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184_notes:what_happens [2021/04/08 13:50] dmcpadden184_notes:what_happens [2021/06/17 15:24] (current) bartonmo
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 Thus far in this course, we have considered the [[184_notes:pc_efield|electric]] and [[184_notes:moving_q|magnetic]] fields completely separately, either only looking at the effects of an electric field by itself or a magnetic field by itself. However, there are many real-world contexts where a charge may be moving in a magnetic field and also near other charges. This means the charge would feel both an [[184_notes:pc_force|electric force]] and a [[184_notes:q_b_force|magnetic force]]. Through Newton's second law ($\vec{F}_{net}=\vec{F}_1+\vec{F}_2+...$), we can think about how the combination of these forces affects individual charges. Using the magnetic and electric force is one way that we can think about combining electric and magnetic fields. Note that this is not a direct relationship between electric field and magnetic field, but rather relies on using force.  Thus far in this course, we have considered the [[184_notes:pc_efield|electric]] and [[184_notes:moving_q|magnetic]] fields completely separately, either only looking at the effects of an electric field by itself or a magnetic field by itself. However, there are many real-world contexts where a charge may be moving in a magnetic field and also near other charges. This means the charge would feel both an [[184_notes:pc_force|electric force]] and a [[184_notes:q_b_force|magnetic force]]. Through Newton's second law ($\vec{F}_{net}=\vec{F}_1+\vec{F}_2+...$), we can think about how the combination of these forces affects individual charges. Using the magnetic and electric force is one way that we can think about combining electric and magnetic fields. Note that this is not a direct relationship between electric field and magnetic field, but rather relies on using force. 
  
-The notes this week are going to focus on a more fundamental (and direct) relationship between electric and magnetic fields, which hinges on a **changing** magnetic field rather than a constant magnetic field. So our starting question is: what happens when you have a changing magnetic field? The following video demonstrates what happens when you move a permanent magnet towards a coil of wire. The coil is connected to a [[https://en.wikipedia.org/wiki/Galvanometer|galvanometer]], which is a device that measures small currents (on the order of $\mu A$), but it is not connected to a battery.  +The notes this week are going to focus on a more fundamental (and direct) relationship between electric and magnetic fields, which hinges on a //changing// magnetic field rather than a //constant// magnetic field. So our starting question is: what happens when you have a changing magnetic field? The following video demonstrates what happens when you move a permanent magnet towards a coil of wire. The coil is connected to a [[https://en.wikipedia.org/wiki/Galvanometer|galvanometer]], which is a device that measures small currents (on the order of $\mu A$), but it is not connected to a battery.  
  
 {{youtube>FLiEX076vQU?large}} {{youtube>FLiEX076vQU?large}}
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 /*{{youtube>3_JjojLl_mA?large}}*/ /*{{youtube>3_JjojLl_mA?large}}*/
  
-==== Where does this current come from? ====+===== Where does this current come from? =====
 [{{  184_notes:week12_1.png?200|Fig. 1: Proposed charge gradient around a wire loop  }}] [{{  184_notes:week12_1.png?200|Fig. 1: Proposed charge gradient around a wire loop  }}]
  
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 [{{  184_notes:week12_4.png?200|Fig. 4: Curly electric field (or non-Coulombic Electric Field)}}]  [{{  184_notes:week12_4.png?200|Fig. 4: Curly electric field (or non-Coulombic Electric Field)}}] 
  
-Notationally, because this is a curly electric field, which is very different than the other kinds of electric fields that we have talked about, we will write a curly electric field from a changing magnetic field as $\vec{E}_{nc}$ because it is this as the "non-coulombic" electric field, whereas we will keep $\vec{E}$ to be for an electric field from static charges. +Because this is a curly electric field, which is very different than the other kinds of electric fields that we have talked about, we will write a curly electric field from a changing magnetic field as $\vec{E}_{nc}$. The "nc" is because this is referred to as the "non-coulombic" electric field, whereas we will keep $\vec{E}$ to be for an electric field from static charges. 
  
 Mathematically, we represent this relationship using [[https://en.wikipedia.org/wiki/Curl_(mathematics)|a vector operation called curl]]; however, for this class, we will generally simplify the curl operation (using [[https://en.wikipedia.org/wiki/Stokes%27_theorem|Stokes' Theorem]]) to be: Mathematically, we represent this relationship using [[https://en.wikipedia.org/wiki/Curl_(mathematics)|a vector operation called curl]]; however, for this class, we will generally simplify the curl operation (using [[https://en.wikipedia.org/wiki/Stokes%27_theorem|Stokes' Theorem]]) to be:
-$$-\int \vec{E}_{nc} \bullet d\vec{l} = \frac{d\Phi_{B}}{dt}$$ +$$\int \vec{E}_{nc} \bullet d\vec{l} = \frac{d\Phi_{B}}{dt}$$ 
-Where this equation (called [[https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction|Faraday's Law]]) says that the electric field around a loop is equal to the change in the magnetic flux through that loop. The next few pages of notes will go through each part of this equation in detail (including where the negative sign came from) and how to use it, but it's important to remember that the relationship behind this equation is that a changing magnetic field will create a curly electric field.+Where this equation (called [[https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction|Faraday's Law]]) says that the curly electric field ($ \vec{E}_{nc}$) around a loop ($d\vec{l}$) is equal to the //opposite// change in the magnetic flux ($\frac{d\Phi_{B}}{dt}$) through that loop. The next few pages of notes will go through each part of this equation in detail (including where the negative sign came from) and how to use it, but it's important to remember that the relationship behind this equation is that a changing magnetic field will create a curly electric field.
  
 ==== Examples ==== ==== Examples ====
 Video example showing a constant flux, rotating loop, and the magnetic being pushed towards and away from the loop: Video example showing a constant flux, rotating loop, and the magnetic being pushed towards and away from the loop:
 {{youtube>mMh9pyxa0o8?large}} {{youtube>mMh9pyxa0o8?large}}
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