An amazing circuit that can produde 1000s of volts from a simple battery

I wished I was taught in High school or even college. Sigh! I found out when I was tutoring a student from a premier European university. 

It is used in ignition coil to generate 1000s of voltage to produce the spark that makes controlled explosion in the car/motorcycle/scooter  (remember spark plug)

The Amazing Simple Circuit

This is that simple circuit:

The Problem:

Let me state the problem: The circuit initially has the switched closed for quite long time and steady current flows. (the inductor is fully loaded with magnetic field). When this switch is open, what is the voltage across AB at that instant of time?

Here is the answer:

In the initial condition, at steady state, when the switch is closed. It acts like simple DC circuit with battery and two resistances. Inductor acts like a simple wire (because the current is not changing) with zero resistance . From simple circuit analysis, the currents are given as follows (Note A is positive):  

$$  I_1 = \frac{V_0}{R_1},  I_2 = \frac{V_0}{R_2}$$

Now, when the switch is open, some thing amazing happens at that very instant. The inductance opposes the action (Lenz's Law) and responds by maintaining the same current in inductance, which is  \( I_2 \).

But the circuit is different now (also note the A has become negative, earlier it was positve. In short, A and B has changed the polarity) and shown below:

The current \( I_2 \) now flows across \( R_1 \) and \( R_2 \). The voltage across AB is same as voltage across \( R_1 \), which is \( I_2 \times R_1 \), the voltage across AB then is:

$$  V_{AB}= I_2 \times R_2 = \frac{V_0}{R_2} \times R_1 $$

What is amazing is the voltage does not depend on the value of inductance!! Such a simple circuit. Don't you think it deserves to be taught in High school or Junior college? The fact that this beautiful question is taughter in European university, explain why Europe leads in many area of science!

Then what the purpose of having inductance with large value? Well, after the circuit is open, the current \(   I_2  \) is generated by inductance coil. And if inductance is large, it can maintain the current for a longer time. From the point of energy, large inductance mean, the coil contains more magnetic field. The magnetic field takes more time to exhaust itself.

THE END