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--- 184_notes:induced_current [2021/04/05 20:44] – [Concluding] stumptyl
+++ 184_notes:induced_current [2021/11/12 23:15] (current) – [Step 1.) Draw a picture of your situation] stumptyl
@@ Line 1: / Line 1: @@
-=====Induced Current Directionality Prediction=====
+=====Finding the Induced Current Direction=====
-Now that we have taken some time to discuss how magnetic fields and induced current are able to interact, it is important to take some time a refine the skills needed to understand how to predict the direction of the induced current around solenoids, through a generalized approach. **In these notes, we will discuss the approach needed to predict the direction of the induced current through various situations.**
+So far, we've been talking about the different pieces of Faraday's Law - namely how a changing magnetic flux will create an induced current. In this page of notes, we will outline the steps that we need to take to determine the direction of the induced current, which are based off of Faraday's Law itself:
-[{{ 184_notes:inducedcurrent_blank.png?600|This is the chart used for predicting the directionality of the Induced Current}}]
+$$V_{ind} = -\frac{d \Phi_B}{dt}$$
+We will use the example of a bar magnet moving away from a wire coil to highlight these steps and to show how you can use a table to keep track of your work. While the specifics of the table will change depending on the context, the structure and steps will work no matter what problem you are solving. So to get started you should make a table like the one shown to the right with 8 columns. The first column will be for a picture/diagram of your situation, the second will be for the B-field direction, the third will be for the dA direction, the fourth will be for your initial magnetic flux, the fifth will be for the final magnetic flux, the sixth will be the change in magnetic flux, the seventh will be for your induced voltage, and the eighth will be for your induced current.
+[{{184_notes:inductionchart_updated_11_12_2021.png?600|This is the table used for predicting the directional of the induced current.  }}]
-====Right Hand Rule====
+This video will walk you through an example of how to use this table or you can read about it in the notes below.
-\\
-As we have learned from previous notes, when discussing the right-hand rule for the induced current the focus is on the $d\vec{A}$. In your right hand, make a fist and place the thumb out. The thumb in this situation will be representing your $d\vec{A}$. The fist itself will be used to predict the induced current flow. Normally, these situations could be described as clockwise and counterclockwise when describing the direction of flow; however, as you may notice that by simply saying “Counter-clockwise” another observer would still be confused because this statement involves assumptions about the observation point. By utilizing this Right Hand Rule model, you will be able to grasp a better understanding of the direction of flow of the Induced Current as well as be able to articulate the direction to the best of your abilities also.
+{{youtube>Qlg0Iu1Do94?large}}
 \\
+**NOTE: In this chart there is a mistake in the chart being used: The variables "V" and "I" are not vectors. Please make a note of this when watching the video to ensure there is no confusion.**
+==== Right Hand Rule Steps ====
+==== Step 1.) Draw a picture of your situation ====
+Before anything, you should start with a picture of your situation. We'll draw this in the first column of the table. For our example, we'll have a bar magnet that is moving away from a set of coils. In the picture, we have marked the coils, the orientation of the magnet (which side is north/south), and which way the magnet is moving (marked with the velocity $\vec{v}$ arrow).
-[{{ 184_notes:ic_scenario.png?440| The chart now includes the specific situation that the notes will walk students through.}}]
+[{{184_notes:inductionchart_updated_11_12_2021.png?440| Step 1: First make a diagram of the particular situation include the coil, magnet, and relevant directions.
+ }}]
-\\
-=====Approach======
+====Step 2.) Draw the direction of the B-field through the relevant area====
-Let's take a moment and walk through how to solve the following situation utilizing the chart that we have laid out.
+Next we need to determine the direction of the magnetic field through the relevant area. For this situation, the relevant area is going to be our coils, so we are particularly interested in the direction of the B-field through the coils. Remember for a bar magnet, the magnetic field should point out from the north side of the magnet, wrap around, and point into the south side of the magnet. Since our coil is next to the south side of the magnet, this means the magnetic field inside the coil will mostly be pointing to the left (in towards the south side of the magnet). So in the second column we will put an arrow to the left.
-\\
+[{{184_notes:inductionchart_partb.png?440| Step 2: isolates the direction of the magnetic field and now places that corresponding vector into the chart. }}]
-[{{ 184_notes:ic_bfield.png?440| Step 1 of the solution process. This step isolates the direction of the magnetic field and now places that corresponding vector into the chart.}}]
-\\
+====Step 3.) Choose the $d\vec{A}$ ====
+Remember that the $d\vec{A}$ is perpendicular to the cross section area of the coils. Meaning, that you can think of the $d\vec{A}$ as pointing "out of” the coil. For our set up, this means that $d\vec{A}$ could point either to the left or right (-x or +x direction). It doesn't matter which way you pick, as long as the $d\vec{A}$ is perpendicular to the area. For this example, we'll pick the $d\vec{A}$ to point to the left, so we draw an arrow in the third column that points to the left.
