B} \cdot {\rm d} \vec{A} = 0$. \\ +Since the magnetic field is a source-free vortex field, the flux over a closed area is always zero: $\Phi_{\rm m} = {\rlap{\Large \rlap{\int} \int} \, \LARGE \circ}_{A} \vec{B} \cdot {\rm d} \vec{A} = 0$. \\ 
 By this, it can be shown that any open surface bounded by the circuit in question can be used to evaluate $\Phi_{\rm m}$ (similar to the [[:electrical_engineering_2:the_stationary_electric_flow#gauss_s_law_for_current_density|Gauss's law for current density]]).  By this, it can be shown that any open surface bounded by the circuit in question can be used to evaluate $\Phi_{\rm m}$ (similar to the [[:electrical_engineering_2:the_stationary_electric_flow#gauss_s_law_for_current_density|Gauss's law for current density]]). 
 For example, $\Phi_{\rm m}$ is the same for the various surfaces $S$, $S_1$, $S_2$ of the figure. For example, $\Phi_{\rm m}$ is the same for the various surfaces $S$, $S_1$, $S_2$ of the figure.
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   - Make a sketch of the situation for use in visualizing and recording directions.   - Make a sketch of the situation for use in visualizing and recording directions.
   - Determine the direction of the applied magnetic field $\vec{B}$.   - Determine the direction of the applied magnetic field $\vec{B}$.
-  - Determine whether its magnetic flux is increasing or decreasing.+  - Determine whether the magnitude of its magnetic flux is increasing or decreasing.
   - Now determine the direction of the induced magnetic field $\vec{B_{\rm ind}}$. The induced magnetic field tries to reinforce a magnetic flux that is decreasing or opposes a magnetic flux that is increasing. Therefore, the induced magnetic field adds or subtracts from the applied magnetic field, depending on the change in magnetic flux.   - Now determine the direction of the induced magnetic field $\vec{B_{\rm ind}}$. The induced magnetic field tries to reinforce a magnetic flux that is decreasing or opposes a magnetic flux that is increasing. Therefore, the induced magnetic field adds or subtracts from the applied magnetic field, depending on the change in magnetic flux.
   - Use the right-hand rule to determine the direction of the induced current $i_{\rm ind}$ that is responsible for the induced magnetic field $\vec{B}_{\rm ind}$.   - Use the right-hand rule to determine the direction of the induced current $i_{\rm ind}$ that is responsible for the induced magnetic field $\vec{B}_{\rm ind}$.
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 <panel type="info" title="Exercise 4.3.4 Calculating the Potential Difference Induced in a Generator Coil"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 4.3.4 Calculating the Potential Difference Induced in a Generator Coil"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-The generator coil shown in <imgref ImgNr13> is rotated through one-fourth of a revolution (from $\phi_0=0°$ to $\phi_1=90°$) in $5.0 ~\rm ms$. +The generator coil shown in <imgref ImgNr13> is rotated through one-fourth of a revolution (from $\varphi_0=0°$ to $\varphi_1=90°$) in $5.0 ~\rm ms$. 
 The $200$-turn circular coil has a $5.00 ~\rm cm$ radius and is in a uniform $0.80 ~\rm T$ magnetic field. The $200$-turn circular coil has a $5.00 ~\rm cm$ radius and is in a uniform $0.80 ~\rm T$ magnetic field.
  
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 So, the course of the voltage when entering or exiting is not uniquely given. So, the course of the voltage when entering or exiting is not uniquely given.
  
-<WRAP> <imgcaption ImgNrEx04s| Solution> </imgcaption> <WRAP> {{drawio>WindingPolePieces2solution}}  \\ </WRAP></WRAP>+<WRAP> <imgcaption ImgNrEx04s| Solution> </imgcaption> <WRAP>{{drawio>WindingPolePieces2solution.svg}}  \\ </WRAP></WRAP>