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R1 = 2 ohm

R2 = 5 ohm

What value of R3 maximizes the dissipation rate in resistance 3? (battery is ideal -> no internal resistance)

DIAGRAM:

_______________

|.........|...........|

R1.......R2.........R3

|.........|...........|

_ +......|...........|

- -.......|...........|

|.........|...........|

_______________

(I hope the diagram is clear...please ignore the dots... they are just there as placeholders... below R1 is the battery... the + side is on the top, the - side is on the bottom)

MY TAKE ON THE PROBLEM:

We know that the dissipation rate for a resistor is P = i^2*R.

In order to maximize the dissipation rate, I figured the best way would be to find an equation relating i3 to R3 (and then subbing it in for i3 in order to get only R3s on the right side)... and then simply using calculus to maximize the rate in terms of resistance.

But therein lies the problem... could not come up with such an equation... I tried using Kirchhoff's loop rule to get these:

-i3R3 + E - i1R1 = 0 (clockwise traversal through the BIG loop)

-i3R3 + i2R2 = 0 (clockwise traversal through the RIGHT loop)

-i2R2 + E - i1R1 = 0 (clockwise traversal throught the LEFT loop)

needless to say, I had assumed that i1 (going through R1) goes upwards, and that r2 and r3 (through R2 and R3, respectively) are pointing downwards.

E is the emf of the battery

Also, I know that i1 = i2 + i3 (junction rule)

the PROBLEM:

As I have mentioned, I am unable to find such an equation...

I need some pointers... if you could PLEASE tell me whether at least I am moving in the right direction...

I am all out of ideas...

Thank you

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# Maximizing dissipation rate in a simple circuit

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