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184_notes:loop [2018/11/01 21:27] – dmcpadden184_notes:loop [2022/04/04 12:46] (current) – hallstein
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 Section 21.6 in Matter and Interactions (4th edition) Section 21.6 in Matter and Interactions (4th edition)
  
-[[184_notes:i_thru|Next Page: Current through a Loop]]+/*[[184_notes:i_thru|Next Page: Current through a Loop]]
  
-[[184_notes:motiv_amp_law|Previous page: Motivating Ampere's Law]]+[[184_notes:motiv_amp_law|Previous page: Motivating Ampere's Law]]*/
  
 ===== Magnetic field along a closed loop ===== ===== Magnetic field along a closed loop =====
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 As you have seen before, the [[184_notes:moving_q|magnetic field circulates around a moving charge]]. Below is a simulation that shows this for a series of moving point charges. As you have seen before, the [[184_notes:moving_q|magnetic field circulates around a moving charge]]. Below is a simulation that shows this for a series of moving point charges.
  
-{{url>http://web.pa.msu.edu/people/caballero/teaching/simulations/MagneticFieldCirculation.html 675px,425px|Charges moving along a line}}+{{url>https://web.pa.msu.edu/people/caballero/teaching/simulations/MagneticFieldCirculation.html 675px,425px|Charges moving along a line}}
  
 As you can see, the sizes of the magnetic field vectors (the arrows) get larger as the charges get closer to the observation locations and get smaller as they pass and move away. The direction of the magnetic field is always around the path of the charges. Now, imagine a constant stream of particles very close together, that is, a current of many electrons moving in a wire. In that case, the direction of the magnetic field is still around the wire, but the magnitude stays constant. That is, **for a steady current, the magnetic field at any one point is a constant in time**. Furthermore, **for cylindrical wires (the ones we will focus on), the magnitude of magnetic field is constant for a given distance from the wire**. You saw this with the long wire when we solved the problem using Biot-Savart.  As you can see, the sizes of the magnetic field vectors (the arrows) get larger as the charges get closer to the observation locations and get smaller as they pass and move away. The direction of the magnetic field is always around the path of the charges. Now, imagine a constant stream of particles very close together, that is, a current of many electrons moving in a wire. In that case, the direction of the magnetic field is still around the wire, but the magnitude stays constant. That is, **for a steady current, the magnetic field at any one point is a constant in time**. Furthermore, **for cylindrical wires (the ones we will focus on), the magnitude of magnetic field is constant for a given distance from the wire**. You saw this with the long wire when we solved the problem using Biot-Savart. 
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 === Symmetry is critical === === Symmetry is critical ===
  
-The structure (or symmetry) of this field is very useful for Ampere's Law. That is, the fact that the B-field always circulates around the wire and is the same magnitude for a given distance makes this problem of a steady state line of charge an excellent candidate for using Ampere's Law. //In fact, this symmetry holds inside the wire as well//. That is, there will be a constant magnitude magnetic field circulating around every point inside the wire as well! This is a really important result that we can explore with Ampere's Law, but that can be a real challenge to deal with using [[184_notes:b_current|Biot-Savart]].+The structure (or symmetry) of this field is very useful for Ampere's Law. That is, the fact that the B-field always circulates around the wire and is the same magnitude for a given distance makes this problem of a steady state current an excellent candidate for using Ampere's Law. //In fact, this symmetry holds inside the wire as well//. That is, there will be a constant magnitude magnetic field circulating around every point inside the wire as well! This is a really important result that we can explore with Ampere's Law, but that can be a real challenge to deal with using [[184_notes:b_current|Biot-Savart]].
  
 We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere's Law: We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere's Law:
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 This means we want to choose a closed loop that: 1) is always following the B-Field (to make $d\vec{l}$ parallel to $\vec{B}$) and 2) has a constant B-Field everywhere around the loop. So if the magnetic field points in circle around the wire, you want to pick a circular loop. If the magnetic field is constant, you want to pick a loop with at least one straight edge (like a rectangular loop). This means we want to choose a closed loop that: 1) is always following the B-Field (to make $d\vec{l}$ parallel to $\vec{B}$) and 2) has a constant B-Field everywhere around the loop. So if the magnetic field points in circle around the wire, you want to pick a circular loop. If the magnetic field is constant, you want to pick a loop with at least one straight edge (like a rectangular loop).
  
-//It's important to note that this loop isn't real// - there is not a wire or anything physically around the wire. It's an imagined loop that helps guide our work with Ampere's Law. We refer to this imagined loop as the "Amperian Loop."+**It's important to note that this loop isn't real** - there is not a wire or anything physically around the wire. It's an imagined loop that helps guide our work with Ampere's Law. We refer to this imagined loop as the "Amperian Loop."
  
 [{{  184_notes:week10_3.png?400|Amperian loop around a line of current}}] [{{  184_notes:week10_3.png?400|Amperian loop around a line of current}}]
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 \end{align*} \end{align*}
  
-where the first two steps listed come from the fact that $\vec{B}$ points in the direction of $d\vec{l}$ everywhere on the loop (so the dot product simplifies to multiplication) and the last two steps listed come from the fact that the B-field is the same magnitude along that loop (so you can pull it out of the integral). The resulting integral of $dl$ gives just $l$. Because this length comes from the integration of the $dl$, the length $l$ here represents the length of the Amperian loop. For our example, the length of the loop would just be the circumference of the loop. Since we said that the loop would have a radius R, this means $l=2\pi R$, so we get:+where the first two steps listed come from the fact that $\vec{B}$ points in the direction of $d\vec{l}$ everywhere on the loop (so the dot product simplifies to multiplication) and the last two steps listed come from the fact that the B-field is the same magnitude along that loop (so you can pull it out of the integral). The resulting integral of $dl$ gives just $l$. Because this length comes from the integration of the $dl$, **the length $l$ here represents the length of the Amperian loop**. For our example, the length of the loop would just be the circumference of the loop. Since we said that the loop would have a radius R, this means $l=2\pi R$, so we get:
 $$\oint \vec{B} \bullet d\vec{l} = B(2 \pi R)$$ $$\oint \vec{B} \bullet d\vec{l} = B(2 \pi R)$$
  
 Remember that the $B$ in this equation is the magnitude of the B-field on that loop (so this would be the magnetic field at a distance $R$ away from the straight wire). Remember that the $B$ in this equation is the magnitude of the B-field on that loop (so this would be the magnetic field at a distance $R$ away from the straight wire).
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