Current through a circuit when the key is open

[Darshit Sharma]

Homework Statement

Consider the circuit diagram given below and answer the questions.

Relevant Equations

Ohm's law. V= IR

The equivalent resistance of the parallel combination is 2.4 ohm.

This implies that if the key is closed the total resistance of the circuit = R + r = 2.4 + 0.6 = 3 ohm

However, if the key is open, no current flows through the circuit and thus, by Ohm's law (R = V/I), as I tends to 0, R tends to infinity.
Thus, if the external resistance of the circuit is infinite, even if we add up the internal resistance of the battery ( 0.6 ohm), the resistance would still be infinity.

However, some of my fellow batchmates were fooling around with their reasoning as only the internal resistance of the battery would be taken into account and the final answer would be 0.6 ohm. They didn't provide a complete reasoning though.

Thus, the final question of this thread is what will be the resistance of the circuit, as the question questions and why?

 

[BvU]

Hi,

So it comes down to: is the 0.6 $\Omega$ part of the circuit or not ?
I would say it is not.

I even find it questionable to say it's part of the circuit when the switch is closed, but this internal resistance must be taken into account to calculate the current when the switch is closed.

The reason for this is that one can for example investigate this 'circuit' with some other setup that doesn't include this particular battery.

However, if the key is open, no current flows through the circuit and thus, by Ohm's law (R = V/I), as I tends to 0, R tends to infinity.

No, because then V=0 too.

$\$

 

[Darshit Sharma]

No, because then V=0 too.

$\$

Sir, please elaborate.

I am unable to make out why V =0.

 

[Darshit Sharma]

Hi,

So it comes down to: is the 0.6 $\Omega$ part of the circuit or not ?
I would say it is not.


$\$

So sir final answer should be infinite or zero or 2.4 or 0.6?

 

[Mister T]

When no current flows through the battery, it has effectively zero internal resistance. This is the case when the key is open.

 

[Darshit Sharma]

When no current flows through the battery, it has effectively zero internal resistance. This is the case when the key is open.

Ohk but V is still there?

 

[Mister T]

Ohk but V is still there?

Yes. You still have the full 6 V across the battery.

 

[BvU]

Sir, please elaborate.
...
I am unable to make out why V =0.

If the switch is open, the voltage over the circuit is zero

It's all about what you consider to be part of "the circuit" and what not. My subjective and somewhat inconsistent take is that "the circuit" under consideration does not include the battery if it is not connected.

It's more about communication than about physics.

$\$

 

[Steve4Physics]

The question asks for the 'resistance of the circuit'.

When the key is in the open position, the 'gap' is a resistor of infinite resistance. This makes the overall circuit resistance = infinity.

IMO.

 

[jbriggs444]

If the switch is open, the voltage over the circuit is zero

Maybe there is some convention of which I am unaware. But we have an oval labelled "A" on the schematic. It has one terminal facing the bottom leg of the circuit and one terminal facing the leg on the top.

To me, the natural reading is that "the circuit" is everything between those two terminals. The oval labelled A is an unspecified sensor that is attempting to measure the circuit, trying to treat it as a voltage source in series with a resistor (Thevenin equivalent) or a current source in parallel with a resistor (Norton equivalent).

The wording of the question and the layout of the diagram seems to beg for the Thevenin equivalent interpretation.

With this interpretation, the voltage "over the circuit" can be whatever the sensor drives it to be.

 

[BvU]

I know I'm not going to win this. To my worn-out brainwashed mind 'the circuit' is what is between A and B on top and the remainder is plumbing.

Usually A (black, with a circle around it) stands for an Ammeter (and I haven't yet seen any of those 'attempting to do' anything or 'trying to treat' something.

As I said, it's more about communication than anything else. All in good spirit.

Consultation with an authority proves me formally wrong. Somehow I don't feel inclined to repent.

