184_notes:b_flux_t

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184_notes:b_flux_t [2017/12/01 00:38] – dmcpadden184_notes:b_flux_t [2021/07/13 12:40] (current) – schram45
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 Section 22.2 in Matter and Interactions (4th edition) Section 22.2 in Matter and Interactions (4th edition)
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 +/*[[184_notes:ac|Next Page: Alternating Current]]
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 +[[184_notes:relating_e|Previous Page: Putting Faraday's Law Together]]*/
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 ===== Changing Magnetic Fields with Time ===== ===== Changing Magnetic Fields with Time =====
 So far, we talked about how you can create a curly electric field (and thus an induced voltage/induced current) from a changing magnetic flux. We have gone through examples of what happens when the area changes and in your last project you worked through what happens when the coil is rotating. You also had a demo video that showed what happened when a strong magnet was brought towards or away from the coil, which is one way that the strength of the magnetic field can change. If instead, you have a magnetic field that is produced by a current, another way that the magnetic field can change is if the current (that is producing the magnetic field) changes. This is actually a fairly common situation for two reasons: 1) we are often turning on and off electrical devices and 2) the current/voltage coming from the wall outlets is an alternating current (meaning it is constantly changing from positive to negative). These notes are going to focus on the first of these, that is, how an induced current develops from a magnetic field that is changing with time that is not the result of the physical motion of a magnet. (The [[184_notes:ac|next page of notes]] will discuss the second case.) So far, we talked about how you can create a curly electric field (and thus an induced voltage/induced current) from a changing magnetic flux. We have gone through examples of what happens when the area changes and in your last project you worked through what happens when the coil is rotating. You also had a demo video that showed what happened when a strong magnet was brought towards or away from the coil, which is one way that the strength of the magnetic field can change. If instead, you have a magnetic field that is produced by a current, another way that the magnetic field can change is if the current (that is producing the magnetic field) changes. This is actually a fairly common situation for two reasons: 1) we are often turning on and off electrical devices and 2) the current/voltage coming from the wall outlets is an alternating current (meaning it is constantly changing from positive to negative). These notes are going to focus on the first of these, that is, how an induced current develops from a magnetic field that is changing with time that is not the result of the physical motion of a magnet. (The [[184_notes:ac|next page of notes]] will discuss the second case.)
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 {{youtube>7jyFztBVBjw}} {{youtube>7jyFztBVBjw}}
  
-==== Increasing Current to Steady State ==== +===== Increasing Current to Steady State ===== 
-{{ 184_notes:Week14_1.png?200}}+[{{ 184_notes:Week14_1.png?200|Approximating current vs time graph for initially connecting a circuit}}]
  
 When you initially connect a resistor to a battery, the current in the circuit is initially zero and then increases to its steady state value. If we made a graph of the magnitude of the current in the wires around the circuit, it would look something like one shown to right. In this case we would mathematically model the current as $I=I_0(1-e^{-t/\tau})$, where $I_0$ is the steady state current and $\tau$ is a constant that tells you how fast the current reaches the steady state. If you only have a resistor in the circuit, it takes nano-seconds to micro-seconds to reach the steady state current. When you initially connect a resistor to a battery, the current in the circuit is initially zero and then increases to its steady state value. If we made a graph of the magnitude of the current in the wires around the circuit, it would look something like one shown to right. In this case we would mathematically model the current as $I=I_0(1-e^{-t/\tau})$, where $I_0$ is the steady state current and $\tau$ is a constant that tells you how fast the current reaches the steady state. If you only have a resistor in the circuit, it takes nano-seconds to micro-seconds to reach the steady state current.
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 When you have a typical circuit (like a battery connected to a light bulb), the change in the magnetic field is small enough that the induced current in any nearby loop is probably negligible. However, as circuit components become smaller and smaller and are placed closer and closer together (with roughly the same amount of current), the induced currents (because of changing magnetic fields) can become an important consideration in the design of electronics. When you have a typical circuit (like a battery connected to a light bulb), the change in the magnetic field is small enough that the induced current in any nearby loop is probably negligible. However, as circuit components become smaller and smaller and are placed closer and closer together (with roughly the same amount of current), the induced currents (because of changing magnetic fields) can become an important consideration in the design of electronics.
  
