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184_notes:b_flux_t [2017/11/15 01:06] – [Increasing Current to Steady State] caballero | 184_notes:b_flux_t [2020/08/24 17:50] – dmcpadden | ||
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+ | Section 22.2 in Matter and Interactions (4th edition) | ||
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+ | [[184_notes: | ||
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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/ | So far, we talked about how you can create a curly electric field (and thus an induced voltage/ | ||
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+ | {{youtube> | ||
==== Increasing Current to Steady State ==== | ==== Increasing Current to Steady State ==== | ||
- | FIXME Add Graph | + | [{{ 184_notes: |
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/ | 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/ | ||
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While this is a realistic representation of the current in the circuit, mathematically, | While this is a realistic representation of the current in the circuit, mathematically, | ||
- | When the current is at its steady state value, there would not be an induced current in a nearby | + | When the current is at its steady state value, there would not be an induced current in a nearby coil because the magnetic field would not be changing (constant current would mean the magnetic field would also be constant over time). However, when the current is increasing, you would have an increasing magnetic field around the wires. If you were to put a loop of wire near this increasing magnetic field, you would see an induced current in the loop, even though |
- | When you have large scale circuits | + | When you have a typical circuit |
==== Flux through a Loop ==== | ==== Flux through a Loop ==== | ||
- | FIXME Add figure | + | [{{ 184_notes: |
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' | We know from Faraday' | ||
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+ | [{{184_notes: | ||
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.) | ||
- | 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. | + | [{{ 184_notes: |
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+ | 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. | ||
$$\Phi_{B_{sm}}=\int B_{lg} dA_{sm}$$ | $$\Phi_{B_{sm}}=\int B_{lg} dA_{sm}$$ | ||
Then because the magnetic field is constant in the small loop (based on our assumption above), we can pull it out of the integral. Then integrating $dA_{sm}$ gives the area of the small loop: | Then because the magnetic field is constant in the small loop (based on our assumption above), we can pull it out of the integral. Then integrating $dA_{sm}$ gives the area of the small loop: | ||
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$$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$ | $$-V_{ind}=\frac{\mu_0N_{lg}1\cdot 10^9 }{2 R_{lg}} A_{sm}N_{sm}$$ | ||
- | FIXME Add figure | + | [{{184_notes: |
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. | ||
Depending on the area of the loops and the number of coils, this potential may be small and negligible or it may be large enough to cause problems in the circuit. However, the induced potential (and thus induced current) would only exist in the small coil for a very short period of time - only during the time period where the current is increasing in the large coil. When the current reaches a steady state in the larger coil, there is no longer a changing magnetic flux through the small coil so there is no induced current or potential. The [[184_notes: | Depending on the area of the loops and the number of coils, this potential may be small and negligible or it may be large enough to cause problems in the circuit. However, the induced potential (and thus induced current) would only exist in the small coil for a very short period of time - only during the time period where the current is increasing in the large coil. When the current reaches a steady state in the larger coil, there is no longer a changing magnetic flux through the small coil so there is no induced current or potential. The [[184_notes: | ||
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+ | ==== Examples ==== | ||
+ | [[: |