Braess's Paradox

Braess's paradox states that when we add a new path into a network, it may decrease the total flow through the network.

The paradox can be nicely illustrated on a spring. However, beware that Nature's illustration of the spring experiment is flown:

The lengths of the springs should be different when we cut the rope (while the lengths of the remaining ropes should remain unchanged). The corrected illustration:

On Wikipedia, you can read that you can also observe it in electrical networks "at low temperatures using a scanning gate microscopy":

But you don't need a fancy technology to observe the paradox in electrical circuits. Nature's article actually got close to a "simple circuit":

But it requires a source of constant current. A simpler schema, which utilizes a source of constant voltage, is:

We just need 2 light bulbs, 2 resistors, a 9V battery (or any other source of constant voltage) and an ampermeter to measure the change of the current. Optionally, we may include a button or switch. But a piece of wire is enough.

How does it work? Incandescent bulbs have an interesting property: when they are cold, they work almost like a short-circuit. Only once they heat up, their resistance increases to the nominal value (the diagram is for a 60W light-bulb):

The nominal resistance of the light bulb in our schema is 5V/(0.45W/5V)=56Ω. When the button is open, the nominal current thru a single light bulb is 9V/(56Ω+68Ω)=0.073A. But once we close the button, more current goes thru the light bulbs than thru the resistors, because the light bulbs have nominal resistance of 56Ω while the resistors have 68Ω. But the increased current thru the light bulbs heats up the filaments, the light bulb resistance increases and the total current thru the network decreases.

Experimental currents thru the whole network for different resistors:

resistance [Ω]	open [mA]	close [mA]
32	160	160.5
46	140	140
68	125	124
100	112	113

The paradox was observed only for one value of the resistors' resistance (68Ω). The optimal resistance for maximization of the paradox effect is somewhere around 62Ω. If you have a tandem potentiometer that can withstand ~1 Watt (e.g.: an old wire potentiometer), I would be happy to hear what is the actual optimal value and the size of the effect.

Applications: A teaser on an open-day/science-fair in a school. You ask visitors to guess when the circuit is going to consume more energy:

1. When the button is visibly closed and the light bulbs are shining brightly?
2. Or when the button is visibly open and the light bulbs are just dim?

The paradoxical answer is that the circuit consumes more energy when the light bulbs are dim. 

To puzzle the visitors even more, you may stress that the circuit contains only passive components and that the trick is not in the power supply.