-==== Flux through a Loop ==== +===== Flux through a Loop ===== 
-{{  184_notes:Week14_2.png?200}}+[{{  184_notes:Week14_2.png?200|Concentric coils, with the larger coil connected to a battery}}]
  
 As an example, let's consider a set of concentric coils, where the larger outside coil is initially connected to a battery so that it's current increases from 0 A to 1 A in 1 ns. What would be the induced potential in the smaller inner coil? As an example, let's consider a set of concentric coils, where the larger outside coil is initially connected to a battery so that it's current increases from 0 A to 1 A in 1 ns. What would be the induced potential in the smaller inner coil?
  
-We know from Faraday's law that the induced voltage in the small loop should be equal to the change in magnetic flux through the small coil. Since the small coil is at the center of the large coil, the magnetic flux through the small coil would be due to the magnetic field from the large coil.+We know from Faraday's law that the //induced voltage in the small loop should be equal to the change in magnetic flux through the small coil//. Since the small coil is at the center of the large coil, the magnetic flux through the small coil would be due to the magnetic field from the large coil.
  
-{{184_notes:Week14_3.png?150  }}+[{{184_notes:Week14_3.png?150|Magnetic field from the current in the outer loop  }}]
  
 This means we want to start by writing out what would be the magnetic field at the center of the larger loop (and therefore the magnetic field going through the small coil). We found in an earlier example that the magnetic field through a coil (with a counter-clockwise current) would be: This means we want to start by writing out what would be the magnetic field at the center of the larger loop (and therefore the magnetic field going through the small coil). We found in an earlier example that the magnetic field through a coil (with a counter-clockwise current) would be:
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 We will //__assume our $d\vec{A}_{sm}$ points out of page__// (as opposed to into the page) because this will mean that our magnetic flux is positive. (We could have used a $d\vec{A}$ into the page - it would give the same result, but then the flux would have been negative and we would have to keep track of that extra negative sign.) We will //__assume our $d\vec{A}_{sm}$ points out of page__// (as opposed to into the page) because this will mean that our magnetic flux is positive. (We could have used a $d\vec{A}$ into the page - it would give the same result, but then the flux would have been negative and we would have to keep track of that extra negative sign.)
  
-{{  184_notes:Week14_4.png?200}}+[{{  184_notes:Week14_4.png?175|Direction of B and dA inside the small loop}}]
  
 If $d\vec{A}_{sm}$ points out of the page and $\vec{B}$ points out of the page, then the dot product in our flux equation simplifies to the multiplication of the vector magnitudes.  If $d\vec{A}_{sm}$ points out of the page and $\vec{B}$ points out of the page, then the dot product in our flux equation simplifies to the multiplication of the vector magnitudes. 
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 $$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$ $$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$
  
-{{184_notes:Week14_5.png?250  }}+[{{184_notes:Week14_5.png?225|Graphs of B/I vs time for the large loop and induced V vs time for the small loop  }}]
  
 This equation tells us that as the magnetic field increases linearly in the large coil, a constant potential (and thus a constant current) is induced in the smaller coil. Remember that the negative sign on the induced potential says that the **induced current will generate a magnetic field that will be in the direction that opposes the change in magnetic flux**. In this case, we have an increasing positive magnetic flux, where we defined a positive flux to be out of the page following the magnetic field. Since the change in flux is increasing, the induced current will oppose this change and thus create a magnetic field that points into the page. So using the right hand rule for induction, we point our thumb into the page, and curl your fingers, which gives the direction of the induced current in the small coil - which would be clockwise.   This equation tells us that as the magnetic field increases linearly in the large coil, a constant potential (and thus a constant current) is induced in the smaller coil. Remember that the negative sign on the induced potential says that the **induced current will generate a magnetic field that will be in the direction that opposes the change in magnetic flux**. In this case, we have an increasing positive magnetic flux, where we defined a positive flux to be out of the page following the magnetic field. Since the change in flux is increasing, the induced current will oppose this change and thus create a magnetic field that points into the page. So using the right hand rule for induction, we point our thumb into the page, and curl your fingers, which gives the direction of the induced current in the small coil - which would be clockwise.  
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 ==== Examples ==== ==== Examples ====
-[[:184_notes:examples:Week14_changing_current_rectangle|Changing Current Induces Voltage in Rectangular Loop]]+  * [[:184_notes:examples:Week14_changing_current_rectangle|Changing Current Induces Voltage in Rectangular Loop]] 
 +    * Video Example: Changing Current Induces Voltage in Rectangular Loop 
 +{{youtube>wlEjFcmfD50?large}}
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  • Last modified: 2017/12/01 00:38
  • by dmcpadden