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--- 184_notes:motiv_amp_law [2017/10/04 10:25] – caballero
+++ 184_notes:motiv_amp_law [2018/07/24 14:36] – [Motivating Ampere's Law] curdemma
@@ Line 1: / Line 1: @@
-====== Motivating Ampere's Law ======
+Sections 21.5 and 21.6 in Matter and Interactions (4th edition)
-Earlier you learned about the [[Biot-Savart law|184_notes:b_current]], which is a way of determining the magnetic field due to moving charges. The Biot-Savart law relies on the [[principle of superposition|184_notes: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 [[Ampere's law|https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law]] can be conceptually more difficulty than Biot-Savart, but is often, mathematically more straight-forward.
+[[184_notes:loop|Next Page: Magnetic Field along a Closed Loop]]
-===== Why Ampere's Law? =====
+====== 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: Gauss's Law and Ampere's Law. We'll start by talking about Ampere's Law, which is an alternative method for calculating the magnetic field. We require highly symmetric magnetic fields to be able to solve Ampere's Law. Next week we will learn about Gauss's Law, the electric field equivalent of Ampere's Law.
+{{youtube>M6FRK4aY90E?large}}
-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. And in that case, it can be used in general because it's just a procedure that exploits superposition (adding up of small contributions).
+===== Why Use Ampere's Law? =====
-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 things a bit easier (like [[Gauss' Law for electric fields|184_notes:gauss_motive]]).
+Before 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!
+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]]).
 ===== What is Ampere's Law? =====
-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 ([[and people are looking!|http://physicstoday.scitation.org/doi/full/10.1063/PT.3.3328]]), there are no single magnetic 'charges', termed '[[magnetic monopoles|https://en.wikipedia.org/wiki/Magnetic_monopole]].' 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, no closed volume has net magnetic flux.
+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. For example, if we think about enclosing a bar magnet with a Gaussian-like surface (an imaginary bubble), the magnetic flux through the whole bubble would be zero! For every magnetic field vector pointing into the bubble, there is also a magnetic field vector pointing out.
+[{{184_notes:Week11_flux_vol.png?300|Permanent magnet enclosed in an "imaginary bubble"  }}]
-$$\int\int\mathbf{B}\cdot d\mathbf{A} = 0$$
+It turns out that this is true no matter what the source of your magnetic field is (bar magnet, moving charge, or current-carrying wire) - 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__//. Mathematically, we would write this as:
-So, what is Ampere's Law? It must relate the field ($\mathbf{B}$) to the source of the field (i.e., moving charges, $I$). Because these two are related through [[a vector operation called a curl|https://en.wikipedia.org/wiki/Curl_(mathematics)]], the mathematical relationship between them in an integral over a closed loop,
+$$\int\int\vec{B}\cdot d\vec{A} = 0$$
-$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$
+So then, what is Ampere's Law? We want it to 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 an **imagined closed loop** rather than an imagined closed surface (like with Gauss's Law). Mathematically, we will represent this relationship as:
-where the loop encloses an some current that passes through the surface of the loop (see below).
+$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
-FIXME Add figure of current and loop
+{{  184_notes:week10_1.png?400}}
-The situation above and the accompanying formula is what we will unpack over the next few pages.
+where the imagined loop that we pick has some enclosed current that passes through the surface of the loop (like the example to the right). The next few pages of notes will unpack the parts of this equation and how we can use it to find the magnetic field from a current.