Inductors and back EMF
Don't think of the back emf as being caused by the source voltage. Think of the back emf as being caused by the changing current.

The source produces a current. That (changing) current causes the inductor to produce a back emf. The back emf limits (but doesn't eliminate) the current produced by the source. If it eliminated the current, then there would be no back emf. The two things don't oppose each other so much as they balance each other.

By KVL, the back-emf of the inductor must be be equal and opposite to the voltage produced by the source, or else energy conservation would be broken.


  Isn't there a cancellation of voltage to zero volts?


There is, in the sense that if you follow the path of the circuit and add up all the voltages you pass (the source and the inductor, in a simple case), then the sum of all those voltages will be zero once you've followed the complete circuit. 

But of course that's just KVL, and it applies to any circuit elements (resistors, capacitors, or whatever) and not just to inductors.

I think what's confusing (at least to me) is the use of the word "opposite". In a normal drawing of this circuit, we draw the source with its positive terminal at the top of the page, and then the inductor will also produce a back emf that's positive at the top of the page. But if you consider the voltage in the direction of traversing the circuit (for example, always going in a clockwise direction around the circuit), then the two potential differences (source emf and inductor back-emf) will be opposite.

edit RE the video

I think the video is confusing because he never defines the sign convention he is using to define his voltages across the different elements.

In order for the description to be correct, he must have defined opposite conventions for the two types of elements, like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Keep in mind Kirchoff's voltage law. With these conventions, \$V_r\$ must be equal to the V1 source voltage (regardless of what type of resistor is used or even if you replaced the resistor with a different element like a capacitor or inductor). And \$V_l\$ must be the opposite of the V2 source voltage (again, it would still be opposite even if you replaced the inductor with a different element like a resistor or capacitor.

Again, this is simply because of KVL and the choice of sign convention for the voltage across the two elements. But in the video he never shows the chosen sign convention, so this makes it impossible to understand what he's trying to teach.

Either that or the guy in the video has no understanding of circuit theory and believes that a circuit with an inductor in it can violate KVL, which is simply not correct.