-====Step 1.) Direction of B-field====
+[{{184_notes:inductionchart_partc.png?440| Step 3: Pick a direction for dA that points perpendicular to the coil. In this example, we pick  dA to be to the left.  }}]
-\\
-When given a magnetic bar, remember how the B-field flows. The B-field moves from the south pole to the north pole, then wraps around the magnetic bar as it flows back to the south. This B-field is also mirrored across the bar and will show a symmetrical flow to the B-field through a magnet.
+====Step 4.) $\Phi_{B,i}$, $\Phi_{B,f}$ and $\frac{d\Phi_{B}}{dt}$====
+Now, we're ready to figure out what is happening to the initial flux, final flux, and therefore what is happening to the change in flux for the scenario. For this step, we don't need to do exact calculations. Instead, what we really care about is the relative magnitude of the flux (is it a big or small flux) and the sign of the flux (is it positive or negative?).
-\\
+The magnitude of the flux will depend on the size of the B-field and area at that particular time. For example if your bar magnet is close to your loop, you'd expect a big flux since the magnetic field is stronger closer to the magnet. The sign of the flux will be determined by the dot product of $\vec{B}$ and $d\vec{A}$. Remember we can simplify the flux equation by saying:
-[{{ 184_notes:ic_da.png?440| Step 2 of the solution process. This step isolates the dA of the scenario and allows the students to include that in the chart.}}]
+$$\Phi_B = \int \vec{B} \bullet d\vec{A} = \int B *dA *cos(\theta)$$
-\\
+So if $\theta$ is between $0^\circ-90^\circ$, then we know the flux will be positive. If $\theta$ is between $90^\circ-180^\circ$, then the flux will be negative. So let's go through this for our example.
-====Step 2.) Choose the $d\vec{A}$ ====
+===Initial flux===
-\\
+The initial flux is determined for the "beginning of the scenario." For our situation, we start with the bar magnet initially close to the coil (and it will be moving away). This means that the initial magnetic flux should be //relatively big// through the coil since the magnetic field will be stronger closer to the magnet. The sign of the initial flux will be //positive// since both the B-field and the area vector point to the left ($\theta = 0^\circ$). So in the fourth column, we write that the initial flux will be "big and positive".
-Remember that the $d\vec{A}$ is perpendicular to the cross section area of the solenoid. Meaning, that you can think of the $d\vec{A}$ as pointing “through” the solenoid. As far as what direction the $d\vec{A}$ should be, that is something you can choose between which end of the solenoid the $d\vec{A}$ will be pointed towards.
-\\
+===Final Flux===
-[{{ 184_notes:ic_flux.png
+The final flux is determined after some time has passed. Think about this as looking at the “End of the Scenario”. For our situation, this means the the bar magnet will have moved further away from the coil to the left. This means that magnetic field through the coil will still be pointing to the left (toward the south end of the magnet) but it will be smaller in magnitude since the magnet is further away. (Nothing has changed about our coil so the dA vector remains the same). This means that our final magnetic flux will be relatively small (compared to our initial flux), but the flux should still be positive since B and dA still point in the same direction. Thus in the fifth column, we write that the final flux is "small and positive."
-?440|Step 3 of the solution process. This step takes the movement of the object and generalizes the change in flux of the scenario. These are then placed into the chart.}}]
-\\
+===Change in Flux===
+Now we can determine the sign of the change in flux. Since we already have the initial and final flux values, this step is simple. We just use:
+$$\Delta \Phi_B = \Phi_{B,f} - \Phi_{B,i}$$
-====Step 3.) $\frac{d \Phi_B}{dt}$====
+In our case, this means we'd be taking a small positive number minus a big positive number. This will result in a //negative// change in flux. (If it helps, you can assign numbers to help you think through this. For example, we could take $2-10 = -8$.) So we write down in the sixth column that the change in flux is negative.
-\\
-This portion of the approach is where the generalization of the situation begins. Meaning now we are going to talk about what the items: Big/small and Positive/Negative will mean. When we refer to Big and Small, we are using these terms to understand the relative impact that the situation has before and after with whatever object is moving. As we've seen before, the closer to the source the stronger the B-field. The farther away the observation point is, the smaller the B-field.
-When referring to positive/negative we will be talking about directionality. If the $d\vec{A}$ that you choose is in the same direction as the B-field, then the flux will be deemed a positive flux. If $d\vec{A}$ and the B-field are pointing in the opposite direction (anti-parallel) then there will be negative flux.
+[{{184_notes:inductionchart_partd.png
+?440|Step 4: Determine the sign of the change in flux based on the initial and final flux for the situation.  }}]
-\\
+====Step 5: Determining $V_{ind}$====
+Now that the direction of the $\frac{d \Phi_B}{dt}$  has been identified, we can simply flip the sign for the V-induced. We have to do this step because there is a negative sign in Faraday's Law:
+$$V_{ind} = -\frac{d \Phi_B}{dt}$$
-====Initial flux====
+For our example, the change in flux was negative. So we write down "positive" for the $V_{ind}$ column.
