Resistance Association - How to solve this exercise in the proper way?

In summary: The correct equation should be mϵ=(mrn+R)iIn summary, the problem involves 40 identical generators with an electromotive force of 1.5V and a resistance of 0.25 Ohms, along with one resistor with a resistance of 2.5 Ohm. The goal is to find the group configuration that results in the greatest energy dissipation per second, which is 90W. The solution involves creating 2 groups of 20 cells, ensuring equal current and power sharing among the cells. The method used is similar to the Maximum Power Transfer theorem, but with the use of mean-inequality. The proposed equation for the problem is incorrect and should be revised to mϵ=(mrn
  • #1
jaumzaum
434
33
Homework Statement
See below
Relevant Equations
See below
This was an exercise from the 2011 admission test of an university from Brazil:
40 identical generators with electromotive force 1,5V and resistance 0,25 Ohms and one resistor with resistance 2,5 Ohm are associated in such a way that the 2.5- resistor dissipates the greatest energy per second, that is? Answer: 90W.

My solution:
If we make m groups of n generators we would have:
$$m\epsilon=(\frac{mr}{n}+R)i$$
$$i=\frac{1,5mn}{0,25m+2,5n}=\frac{6mn}{m+10n}=\frac{240}{40/n+10n}=\frac{24}{4/n+n}$$
We know by the mean-inequality that
$$4/n+n>=2\sqrt{4} =4 $$
This is possible because it occurs when:
$$4/n=n \rightarrow n=2, m=20$$
$$i=6A \rightarrow P=90W$$

But why do all the groups have the same number of resistors? Why does the circuit have only series or parallel associations? Why isn't there any Delta/Star or any other more complex associations? Is there any way to solve this exercise in the right way?

P.S. I'm really not sure if the best way to post this is in the Physics or Math Forum. I'm posting in the Physics one. Sorry if I'm wrong by that/
 
Physics news on Phys.org
  • #2
I didn't understand your method of using mean-inequality, but I got 90W using Maximum Power Transfer theorem.
I believe you have interchanged the values of m and n. As per my calculations, you need to have 2 groups of 20 cells.
jaumzaum said:
But why do all the groups have the same number of resistors?
To ensure equal current and power sharing among the cells, thereby avoiding under/overloading of any cell.
If the power sources are identical, they should practically share equal power.
 
  • Like
Likes berkeman
  • #3
cnh1995 said:
I didn't understand your method of using mean-inequality,
Ok I got your method. It is almost the same as Maximum Power Transfer theorem, except that you are not using Thevenin resistance in your calculations.
jaumzaum said:
My solution:
If we make m groups of n generators we would have:

mϵ=(mrn+R)i​
I believe this equation is incorrect. Put m=1, n=40 and see if you get the correct current for 1 group of 40 cells.
 

FAQ: Resistance Association - How to solve this exercise in the proper way?

What is the Resistance Association exercise?

The Resistance Association exercise is a problem-solving exercise that involves finding the resistance of a circuit using Ohm's Law and Kirchhoff's Laws.

What are the steps to solving the Resistance Association exercise?

The steps to solving the Resistance Association exercise are as follows: 1) Identify the known values and unknown values in the circuit. 2) Apply Ohm's Law to calculate the resistance of individual resistors. 3) Use Kirchhoff's Laws to set up and solve a system of equations to find the total resistance of the circuit. 4) Double check your calculations and answer to ensure accuracy.

What are some common mistakes made when solving the Resistance Association exercise?

Some common mistakes made when solving the Resistance Association exercise include: forgetting to convert units, using incorrect values for resistors, and making errors in calculations. It is important to double check your work and use the correct formulas and values to avoid these mistakes.

How can I improve my problem-solving skills for the Resistance Association exercise?

To improve your problem-solving skills for the Resistance Association exercise, it is important to have a strong understanding of Ohm's Law and Kirchhoff's Laws. Practice solving similar problems and double check your work for accuracy. Additionally, seeking help from a tutor or classmate can also improve your skills.

Are there any alternative methods for solving the Resistance Association exercise?

Yes, there are alternative methods for solving the Resistance Association exercise, such as using Thevenin's or Norton's Theorems. These methods can be useful for more complex circuits, but it is important to have a strong understanding of the basics before using them.

Back
Top