Central question
I think your question may center on this:
and therefore back emf be 10v and then the current would be 0 since
those two voltages would cancel each other out
I think you are curious why, if the back-EMF is the same as the applied EMF, things don't just "cancel out" and therefore why does any change take place, at all.
Short overview
An inductance is defined by its design and implementation. Just as a capacitance is defined by its design and implementation. Capacitors hold charge, \$q\$, and are defined such that \$C=\frac{\text{d}\,q}{\text{d}\,V}\$ (or, the infinitesimal change of charge with respect to some infinitesimal change in voltage.) Inductors hold flux, \$\phi\$, and are defined such that \$L=\frac{\text{d}\,\phi}{\text{d}\,I}\$ (or, the infinitesimal change of flux with respect to some infinitesimal change in current.)
You can think of the flux (\$\phi=L\cdot I=\int V\:\text{d}\,t\$) of an inductor, in Webers, as being the dual of charge (\$q=C\cdot V=\int I\:\text{d}\, t\$) on a capacitor, in Coulombs. In mechanical physics, these are the equivalent of momentum:
\$p=m\cdot v=\int F\:\text{d}\,t\$.
The equivalent of an external mechanical force (\$F\$) for an inductor is voltage (\$V=\frac{\text{d}\,\phi}{\text{d}\,t}=L\,\frac{\text{d}\,I}{\text{d}\,t}\$) and for a capacitor it is current (\$I=\frac{\text{d}\,q}{\text{d}\,t}=C\,\frac{\text{d}\,V}{\text{d}\,t}\$). The equivalent for mechanical velocity, \$v\$, for an inductor is current, \$I\$, and for a capacitor it is voltage, \$V\$. (The equivalent for mechanical acceleration, \$a\$, for an inductor is \$\frac{\text{d}\,I}{\text{d}\,t}\$ and for a capacitor it is \$\frac{\text{d}\,V}{\text{d}\,t}\$.)
An answer
I'd earlier written that \$L=\frac{\text{d}\,\phi}{\text{d}\,I}\$. But this is also \$L=\frac{\text{d}\,\phi}{\text{d}\,I}=\frac{\frac{\text{d}\,\phi}{\text{d}\,t}}{\frac{\text{d}\,I}{\text{d}\,t}}\$.
The inductance is then also defined as the rate of change of flux with respect to the rate of change in its current. A changing current in the inductor implies there must be a changing magnetic flux. A changing flux yields a non-Coulomb electric field that curls around the region of changing flux in the inductor. This non-Coulomb electric field sets itself up to oppose the changing current and is called back EMF. This back EMF must be met by exactly the same magnitude of applied EMF (the forces are equal, but opposite, just as in mechanical physics every force has an equal but opposite force) in order to yield some rate of change in the flux (and current.) The inductance value itself determines the ratio.
In some sense, you can think of the back EMF as similar to the inertia of mechanical mass. An inductor is like mass and when you apply a force (here, a voltage \$V\$) to it, it responds with an equal but opposite force (here, a back EMF \$\epsilon\$.) It's just the response to applying a force on the inductor and it must be equal, but opposite. So there is no conflict where these voltages "cancel out and nothing happens." Instead, back EMF, \$\epsilon\$, is simply the inertial counter-force that opposes some applied external force, \$V\$, on the inductor.