-Now that we have an understanding of the key terms we can start to walk through the table. The initial flux will be done __BEFORE THE POSITION__ CHANGE of the object. Think about this as looking at the “Beginning of the Scenario”. From this observation, consider how much of an impact the B-field will have on the observation point. Are the objects closer before or after the position update? If CLOSER BEFORE, then we would deem this initial flux to be “large”. If CLOSER AFTER, then we would deem this initial flux to be “small”.
-\\
-====Final Flux====
-The final flux is calculated __AFTER THE POSITION CHANGE__ of the object. Think about this as looking at the “End of the Scenario”. Compared to the initial position, if the objects are FARTHER, then we would deem this final flux to be “small”. If the objects are now CLOSER, then we would deem the final flux to be “large”.
+[{{184_notes:inductionchart_parte.png
+?440|Step 5: The sign of the V-induced is the **opposite** of the change in flux.  }}]
+====Step 6: Determining the direction of $I_{ind}$====
+Our final step then is to determine the direction of the induced current! This is where the right hand rule part comes in. But what does it mean for $V_{ind}$ to be positive or negative? Ultimately this goes back to our choice of $d\vec{A}$ at the beginning. So there will be two scenarios:
+  * If $V_{ind}$ is positive, you should put the thumb of your right hand in the **same direction as your $d\vec{A}$**. Then the way that you curl your fingers will show the direction of the induced current.
+  * If $V_{ind}$ is negative, you should put the thumb of your right hand in the **opposite direction as your $d\vec{A}$**. Then the way that you curl your fingers will show the direction of the induced current.
-[{{ 184_notes:ic_vinduced.png
+For our example, we found that $V_{ind}$ was positive. So this means that we would stick our thumb in the -x direction (pointing to the left) and then curl our fingers around in a circle. You should find the the current would go into the page at the top of the coil and would come back out of the page at the bottom of the coil. Just because it's often hard to draw the current direction in 3D, we often will just notate where on the coil the current is into/out of the page.
-?440|Step 4 of the solution process. The sign of the V-induced is the **opposite** of the change in flux.}}]
-[{{ 184_notes:inducedcurrentscenario_final.png
-?440|Step 5 of the solution process. This allows for the description of the motion of the induced current to be highlighted and place within the chart.}}]
-\\
-=====Concluding=====
-$\frac{d \Phi_B}{dt}$, remember, is the change in flux over a given time. Meaning that we need to calculate and understand the change of the flux as we start and end the scenario. Since these were generalizations given to find flux, the only thing we need for this approach is the sign of the change in flux. Meaning if I have a “Big Positive Initial Flux” (i.e. the B-Field and the $d\vec{A}$ are in the same direction and very close together at the beginning of the scenario) and then have a “Small Positive Final Flux” (i.e. the B-Field and the $d\vec{A}$ are in the same direction and relatively farther away at the end of the scenario). This change would result in a final $\frac{d \Phi_B}{dt}$  of a negative number. (Final – initial with the large value being in the latter.) Now that the direction of the $\frac{d \Phi_B}{dt}$  has been identified, we can flip the sign for the V-induced.
-\\
-\begin{align*}
--\int \vec{E}_{nc} \bullet d\vec{l} &= \frac{d \Phi_B}{dt} &&&& (1) \\
--V_{ind} &= \frac{d \Phi_B}{dt} &&&& (2) \\
-\end{align*}
-\\
-This is now a built in check to see if the predicted $d\vec{A}$ was correct! __**If the V-induced value is positive, then your right hand will stay in the same direction that you started with. If the V-induced value is negative, then you must flip your hand to go in the opposite direction.**__ Now that we have the properly identified $d\vec{A}$, we can utilize the Right Hand Rule Method to describe the direction of the I-induced.
-\\
-Remember that the thumb represents the $d\vec{A}$ and the curl of your fingers describes the rotational direction that the current will flow. In our example, since the $d\vec{A}$ points in the positive x-direction and the V-induced was positive, Meaning that if the induced current was occurring in the z-direction, we would label the answers as either “Into the page, then Out of the Page” or vice versa to describe the direction of flow. By utilizing this approach, students are able to have a formal methodology for approaching how to describe the directionality of the Induced Current.
+[{{184_notes:inductionchart_partf.png
+?440|Step 6: Find the direction of the induced current based on the right hand rule. }}]
+The right hand rule for determining the induced current direction is definitely more complicated than our previous right hand rules; however, this table helps break the process down into manageable chunks. We definitely recommend writing out each step in this way - otherwise it is easy to miss a step or a negative sign, which will throw off your final result.