184_notes:ind_i

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
184_notes:ind_i [2021/04/08 15:23] – dmcpadden184_notes:ind_i [2022/11/15 16:21] (current) – valen176
Line 10: Line 10:
 {{youtube>sRBN8pvbHMk?large}} {{youtube>sRBN8pvbHMk?large}}
  
-==== Induced Voltage and the Electric Field ==== +===== Induced Voltage and the Electric Field ===== 
-We know from Faraday's Law that a changing magnetic field will create a curly electric field. If we put a loop of wire nearby, we can add up the little bits of length along that loop with the curly electric field, which tells us how curly that electric field is. (This is very similar to what we did with [[184_notes:loop|Ampere's Law]].) The units of this integral (∫→Enc∙d→l) will give us the same units of electric potential (volts) - Vm⋅m=V. However, because we have curly electric field from the changing magnetic field, this is not technically an electric potential. (When we defined the electric potential we made an assumption that all the charges were stationary or not moving, which is no longer the case.)+We know from Faraday's Law that a changing magnetic flux will create a curly electric field. If we put a loop of wire nearby, we can add up the little bits of length along that loop with the curly electric field, which tells us how curly that electric field is. (This is very similar to what we did with [[184_notes:loop|Ampere's Law]].) The units of this integral (∫→Enc∙d→l) will give us the same units of electric potential (volts) - Vm⋅m=V. However, because we have curly electric field from the changing magnetic flux, this is not technically an electric potential. (When we defined the electric potential __//we made an assumption that all the charges were stationary or not moving,//__ __//**which is no longer the case.)**//__
  
 [{{  184_notes:week12_9.png?250|The curly electric field around a loop}}] [{{  184_notes:week12_9.png?250|The curly electric field around a loop}}]
Line 31: Line 31:
 Depending on what you want to focus on or calculate you may pick one of these equations, but they are saying the exact same thing. Depending on what you want to focus on or calculate you may pick one of these equations, but they are saying the exact same thing.
  
-==== Why do we need the negative sign? ====+===== Why do we need the negative sign? =====
  
 {{youtube>a9DYMs00348?large}} {{youtube>a9DYMs00348?large}}
  
-[{{  184_notes:week12_5.png?300|Flux through loop}}]+[{{  184_notes:week12_5.png?300|Initial flux through loop}}]
  
-When we're talking about Faraday's Law, the negative sign plays an important role. It must be there to satisfy momentum and energy conservation, which we'll talk about in the next few paragraphs. If we think about the third equation above, the negative sign says that **the induced current generates a magnetic field that will be in the direction that opposes the change in magnetic flux**. The curly electric field that the changing magnetic field generates points in the same direction as this current.  This should make some intuitive sense because the conventional current always points in the same direction as the electric field that is driving it. However, we generally talk about the direction of the induced current (rather than the electric field or induced voltage) since it make more intuitive sense and is easier to measure. +When we're talking about Faraday's Law, the negative sign plays an important role. It must be there to satisfy momentum and energy conservation, which we'll talk about in the next few paragraphs. If we think about the third equation above, the negative sign says that **the induced current generates a magnetic field that will be in the direction that opposes the change in magnetic flux**. The curly electric field that the changing magnetic flux generates points in the same direction as this current.  This should make some intuitive sense because the conventional current always points in the same direction as the electric field that is driving it. However, we generally talk about the direction of the induced current (rather than the electric field or induced voltage) since it make more intuitive sense and is easier to measure. 
  
 [{{  184_notes:week12_6.png?300|Increasing Flux}}] [{{  184_notes:week12_6.png?300|Increasing Flux}}]
Line 43: Line 43:
 As an example of how to figure out which direction the induced current flows, let's say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page (shown in the top figure to the right). Initially there would be some magnetic flux through the loop. At a later time (shown in the second picture to the right), after the bar has moved down the rails, there would be a larger magnetic flux through the loop because the area of the loop will have increased. Since the magnetic flux increased, we know that there should be an induced current in the loop - but what direction should it flow around the loop? As an example of how to figure out which direction the induced current flows, let's say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page (shown in the top figure to the right). Initially there would be some magnetic flux through the loop. At a later time (shown in the second picture to the right), after the bar has moved down the rails, there would be a larger magnetic flux through the loop because the area of the loop will have increased. Since the magnetic flux increased, we know that there should be an induced current in the loop - but what direction should it flow around the loop?
  
