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| ====== Motivating Ampere's Law ====== | Sections 21.5 and 21.6 in Matter and Interactions (4th edition) |
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| Earlier you learned about the [[184_notes:b_current|Biot-Savart law]], which is a way of determining the magnetic field due to moving charges. The Biot-Savart law relies on the [[184_notes:superposition|principle of superposition]], each little charge creates a small contribution to the net magnetic field and we add up all the contributions, often by integrating along the wire, to find the net magnetic field. Here, you will read about another way to determine the magnetic field that exploits the symmetry (or shape) of the situation. This approach called [[https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law|Ampere's law]] can be conceptually more difficult than Biot-Savart, but is often, mathematically more straight-forward. | /*[[184_notes:loop|Next Page: Magnetic Field along a Closed Loop]]*/ |
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| | ====== Motivating Ampere's Law ====== |
| | So far in this course, we have talked about the sources of electric fields, how electric fields are applied to circuits, and the sources of magnetic fields. Over the next two weeks, we are going to talk about two mathematical shortcuts for calculating the electric and magnetic fields: Ampere's Law and Gauss's Law. In both Ampere's Law and Gauss's Law, we will require that either the magnetic field (for Ampere's Law) or the electric field (for Gauss's Law) be highly symmetric. We'll start by talking about Ampere's Law, which is an alternative method for calculating the magnetic field (though there will be many parallels between Ampere's Law and Gauss's Law). Next week we will learn about Gauss's Law, the electric field equivalent of Ampere's Law. |
| {{youtube>M6FRK4aY90E?large}} | {{youtube>M6FRK4aY90E?large}} |
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| ===== Why Use Ampere's Law? ===== | ===== Why Use Ampere's Law? ===== |
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| The Biot-Savart law, is always true, but it is not always useful for solving problems with pencil-and-paper. Solving problems with Biot-Savart using pencil-and-paper techniques requires that the integral that you construct have a known anti-derivative. That is, the integral that you construct can be integrated into a known function. Now, this might seem to suggest that Biot-Savart is only useful when that's that case, but Biot-Savart can also be used in a computer program. The computer is very good at adding many little bits of something and is able to do so with or without a known anti-derivative. The Biot-Savart Law (either analytically or computationally) can be used in almost any case because it's just a procedure that exploits superposition (adding up of small contributions). | Last week when solving for the [[184_notes:b_current|magnetic field from a current]], we used an integral (and superposition) to add up the magnetic field from the current in each little chunk of wire, which we called the Biot-Savart Law. The Biot-Savart law is always true, but the integral can sometimes be impossible to solve by hand. In these cases, you can always program the [[184_notes:b_sup_comp|computer to do the superposition]] for you - the computer is very good at repeatedly adding many little bits of something! |
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| Where Biot-Savart becomes a bit unwieldy is when we want to find the magnetic field due to distributions of currents (termed volume currents) where the symmetry is really simple (e.g., a long thick wire with current throughout). In this case, we will find that Ampere's Law is a nice shortcut to solving these problems. So think of Ampere's Law as a useful analytical technique that can be used in some cases where the symmetry of the situation (as we will see) suggests it's a better choice than Biot-Savart. Ultimately both give you the magnetic field at a location, but sometimes one approach makes the math a bit easier (like [[184_notes:gauss_motive|Gauss' Law for electric fields]]). | Where Biot-Savart becomes a bit unwieldy is when we want to find the magnetic field due to distributions of currents (termed volume currents) where the symmetry is really simple (e.g., a long thick wire with current throughout). In this case, we will find that Ampere's Law is a nice shortcut to solving these problems. So think of Ampere's Law as a useful analytical technique that can be used in some cases where the symmetry of the situation (as we will see) suggests it's a better choice than Biot-Savart. Ultimately both give you the magnetic field at a location, but sometimes one approach makes the math a bit easier. |
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| ===== What is Ampere's Law? ===== | ===== What is Ampere's Law? ===== |
| | So then, what is Ampere's Law? Ampere's law (at it's most fundamental) is a way to relate the magnetic field ($\vec{B}$) to the source of the field (i.e., moving charges, $I$). In equation form, Ampere's Law is given by: |
| Given the analogy to Gauss' Law, you might think the approach is going to be similar, we integrate the magnetic field over a surface and that tells us something. Unfortunately, as far as a we know ([[http://physicstoday.scitation.org/doi/full/10.1063/PT.3.3328|and people are looking!]]), there are no single magnetic 'charges', termed '[[https://en.wikipedia.org/wiki/Magnetic_monopole|magnetic monopoles]].' That is, everywhere we look magnetic poles come in pairs - a north pole and a south pole. So the integral of the magnetic field over a closed surface is always zero. That is, a closed volume never has net magnetic flux //__assuming that the magnetic field is constant__//. | |
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| $$\int\int\vec{B}\cdot d\vec{A} = 0$$ | |
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| So, what is Ampere's Law? It must relate the field ($\vec{B}$) to the source of the field (i.e., moving charges, $I$). Because these two are related through [[https://en.wikipedia.org/wiki/Curl_(mathematics)|a vector operation called a curl]], the mathematical relationship between them is an integral over a closed loop, | |
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| $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$ | $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$ |
| {{ 184_notes:week10_1.png?400}} | {{ 184_notes:week10_1.png?400}} |
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| where the loop encloses an some current that passes through the surface of the loop (see below). This magnetic field is curly, you have seen the pattern for the field around a long wire. How curly we observe the magnetic field to be is related the current that produces it. The more current the more curly the magnetic field (i.e., the great magnitude at a given distance and thus the more it changes in a small circular step around the wire). | This equation says that if we add up little bits of the magnetic field along the length of **any closed path** (the $\oint \vec{B} \cdot d\vec{l}$ part) that should be related to the amount of current that passes through this closed path (the $\mu_0 I_{enc}$ part). This equation is true for any closed path around a current, but we will see that there are some choices of path that are easier to handle mathematically than others. (This is the main reason why we need very symmetric fields to be able to use Ampere's Law.) Because we get to choose the closed path, we will often refer to this path as an "imagined loop" or an "Amperian loop". |
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| The situation above and the accompanying formula is what we will unpack over the next few pages. | The next few pages of notes will unpack each of the parts of this equation and show how we can use it to find the magnetic field from a current. |