Can anyone explain to me why the induced magnetic field will oppose the change in magnetic flux? Is it an energy thing?
I know that the induced emf is $$ emf= - \frac{d\phi}{dt} $$ but my book doesn't give a satisfactory (or any) explanation as to why this is the case. I recognize that an emf source supplies energy and does work on charge carriers, but I dont see what that has to do with the magnetic flux.
Simply consider Maxwell equation :
$$\vec{\nabla}\wedge\vec{E}=-\frac{\partial\vec{B}}{\partial t}$$
If you interger this on a given closed surface $\Sigma$, it follows :
$$\oint_\Sigma \left(\vec{\nabla}\wedge\vec{E}\right) \cdot d\vec{S} =-\frac{\partial}{\partial t}\oint_\Sigma \vec{B}\cdot d\vec{S}$$
where $d\vec{S}=dS\,.\vec{n}$ with $dS$ the differential element of the surface $\Sigma$ and $\vec{n}$ the local normal direction of $\Sigma$, centred on $dS$. Here, $\phi=\oint_\Sigma \vec{B}\cdot d\vec{S}$ stands for the magnetic flux through the surface $\Sigma$.
At this point, Stockes theorem gives you :
$$\oint_\Sigma \left(\vec{\nabla}\wedge\vec{E}\right) \cdot d\vec{S}=\oint_\Gamma \vec{E}\cdot d\vec{l}$$ where $\Gamma$ is a given contour included in $\Sigma$, and $d\vec{l}=dl.\vec{r}$ is the differential length element along $\Gamma$. What is called emf is simply $e=\oint_\Gamma \vec{E}\cdot d\vec{l}$. Then directly follows :
$$e=-\frac{d\phi}{dt}$$
If you want to interprete the $-$ signe, keep in mind that it is linked magnetic flux conservation that follows a moderation law : "Effects are in opposition with their causes".
Maybe you have seen the experiment where a permanent magnet is dropped through a pipe made of conducting metal. The magnetic flux through a cross-section of the pipe will be changing, so a current is induced. The induced magnetic field is such as to oppose the change in magnetic flux, so it will slow down the falling magnet. Eventually an equilibrium is reached where the magnetic force and the gravitational force balance, and the magnet falls at constant velocity until the end of the pipe. You can find many demonstrations on YouTube by searching for "magnet in pipe" or something similar.
Now imagine if it were the opposite way, that is, the induced magnetic field was in the opposite direction, as to enhance the change in magnetic flux. Then the magnet would be accelerated, so the induced current would grow, accelerating the magnet more, and so on. This is absurd from conservation of energy! Not only would the kinetic energy of the magnet increase without a source, the pipe would get hotter and hotter from the increasing current.
(In the first case, the one that is realized in nature, the pipe does get heated. Instead of the magnets gravitational potential energy going to kinetic energy, it becomes heat. So conservation of energy is respected.)
