A rather challeging circuit problem

[DaMastaofFisix]

So the Problem states as follows: Two batteries, each of Voltage V and internal resistance r, are connected in parallel. This parallel connection is then connected in parallel with a Resistor of Value R.
a) For What Value of R is the Power is maximum?
b) If V=12 and r=.03, what is the power dissipated across the resistor?

Okay, so I understand that this involves a little bit of differential calculus, no big deal. I am also aware that Power=(i^2)R, where i is the current through the resistor R. I am also aware the this requires some serious loop rule action in order t be soleved, hence the upcoming problem; I have trouble With the equations and their "plus and minus" consistency so to speak. Can Someone hook it up with a clue or two?

BTW, the internal resistance is to the right of each battery...if that helps at all.

 

[Hammie]

Maximum power is transferred when the load impedance (here resistance) equals source impedance (resistance again)

Try drawing a simpler equivalent circuit.

 

[berkeman]

Maximum power is transferred when the load impedance (here resistance) equals source impedance (resistance again)

When the impedances are complex, the maximum power is transferred when the source impedance is the complex conjugate of the load impedance. But we all work so much with real loads and source impedances that it's easy to forget the conjugate part.

Try drawing a simpler equivalent circuit.

This is good advice. Just draw a source voltage, a source resistor, and a load resistor. Write the equation for the power dissipated in the load resistor as a function of the single variable Rs, and use differentiation to solve for the value of Rs that results in the highest power in the load. Then once you see how the general principle works, you can apply it to your present problem.

 

[DaMastaofFisix]

Excellent, thanks so much. Turns out that the arrangement calls for maximum power at a valur or R=r/2, which in a way makes sense, t=since it's analagous to a equivalent resistor combination. Just needed a little guiding hand.