184_notes:motiv_amp_law

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Sections 21.5 and 21.6 in Matter and Interactions (4th edition)

Next Page: Magnetic Field along a Closed Loop

Motivating Ampere's Law

Last week, we talked about how Gauss's Law gave a nice mathematical shortcut for finding the electric field from symmetric distributions of charge. This week, we are going to talk about the equivalent mathematical shortcut for magnetic fields, called Ampere's Law. While there are definitely some differences between Gauss's Law and Ampere's Law, the process for solving these kinds of problems should seem very similar - starting with the fact that we require highly symmetric magnetic fields to be able to solve Ampere's Law.

Before when solving for the 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 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 Gauss' Law for electric fields).

Given the analogy to Gauss' Law, you might think the approach is going to be similar - we integrate the magnetic field over an imagined closed surface area and that tells us something. Unfortunately, as far as a we know (and people are looking!), there are no single magnetic 'charges', termed '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.

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:

$$\int\int\vec{B}\cdot d\vec{A} = 0$$

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 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:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$

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.

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