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184_notes:loop [2017/10/05 00:16] – [What do we get for the integral?] caballero | 184_notes:loop [2020/08/24 13:28] – dmcpadden |
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===== Magnetic field due to a wire ===== | Section 21.6 in Matter and Interactions (4th edition) |
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Our canonical example for the magnetic field will be a long straight wire with a current I running through it. In this case, we will consider that the wire is a bit thicker than the average wire, so that it has a current density $J=I/A$, which is uniform. That is, at every point in the wire the same amount of charge per unit time per unit area exists. This will help us understand the power of Ampere's Law. The figure below shows a cross-section of the wire. | /*[[184_notes:i_thru|Next Page: Current through a Loop]] |
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FIXME Add figure with cross-section of wire | [[184_notes:motiv_amp_law|Previous page: Motivating Ampere's Law]]*/ |
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==== What is the "shape" of the magnetic field? ==== | ===== Magnetic field along a closed loop ===== |
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As with the [[184_notes:gauss_motive|electric field and Gauss' Law]], the structure and shape of the magnetic field is very important to understanding Ampere's Law. So we will start with the left-hand side of the equation, | For the context of our explanation, we will use a long straight wire with a current $I$ running through it as our example. We'll start by talking about a thin wire and eventually build up to talking about a thick wire. In this class, we will also make the typical //__assumption that the current in the wire is uniform and in a steady state__// (upper-division courses may address non-uniform currents). That is, at every point in the wire the same amount of charge per unit time exists. |
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$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$ | ==== What is the "shape" of the magnetic field? ==== |
| Since we already calculated the B-field from a long straight wire using the [[184_notes:b_current|Biot-Savart law]] (by adding up all the contributions), we already have some insight into to what we think the magnetic field should be from the wire - it should curl around the wire. (We will eventually use the Biot-Savart result at the end to verify what we find using Ampere's Law.) For Ampere's Law, the structure, shape, and symmetry of the magnetic field is very important to understanding how to set up the different parts of the equation. **This means that the first step in Ampere's law is to draw the magnetic field.** |
to understand what it means and how we can calculate it. In doing this, we will always check our work with the result for a long wire we obtained from using the [[184_notes:b_current|Biot-Savart law]], by adding up all the contributions. | |
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=== Magnetic field circulates around moving charges === | === Magnetic field circulates around moving charges === |
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As you have seen, 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. |
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{{url>http://web.pa.msu.edu/people/caballero/teaching/simulations/MagneticFieldCirculation.html 675px,425px|Charges moving along a line}} | {{url>http://web.pa.msu.edu/people/caballero/teaching/simulations/MagneticFieldCirculation.html 675px,425px|Charges moving along a line}} |
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As you can see the size of the magnetic field vector (the arrows) gets larger as the charges get closer to the observation locations and smaller as the pass it and move away. The direction is always around the path of the charges. Now, imagine a constant stream of particles very close together, that is, a current of 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 === |
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The structure (or symmetry) of this field is very useful for Ampere's Law. That is, the fact that it always circulate around the wire and is the same magnitude for a given distance makes this problem 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 results 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 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]]. |
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==== What is $\mathbf{B}\cdot d\mathbf{l}$? ==== | We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere's Law: |
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You have seen these kinds of dot products before, i.e., when we defined [[184_notes:pc_potential|electric potential]]. But in that case, we understood this to be a measure work per unit charge. In the case of the magnetic field there is no such analog. That integral of the magnetic field along a path is not an work. As you will see the [[184_notes:motiv_b_force|magnetic force]] has important properties that make it such that the magnetic field can do no work. So what is this thing? | $$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}$$ |
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This integral formulation comes from the [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|mathematical model]] that the magnetic field seems to obey in every instance we observe it. Namely, how much the magnetic field curls around the wire depends on how much current there is creating the magnetic field. This measure, which is beyond the scope of this course, called the [[https://en.wikipedia.org/wiki/Maxwell%27s_equations#Amp.C3.A8re.27s_law_with_Maxwell.27s_addition|curl of the magnetic field]] is an important property of the field and is an essential quality of what distinguishes it from the electric field, which can have no curly shape when it is created by static charges. | Again, the first thing we will consider will be the shape of the magnetic field for the situation - which is related to the left-hand side of Ampere's Law - namely the $\oint \vec{B} \bullet d\vec{l}$ part. |
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So think of $\mathbf{B}\cdot d\mathbf{l}$ as the little measure of how much the magnetic field curls around its source (the current). Our job is to add up all those little contributions to find out how curl the magnetic field is at a given distance from the current, | {{youtube>XkUeIV31cKs?large}} |
