184_notes:b_sup_comp

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184_notes:b_sup_comp [2018/03/15 14:29] – [How can we use a computer for this?] dmcpadden184_notes:b_sup_comp [2021/06/16 19:16] (current) – bartonmo
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 +/*[[184_notes:b_shapes|Next Page: Shapes of Wire and Magnetic Field]]
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 +[[184_notes:b_current|Previous Page: Currents Make Magnetic Fields]]*/
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 ===== Using Superposition of Magnetic Field and the Computer ===== ===== Using Superposition of Magnetic Field and the Computer =====
  
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 While calculating the integral will give you an exact value for the magnetic field, it is possible to end up with an integral that is difficult or impossible to solve. Instead we can use the fact that the magnetic field obeys the principle of superposition to define a powerful algorithm for computing the magnetic field at any given location from any shape of current/wire. This is very similar to what we did with the [[184_notes:comp_super|electric field from a line before]]. In these notes, you will read about how you can structure code to calculate the magnetic field from any shape of current-carrying wire. While calculating the integral will give you an exact value for the magnetic field, it is possible to end up with an integral that is difficult or impossible to solve. Instead we can use the fact that the magnetic field obeys the principle of superposition to define a powerful algorithm for computing the magnetic field at any given location from any shape of current/wire. This is very similar to what we did with the [[184_notes:comp_super|electric field from a line before]]. In these notes, you will read about how you can structure code to calculate the magnetic field from any shape of current-carrying wire.
  
-==== The Superposition Principle ====+===== The Superposition Principle =====
  
 [[184_notes:b_current|As we said before]], the principle of superposition states that the magnetic field at any given location in space is determined by vector sum of the magnetic field due to the current in each part of wire. [[184_notes:b_current|As we said before]], the principle of superposition states that the magnetic field at any given location in space is determined by vector sum of the magnetic field due to the current in each part of wire.
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 where $\vec{B}_1$ would be the magnetic field from one piece of the wire, $\vec{B}_2$ would be the magnetic field from a second piece of the wire, and so on. This idea of adding together the magnetic field from each chunk of the wire will form the basis of what we want the computer to calculate for us. where $\vec{B}_1$ would be the magnetic field from one piece of the wire, $\vec{B}_2$ would be the magnetic field from a second piece of the wire, and so on. This idea of adding together the magnetic field from each chunk of the wire will form the basis of what we want the computer to calculate for us.
  
-==== How can we use a computer for this? ====+===== How can we use a computer for this? =====
  
 For most real-world situations, the magnetic field integral cannot be solved analytically. That is, you could most likely write down the integral, but it cannot be computed because there's no anti-derivative for the function that you would be trying to integrate. So we have to think of another approach -- one that makes use of the principle of superposition, which we know the magnetic field obeys. For most real-world situations, the magnetic field integral cannot be solved analytically. That is, you could most likely write down the integral, but it cannot be computed because there's no anti-derivative for the function that you would be trying to integrate. So we have to think of another approach -- one that makes use of the principle of superposition, which we know the magnetic field obeys.
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   - Repeat steps 5-7 for another chunk; and continue repeating until you've done this for all chunks of the wire   - Repeat steps 5-7 for another chunk; and continue repeating until you've done this for all chunks of the wire
  
-These somewhat monotonous steps will give us an approximate value for the magnetic field at the point of interest. The smaller the chunks, the better the approximation. You can probably see why setting up a computer to do this makes a lot of sense. Computers are really good at doing the same calculation over and over again!+These somewhat monotonous steps will give us an approximate value for the magnetic field at the point of interest. **The smaller the chunks, the better the approximation.** You can probably see why setting up a computer to do this makes a lot of sense. Computers are really good at doing the same calculation over and over again!
  
 So if we want to compute the magnetic field at a given location due to any length/shape of wire, the algorithm is just splitting the wire into chunks, computing the magnetic field of each chunk, and adding all the contributions together. This is a form of [[https://en.wikipedia.org/wiki/Numerical_integration|numerical integration]], which is a powerful technique in computational science. As a tool for thinking through these computational algorithms, we will sometimes write out the steps we want the computer to take in plain words rather than code - this is called **pseudocode**. The pseudocode for the magnetic field algorithm above is the following: So if we want to compute the magnetic field at a given location due to any length/shape of wire, the algorithm is just splitting the wire into chunks, computing the magnetic field of each chunk, and adding all the contributions together. This is a form of [[https://en.wikipedia.org/wiki/Numerical_integration|numerical integration]], which is a powerful technique in computational science. As a tool for thinking through these computational algorithms, we will sometimes write out the steps we want the computer to take in plain words rather than code - this is called **pseudocode**. The pseudocode for the magnetic field algorithm above is the following:
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