How Is Optimal Load Resistance Calculated in Parallel Battery Circuits?

[w3390]

Homework Statement

Two batteries that have emfs E1 and E2 and internal resistances r1 and r2 are connected in parallel. Prove that if a resistor of resistance R is connected in parallel with combination, the optimal load resistance (the value of R at which maximum power is delivered) is given by R= (r1*r2)/(r1+r2).

Homework Equations

resistor in series: 1/Req=(1/r1)+(1/r2)+...

The Attempt at a Solution

I have drawn out the circuit explained in the problem. Right off the bat I found out the equivalent resistance of the circuit and it looks to have the same form as the answer in the problem statement, but is has the extra R. I'm not sure if I need to take into account the emf or the current in each segment of the wire?

 

[Delphi51]

the value of R at which maximum power is delivered

so you must get an expression for the power delivered to the load (resistor R). Hopefully it will be some function with a maximum value that you can find with differentiation or by finding the peak of a quadratic function.

 

[nlightner]

I would approach this problem by first understanding the concept of parallel and series circuits. In a parallel circuit, the voltage across each branch is the same, while the current is divided between the branches. In a series circuit, the current is the same in each branch, while the voltage is divided between the branches.

Based on this understanding, I would start by using Ohm's Law to calculate the current in each branch of the parallel circuit. The current through the first battery would be I1 = E1/(r1+R) and the current through the second battery would be I2 = E2/(r2+R).

Next, I would use the power formula P=IV to calculate the power delivered by each battery. The power delivered by the first battery would be P1 = I1*E1 and the power delivered by the second battery would be P2 = I2*E2.

To find the total power delivered by the parallel combination of the two batteries, I would add the powers P1 and P2. This would give me the total power delivered by the parallel combination at a given resistance R.

To find the optimal load resistance, I would take the derivative of the total power with respect to R and set it equal to 0. This would give me the value of R at which the maximum power is delivered.

After solving for R, I would substitute the values for r1 and r2 back into the equation to confirm that the optimal load resistance is indeed given by R = (r1*r2)/(r1+r2).

Overall, this conceptual proof shows that the optimal load resistance for a parallel combination of two batteries with internal resistances r1 and r2 is dependent on the individual resistances and is not affected by the emfs of the batteries.