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$$\oint \mathbf{B} \cdot d\mathbf{l}.$$ | ==== What is $\vec{B}\bullet d\vec{l}$? ==== |
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| You have seen these kinds of dot products before - when we defined [[184_notes:pc_potential|electric potential]]. But in the case of electric potential, we understood this to be a measure of work per unit charge. In the case of the magnetic field, there is no such analog. **That integral of the magnetic field along a path is not the considered "work"**. As you saw before, the [[184_notes:q_path|magnetic force]] has important properties that make it such that the magnetic field cannot do work. So what is this $\vec{B}\bullet d\vec{l}$ thing? |
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| This integral formulation comes from the [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|mathematical model]] that the magnetic field seems to obey in every instance we observe it. Namely, how much the magnetic field curls around the wire depends on how much current there is creating the magnetic field. This measure, which is beyond the scope of this course, called the [[https://en.wikipedia.org/wiki/Maxwell%27s_equations#Amp.C3.A8re.27s_law_with_Maxwell.27s_addition|curl of the magnetic field]] is an important property of the field and is an essential quality of what distinguishes it from the electric field (a single //__stationary__// charge has an E-field that points away or towards the charge - it cannot be curly). |
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| **So think of $\vec{B}\bullet d\vec{l}$ as the little measure of how much the magnetic field curls around its source (the current)**. Our job is to add up all those little contributions along a closed path to find the curl of the magnetic field at a given distance from the current, |
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| $$\oint \vec{B} \bullet d\vec{l}.$$ |
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This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. | This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. |
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=== How do we compute it? === | === How do we compute it? === |
| [{{ 184_notes:week11_loop_shapes.png?300|Shapes of Amperian Loops}}] |
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As we will work exclusively with cylindrically shaped wires, the loop you follow the magnetic field around will always be a circle with the wire at the center. This loop isn't real, but rather, it's like the [[184_notes:gauss_ex|Gaussian surface]] that we had before. It's a construct that helps guide our work with Ampere's Law. We refer to it as the "Amperian Loop." | First, **we want to pick a Amperian loop (or the imaginary path) so that the observation point is somewhere along the path**. For example, if we wanted to find the magnetic field a distance of $R$ away from the wire, we would want to pick a loop that has a radius of $R$ around the wire. |
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FIXME Figure of amperian loop | Then, since we get to pick what kind of loop we want, we want to choose a loop that will make our math easy. If we look at the $$\oint \vec{B} \bullet d\vec{l}$$ - there are a couple of steps that will make our calculation MUCH simpler: |
| - **We want $\vec{B}$ to be parallel to $d\vec{l}$ because then the dot product turns into multiplication** $$\oint \vec{B} \bullet d\vec{l} = \oint |\vec{B}||d\vec{l}| $$ |
| - **We want the magnitude of $\vec{B}$ to be constant so we can pull it out of the integral** $$\oint |\vec{B}||d\vec{l}| = |\vec{B}| \oint |dl|$$ |
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It is around this loop that we compute the integral, checking how much of $\mathbf{B}$ lines up with our little $d\mathbf{l}$ and adding it up. Formally, we are doing a path or line integral around a loop with the magnetic field, but for most cases this integral will simplify quite a bit. Also, it's ok if this idea is a bit abstract now, we will put it all together with an example in next set of notes. | 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). |
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==== What do we get for the integral? ==== | **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." |
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If the magnetic field points in the same direction as our Amperian loop and it has the same magnitude along that loop, then the calculation of this integral is relatively straight-forward. Both of these conditions are satisfied for any Amperian loop that is centered on a wire with a uniform distribution of current. | [{{ 184_notes:week10_3.png?400|Amperian loop around a line of current}}] |
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| Formally, we are doing a path or line integral around a loop with the magnetic field, but for most cases this integral will simplify quite a bit. Also, it's ok if this idea is a bit abstract now, we will put it all together with an example in next set of notes. |
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| ==== Long Wire Example ==== |
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| Both of these conditions are satisfied if we pick a circular Amperian loop that is centered on a wire with a uniform distribution of current: 1) the magnetic field points in the same direction as our Amperian loop and 2) it has the same magnitude along that loop. This makes the calculation of this integral is relatively straight-forward. |
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For an Amperian loop of radius $R$ centered on a wire with uniform current, we find that this integral is, | For an Amperian loop of radius $R$ centered on a wire with uniform current, we find that this integral is, |
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$$\oint \mathbf{B} \cdot d\mathbf{l} = B\oint dl = B2\pi R$$ | \begin{align*} |
| \oint \vec{B} \bullet d\vec{l} &= \oint |\vec{B}||d\vec{l}| \\ |
| &= \oint B dl \\ |
| &= B \oint dl \\ |
| &= B l |
| \end{align*} |
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| 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: |
| $$\oint \vec{B} \bullet d\vec{l} = B(2 \pi R)$$ |
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where the last two steps come from the fact that $\mathbf{B}$ is the direction of $d\mathbf{l}$ and is the same magnitude along that loop. Thus, it can pulled out from the integral. The resulting integral of $dl$ is just the circumference of the loop. | 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). |