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BruceW

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For resistors 1 and 2, we know the power dissipated by each is [itex]I_1^2 R_1[/itex] and [itex]I_2^2 R_2[/itex] so the total power being dissipated by the circuit is:

[tex]P= I_1^2 R_1 + I_2^2 R_2 [/tex]

We also know that the sum of currents going into a node equal zero. So let's suppose we had current [itex]I[/itex] going into the circuit, which is then split into [itex]I_1[/itex] and [itex]I_2[/itex], so that:

[tex]I = I_1 + I_2 [/tex]

Let's suppose from here that [itex]I[/itex] is set. So how much current will go through one resistor and how much will go through the other resistor? We could work this out by using Ohm's law, but I tried out something else that comes up with the same answer. If we simply try to minimise the power lost by the circuit, then we will be given the correct answer. So here's my calculation:

[tex]dP = 2I_1 R_1 dI_1 + 2 I_2 R_2 dI_2 [/tex]

And we also know that [itex]dI_2 = - dI_1[/itex] since we are saying the current [itex]I[/itex] is set. So using this, we get:

[tex]dP = 2(I_1 R_1 - I_2 R_2 )dI_1[/tex]

So from this, we see there is a stationary point when [itex]I_1 R_1 = I_2 R_2 [/itex] So is this a minimum, or what? Well, we have

[tex]\frac{dP}{dI_1} = 2(I_1 R_1 - I_2 R_2 )[/tex]

So now, take the total differential of this with respect to [itex]dI_1[/itex], keeping in mind that [itex]dI_2 = - dI_1[/itex] and we get:

[tex]\frac{d^2 P}{d I_1^2} = 2(R_1 + R_2) [/tex]

So as long as the resistances are positive, the stationary point at [itex]I_1 R_1 = I_2 R_2 [/itex] will be a minimum!

So what the heck does this all mean? I started with the equation for power dissipated by a resistor, and the rule saying that currents 'add up' at a node, and I said that the dissipated power must be minimised, and that told me that [itex]I_1 R_1 = I_2 R_2 [/itex] So the current flows through this circuit according to the principle of minimising the dissipated power!

So what do you all think? Is this just a coincidence, or is there some simple explanation that I've overlooked, or is this an example of a principle that I am unaware of? Thanks in advance :)