My understanding is that if you run an electrical current through a coil of wire, interrupting the current will cause the magnetic field to collapse, producing a spike of higher voltage than what you originally put into the system.

Why is this? Why doesn’t the current produced by the collapsing field have the same voltage as before?

V is voltage
L is inductance a property of the loop of wire.
di/dt is the rate of change of the current in the wire.

This is the basic equation for inductors. By breaking the loop you stop the current so the change in the current is very large and thus the voltage goes way up.

As for the magnetic field colapsing causeing a spark.

The engery stored in the magnetic field is 1/2Li*i

This energy has to go somewhere when you break the circiut it is lost in the energy that it takes to create the spark that will happen when you break the circiut.

Simply, the current produced by the collapsing field has the same current as before. When you change the conditions (e.g. interrupt the current), the inductor tries to maintain the same current. Say a 1-Amp current was flowing from an inductor to a 1 Ohm resistor. According to Ohm’s Law, V=IR, so the voltage across the resistor is 1 Volt times 1 Ohm = 1 Volt. Now, let’s say the reistor is a variable resistor, and you increase the resistance to 1000V very very fast. The inductor initially tries to maintain a 1-Amp current, thus forcing 1 Amp through the 1000-Ohm resistor. The risult is 1000 Volts across the resistor! Incidentally, that’s a common method used in high voltage generators.

Now, what happens if instead of replacing the resistor, you just remove the resistor? The inductor tries to force 1 Amp through the air gap, creating a nice big spark.