Stationary charges give rise to electric fields. Moving charges give rise to magnetic fields. In this chapter, we explore how this comes about, starting with currents in wires which give rise to a magnetic field wrapping the wire.

In this chapter, we rewrite the Maxwell equations yet again, this time in the language of actions and Lagrangians that we introduced in the first book in this series. This provides many new perspectives on electromagnetism. Among the pay-offs are a deeper understanding, via Noether’s theorem, of the energy and momentum carried by electromagnetism fields. This will also allow us to explore a number of deeper ideas, including superconductivity, the Higgs mechanism, and topological insulators.

In this chapter, we explore the basics of fluid mechanics. We will think about how to describe fluids and look at the kinds of things they can do.

Unusually, and a little defensively, the title of this chapter highlights what we won’t talk about, rather than what we will. Fluids have a property known as viscosity. This is an internal friction force acting within the fluid as diferent layers rub together. It is crucially important in many applications. In spite of its importance, we will start our journey into the world of fluids by ignoring viscosity altogether. Such flows are called inviscid. This will allow us to build intuition for the equations of fluid mechanics without the complications that viscosity brings. Moreover, the flows that we find in this section will not be wasted work. As we will see later, they give a good approximation to viscous flows in certain regimes where the more general equations reduce to those studied here.

The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.

Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

Take an electric charge and shake it and it will produce light. The purpose of this chapter is to explain how this works.

The Hamiltonian plays the starring role in the standard formulation of quantum mechanics. But, back in the classical world, there are two equivalent ways to write down a theory, one using the Hamiltonian and the other using the Lagrangian. It’s natural to wonder if there might also be another formulation of quantum mechanics, where things are written in terms of the Lagrangian. Happily, there is. And it’s lovely. Its called the path integral

There are four forces in our universe. Two act only at the very smallest scales and one only at the very biggest. For everything inbetween, there is electromagnetism. The theory of electromagnetism is described by four gloriously simple and beautiful vector calculus equations known as the Maxwell equations. These are the first genuinely fundamental equations that we meet in our physics education and they survive, essentially unchanged, in our best modern theories of physics. They also serve as a blueprint for what subsequent laws of physics look like.

This textbook takes us on a tour of the Maxwell equations and their many solutions. It starts with the basics of electric and magnetic phenomena and explains how their unification results in waves that we call light. It then describes more advanced topics such as superconductors, monopoles, radiation, and electromagnetism in matter. The book concludes with a detailed review of the mathematics of vector calculus.

Newtons laws of motion are not the last word in classical mechanics. In the 250 years after Newton, physicists and mathematicians found ways to reformulate classical mechanics, providing powerful tools to solve problems but, equally as importantly, giving us a new perspective on the laws that govern our universe. This chapter takes the first step in this direction. We will introduce the wonderful principle of least action, a simple rule that underlies all known laws of physics. This will give us new insights, not least the wonderful Noethers theorem, relating symmetries to conservation laws.

Over the past hundred years or so, physicists have developed a foolproof and powerful tool that allows us to understand everything and anything in the universe. You take the object that you’re interested in and you throw something at it. Ideally, you throw something at it really hard. This technique was developed around the turn of the 20th century and has since allowed us to understand everything from the structure of atoms, to the structure of materials, to the structure of DNA. In short, throwing stuff at other stuff is the single most important experi- mental method available to science. Because of this, it is given a respectable sounding name. We call it scattering.

This chapter explores what we could do with a computer whose operating system is quantum mechanics, rather than classical mechanics. One of the answers is: factorise primes really quickly. We will explain why this is interesting.

We all know what a wave is. But you may not know just how many different kinds of waves there are and what strange and interesting properties they have. We start this chapter with something very familiar from everyday life: waves on the surface of an ocean. While they may be familiar, their mathematical description is surprisingly subtle. This can be traced, like so many other things in fluid mechanics, to the boundary conditions.

The fundamentals of electromagnetism are simple. Moving electric charges set up electric and magnetic fields. In turn, these fields make the charges move. This dance between charges and fields is described by the Maxwell equations. This brief chapter describes how this comes about. It is, in a sense, everything you need to know about electromagnetism, enshrined in these simple equations. The rest of the book is mere commentary.