-[{{184_notes:week12_7.png?200|Direction of I, B, and v  }}]+[{{184_notes:week12_7.png?200|Direction of induced current, magnetic field, and the velocity vector  }}]
  
 Let's suppose that the induced current flows counter-clockwise in the loop (shown in the figure above). If we use our original right hand rule for magnetic force (→F=∫Id→l×→B), we should get a force on the bar that points in the negative x-direction. This means that the magnetic force on the induce current would act to //slow down// the moving bar. With the bar slowing down, this is actually good for energy conservation. It means that we have to put energy into the system to keep the bar moving, and in turn that mechanical energy is turned into electrical energy by inducing a current. If you stopped moving the moving the bar, it would eventually slow down and come to rest. This tells us by energy conservation - the induced current should flow counter-clockwise around the loop. If we had instead hypothesized that the induced current flowed in a clockwise direction, we would instead get a force in the +x direction. This would mean that the bar would continually speed up, which induces more current, which then causes the bar to speed up even more! This would completely break energy conservation and mean that you are essentially creating energy out of nothing. This simply cannot happen. So we know that the induced current must be counter-clockwise in our loop.  Let's suppose that the induced current flows counter-clockwise in the loop (shown in the figure above). If we use our original right hand rule for magnetic force (→F=∫Id→l×→B), we should get a force on the bar that points in the negative x-direction. This means that the magnetic force on the induce current would act to //slow down// the moving bar. With the bar slowing down, this is actually good for energy conservation. It means that we have to put energy into the system to keep the bar moving, and in turn that mechanical energy is turned into electrical energy by inducing a current. If you stopped moving the moving the bar, it would eventually slow down and come to rest. This tells us by energy conservation - the induced current should flow counter-clockwise around the loop. If we had instead hypothesized that the induced current flowed in a clockwise direction, we would instead get a force in the +x direction. This would mean that the bar would continually speed up, which induces more current, which then causes the bar to speed up even more! This would completely break energy conservation and mean that you are essentially creating energy out of nothing. This simply cannot happen. So we know that the induced current must be counter-clockwise in our loop. 
  
-Determining the direction of the induced current just based on the magnetic flux can be tricky because there are a lot of different parts to the equation. In the next page of notes, we will walk through a new right hand rule for the induced current step-by-step. +Determining the direction of the induced current just based on the magnetic flux can be tricky because there are a lot of different parts to the equation. In the next page of notes, we will walk through a **new right hand rule** for the induced current step-by-step. 
  
-/* For this right hand rule, you want to first point your right thumb in the direction of the **change** in magnetic flux. For our example, the change in magnetic flux would point into the page. (Because ΔΦB=ΦBf−ΦBi, we would have a large flux into the page as the final minus a small flux into the page as the initial, which leaves the change in flux as into the page.) Because the induced current would point opposite to the change, you would flip your thumb in the opposite direction, so pointing out of the page. Finally the direction that your fingers would curl in would be the direction of the induced current. So in our example, the current would flow counter-clockwise around the loop (shown in the figure to the left).  */+/* For this right hand rule, you want to first point your right thumb in the direction of the change in magnetic flux. For our example, the change in magnetic flux would point into the page. (Because ΔΦB=ΦBf−ΦBi, we would have a large flux into the page as the final minus a small flux into the page as the initial, which leaves the change in flux as into the page.) Because the induced current would point opposite to the change, you would flip your thumb in the opposite direction, so pointing out of the page. Finally the direction that your fingers would curl in would be the direction of the induced current. So in our example, the current would flow counter-clockwise around the loop (shown in the figure to the left).  */
  
-[{{  184_notes:week12_8.png?200|Resulting magnetic force}}]+[{{  184_notes:week12_8.png?200|Resulting magnetic force, F}}]
  
-/* As we said before, the fact that the induced current will always generate a magnetic field to oppose the change in flux is an important result and ties back to energy and momentum conservation (it even is sometimes referred to as it's own law: [[https://en.wikipedia.org/wiki/Lenz%27s_law|Lenz's law]]). We can see this by thinking about what actually happens to the bar as it is moving through the magnetic field. Now that there is a induced current flowing through the bar, it would also feel a magnetic force from the magnetic field. Using our right hand rule again, this time with the Iind (point your fingers in the direction of the induced current) and the B-field (curl your fingers toward the B-field), we would get a force (from your thumb) that would point //**opposite**// to the velocity of the bar (shown in the figure to the right). So the magnetic force on the bar with the induced current would actually work to slow the bar down. If this negative sign were not present, the induced current would flow clockwise instead of counter-clockwise, thus making the magnetic force on the bar point in the same direction as the velocity. This would act to speed up the bar and give us a way to get more energy than we started with, which would violate the conservation of energy. */+/* As we said before, the fact that the induced current will always generate a magnetic field to oppose the change in flux is an important result and ties back to energy and momentum conservation (it even is sometimes referred to as it's own law: [[https://en.wikipedia.org/wiki/Lenz%27s_law|Lenz's law]]). We can see this by thinking about what actually happens to the bar as it is moving through the magnetic field. Now that there is a induced current flowing through the bar, it would also feel a magnetic force from the magnetic field. Using our right hand rule again, this time with the Iind (point your fingers in the direction of the induced current) and the B-field (curl your fingers toward the B-field), we would get a force (from your thumb) that would point opposite to the velocity of the bar (shown in the figure to the right). So the magnetic force on the bar with the induced current would actually work to slow the bar down. If this negative sign were not present, the induced current would flow clockwise instead of counter-clockwise, thus making the magnetic force on the bar point in the same direction as the velocity. This would act to speed up the bar and give us a way to get more energy than we started with, which would violate the conservation of energy. */
  
 ==== Examples ==== ==== Examples ====
  • 184_notes/ind_i.1617895402.txt.gz
  • Last modified: 2021/04/08 15:23
  • by dmcpadden