And I'd be surprised if 'Thevenin' means anything for the OP

(by the way, the exercise quality is lacking IMHO)

$\$

 

[Mister T]

I think the intent of the question's author was to probe the student's understanding that internal resistance is present only when current flows through the battery.

 

[kuruman]

I know I'm not going to win this. To my worn-out brainwashed mind 'the circuit' is what is between A and B on top and the remainder is plumbing.

I agree with you that the electrical resistance of anything requires two points A and B between the resistance is to be determined. I disagree with your labeling A and B as you have because it is not very informative. The resistance between A and B going clockwise or counterclockwise? Why not choose the poles of the open switch? What if the switch were closed and put the leads of an ohmmeter at points A and B? The fact that the problem's author didn't supply points A and B makes me agree with you that the quality of this exercise is subpar.

I think the intent of the question's author was to probe the student's understanding that internal resistance is present only when current flows through the battery.

I found this problem here as part of a preparatory exam. Unfortunately, no answers are provided to help us deduce the author's intent.

 

[kuruman]

I looked at some other questions from the preparatory exam and found these two representatives of the quality thereof

 

[jbriggs444]

I know I'm not going to win this.

I did not want to make this an argument. It is just a discussion. No winners or losers. Just interested parties.

It is plausible that the "circuit" is indeed intended to be the pair of resistors in parallel and that the ammeter, switch, voltage source along with its embedded resistor are intended to perform the function of an ohmmeter.

The question then becomes: What reading should this "ohmmeter" display?

The answer then hinges on what assumptions the designer used when he calibrated the gauge. If the designer assumed an ideal 6 volt source without internal resistance then one reading would be expected. If the designer assumed an ideal 6 volt source with a 0.6 ohm internal resistance then a different (and more accurate) reading would be expected.

We need to guess how stupid the designer was.

Yes, I agree that it is a poor question that forces the student to read the mind of the person who posed it.

 

[Mister T]

I found this problem here as part of a preparatory exam. Unfortunately, no answers are provided to help us deduce the author's intent.

Terribly worded question. Mistakes in grammar. Doesn't tell you if you're supposed to read the ammeter when the key is open or closed (it's open in the diagram). Same for the current through the 4-ohm resistor.

 

[Mister T]

We need to guess how stupid the designer was

My guess is that it was a committee of "experts". I taught a mini-course in MCAT physics prep. The questions in the book were an embarrassment. I had to weed out about half of them. After the first time I didn't want to teach it again but the pay was absurdly high.

 

[PhDeezNutz]

In my opinion, there is no “circuit” until there is at least one closed loop. For this reason, I think the resistance when the switch is open is zero.

 

[jbriggs444]

In my opinion, there is no “circuit” until there is at least one closed loop. For this reason, I think the resistance when the switch is open is zero.

The resistance of a short circuit is zero.

If you think that an open circuit is an oxymoron then you should characterize the resistance of an open circuit as "undefined", not "zero".

 

[PhDeezNutz]

The resistance of a short circuit is zero.

If you think that an open circuit is an oxymoron then you should characterize the resistance of an open circuit as "undefined", not "zero".

Fair. Resistance is a property independent of current (provided the voltage isn't too high and actually rips it apart changing it's properties). So even if there is no voltage drop across the element its resistance is the same.

Rethinking this I would say the resistance when the switch is open is $0.6 \Omega$

 

[kuruman]

Rethinking this I would say the resistance when the switch is open is $0.6 \Omega$

Again, the resistance between what point and what other point? Ideally, one defines and measures resistance using a four-point measurement. Here we are given no points! That's my point.

 

[Mister T]

Resistance is a property independent of current

That's Ohm's Law. It's an approximation that fails miserably in many cases.

 

[Darshit Sharma]

Half of the things bounced off my head; sorry to say that and for the delay. I was busy taking some similar quality exams.

So, after all the final verdict is the question, as always, is framed wrong.

However, the answer given is 2.4 ohm. I don't know what they have assumed and why.

