Motivating Ampere's Law

Earlier you learned about the 184_notes:b_current, which is a way of determining the magnetic field due to moving charges. The Biot-Savart law relies on the 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 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.

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).

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 184_notes:gauss_motive).

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), there are no single magnetic 'charges', termed '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.

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

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 https://en.wikipedia.org/wiki/Curl_(mathematics), the mathematical relationship between them in an integral over a closed loop,

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

where the loop encloses an some current that passes through the surface of the loop (see below).

Add figure of current and loop

The situation above and the accompanying formula is what we will unpack over the next few pages.