184_notes:b_flux_t

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184_notes:b_flux_t [2018/08/09 19:13] – curdemma184_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)
  
-[[184_notes:ac|Next Page: Alternating Current]]+/*[[184_notes:ac|Next Page: Alternating Current]] 
 + 
 +[[184_notes:relating_e|Previous Page: Putting Faraday's Law Together]]*/
  
 ===== Changing Magnetic Fields with Time ===== ===== Changing Magnetic Fields with Time =====
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 {{youtube>7jyFztBVBjw}} {{youtube>7jyFztBVBjw}}
  
-==== Increasing Current to Steady State ==== +===== Increasing Current to Steady State ===== 
-[{{ 184_notes:Week14_1.png?200|Current vs Time}}]+[{{ 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|Concentric coils}}]+[{{  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|Magnetic fields and currents induced by a voltage in the outer loop  }}]+[{{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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 $$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$ $$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$
  
-[{{184_notes:Week14_5.png?225|Graphs of B/I vs time (for the large loop) and V vs time (for the small loop)  }}]+[{{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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