Equivalent resistance between two points

Click For Summary

Homework Help Overview

The discussion revolves around calculating the equivalent resistance of multiple resistors connected in parallel between two points, A and B. The original poster attempts to understand the behavior of the equivalent resistance as more resistors are added, particularly questioning the convergence of the series formed by their reciprocals.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore calculating the equivalent resistance for a small number of resistors and observe the trend in values. Some question the nature of the series formed by the resistances and whether it converges or diverges. Others suggest examining the sequence of terms to identify a pattern or general expression.

Discussion Status

There is an ongoing exploration of the series and its implications for the equivalent resistance. Some participants have offered observations about the diminishing returns of adding more resistors, while others emphasize the need to clarify the sequence involved. No consensus has been reached on the nature of the series or the correct approach to the problem.

Contextual Notes

Participants note that the problem may involve assumptions about the convergence of the series, and there are reminders about the forum's guidelines regarding providing direct answers.

Vibhor
Messages
971
Reaction score
40

Homework Statement


?temp_hash=245e5e166ad9cf11402f5697f6433411.png


Homework Equations

The Attempt at a Solution


[/B]
All the resistors are connected in parallel across A and B .The equivalent resistance is 1/Req= (1/R) +(1/2R) + (1/6R) + (1/12R) + ...

I tried to work with few terms and it seems the effective resistance tends to zero as terms increase . But C) is not the correct option .

Now I am not sure how to calculate this sum . Am I missing something very simple ?

Thanks
 

Attachments

  • question3.png
    question3.png
    11.7 KB · Views: 615
Physics news on Phys.org
Calculate the result for up to 4 resistors in parallel.

Look at the results for 1, 2, 3 and then 4 resistors in parallel.

For 1 resistor, Req = R.

For 2 resistors, Req = 2/3 R

continue for the next two.

Observe how much of a change you see from 1 to 2, 2 to 3, etc.

It should be decreasing but it doesn't go to 0.

Another observation is when the added resistor in parallel gets to be about 1/20 R, it doesn't add much to Req.
 
Zero is not the answer for sure, as you can see the values of the parallel resistances go on increasing.

Leave the first R alone. What can you say about the sequence 2R, 6R, 12R, 20R...?
What is the expression for this sequence?
T1=2R, T2=6R...and so on. What is the general expression for this sequence i.e. Tn=?
 
magoo said:
Calculate the result for up to 4 resistors in parallel.

Look at the results for 1, 2, 3 and then 4 resistors in parallel.

For 1 resistor, Req = R.

For 2 resistors, Req = 2/3 R

continue for the next two.

Observe how much of a change you see from 1 to 2, 2 to 3, etc.

It should be decreasing but it doesn't go to 0.

Another observation is when the added resistor in parallel gets to be about 1/20 R, it doesn't add much to Req.
That all may be suggestive but proves nothing. The first step is to figure out the general form of the sequence 1, 2, 6, 12, 20...
Actually, it doesn't have a guessable one unless you omit the initial 1. That seems to be a mistake by the question setter.
 
My suggestion was merely to calculate the results for up to 4 resistors in parallel. If you look at the values or plot the Req versus # of resistors, you'll see that the curve decreases to a value other than 0. You can do it for the 5 values shown also.

If you haven't figured out the series, the fact that the 20R and greater terms are insignificant in the total makes this a reasonably acceptable approach. If the higher terms had greater impact on Req, then this would not be acceptable.
 
  • Like
Likes   Reactions: Greg Bernhardt
I agree with haruspex, the sequence is far from obvious.

There is a website called "Car Talk" run nominally by two MIT graduates (one sadly deceased). This is the sort of puzzle they post. The answer is usually irrelevant to mathemaics to any perceptible extent. This one looks like another such.
 
magoo said:
My suggestion was merely to calculate the results for up to 4 resistors in parallel. If you look at the values or plot the Req versus # of resistors, you'll see that the curve decreases to a value other than 0. You can do it for the 5 values shown also.
I think this approach is simpler (and smarter!). Simply calculate the equivalent resistance up to 4 or 5 resistors and compare that value with the options. The closest option would be the correct one.
 
Last edited:
cnh1995 said:
I think this approach is simpler (and smarter!). Simply calculate the equivalent resistance up to 4 or 5 resistors and compare that value with the options. The closest option would be the correct one.
That sounds like the way to go all right.
 
You can only give a unique answer if you assume going beyond what we are told, that this series which we don't know is convergent. If it is not, if it is divergent, equivalent resistance is 0 (c) .
Assuming then that the series is convergent we can answer the question since we don't need to know what the equivalent resistance is, only what it isn't. It's fairly obviously isn't b. By working out the equivalent resistance of the part we can see we can decide between the two remaining possibilities.
 
  • #10
Sum of 1/r approaches 2R
Rt=R/2 approximate.
 

Similar threads

Total equivalent resistance of a combined circuit
  • · Replies 7 ·
Replies
7
Views
2K
Find the equivalent resistance of this circuit
  • · Replies 31 ·
2
Replies
31
Views
5K
Current through an inductor as a function of time
  • · Replies 10 ·
Replies
10
Views
1K
Calculate the equivalent resistance
  • · Replies 11 ·
Replies
11
Views
2K
How can I calculate the equivalent resistance of this circuit?
  • · Replies 9 ·
Replies
9
Views
2K
Determining the equivalent resistence of a circuit
  • · Replies 4 ·
Replies
4
Views
2K
Equivalent resistance of a circuit
  • · Replies 3 ·
Replies
3
Views
2K
Calculate equivalent resistance
  • · Replies 5 ·
Replies
5
Views
2K
Equivalent resistance between two points
  • · Replies 12 ·
Replies
12
Views
5K
Calculation of equivalent resistance
  • · Replies 4 ·
Replies
4
Views
2K