Fair. Resistance is a property independent of current (provided the voltage isn't too high and actually rips it apart changing its properties). So even if there is no voltage drop across the element its resistance is the same.

Rethinking this I would say the resistance when the switch is open is $0.6 \Omega$

What about that infinite ohm resistor (open switch), Sir?

Again, the resistance between what point and what other point? Ideally, one defines and measures resistance using a four-point measurement. Here we are given no points! That's my point.

I feel this is the sole reason for variable answers.



Anyways, what are some points of variable answers?
Which set of points should I consider to expect different answers?

And Resistance is independent of current ---- ok
but Current is dependent on Resistance ---- right? (Just confirming)


And if resistance is independent of current.....then how do we get infinite resistance across that open switch like all that I was doing R = V/I and since I = 0, R = infinity, was wrong?


I have attached the solution booklet.

 

[kuruman]

Here is a screenshot of this awful question for those still interested in it. I am not.

 

[berkeman]

And Resistance is independent of current ---- ok

I haven't read much of the thread, so sorry if I'm repeating another reply to this question. In an ideal circuit, the answer is yes. But in a real circuit, the resistor heats up more with more current through it, which increases its resistance a bit. The concept to search on is thermal coefficient of resistance. You don't need to worry much about it in your introductory circuit classes in school, but you most definitely need to take it into consideration in your more advanced circuit analysis classes and in real life circuit design.

 

[Mister T]

However, the answer given is 2.4 ohm. I don't know what they have assumed and why.

A 6 ohm and a 4 ohm resistor combined in parallel is equivalent to 2.4 ohms.

And if resistance is independent of current.....then how do we get infinite resistance across that open switch like all that I was doing R = V/I and since I = 0, R = infinity, was wrong?

Because when I = 0, V = 0 also.

 

[Darshit Sharma]

A 6 ohm and a 4 ohm resistor combined in parallel is equivalent to 2.4 ohms.

But that 2.4 ohms should be in series with that infinite resistance open switch....so infinite resistance + 2.4 = infinite resistance

 

[Darshit Sharma]

Because when I = 0, V = 0 also.

So 0/0 = undefined? But I've read somewhere that the resistance of an open switch is infinite



Just confirming once -- Is the answer 2.4 ohms to (i) part correct?


I am sorry to everyone here for both the lack of understanding on my part and the "awful" -ness of the question. One day I'll find a nice question for sure.

 

[Mister T]

So 0/0 = undefined? But I've read somewhere that resistance of an open switch is infinite

Sorry for the confusion. I thought you were talking about an ohmic resistor but obviously you weren't. You were talking about a switch. In this case you have a nonzero voltage across the switch, but you have zero current through the switch. And the switch effctively offers infinite resistance.

But a switch is a nonohmic device, and the relation R = V/I does not apply to a switch.

 

[jbriggs444]

So 0/0 = undefined? But I've read somewhere that the resistance of an open switch is infinite

In this case, with the switch open there would be 6 volts across the switch and no current flowing. That is not $\frac{0}{0}$. The resistance of the open switch would be $\frac{6}{0}$.

Whether you call that "undefined" or "infinite" depends in large part on whether you are a mathematician or an engineer.

 

[Darshit Sharma]

Sorry for the confusion. I thought you were talking about an ohmic resistor but obviously you weren't. You were talking about a switch. In this case you have a nonzero voltage across the switch, but you have zero current through the switch. And the switch effctively offers infinite resistance.

But a switch is a nonohmic device, and the relation R = V/I does not apply to a switch.

Okay, sir. But how do we reach the final 2.4 ohms or 0.6 ohm answer? I am still not able to judge even which one is correct 2.4 or 0.6 or maybe infinite resistance.

 

[Darshit Sharma]

In this case, with the switch open there would be 6 volts across the switch and no current flowing. That is not $\frac{0}{0}$. The resistance of the open switch would be $\frac{6}{0}$.

Whether you call that "undefined" or "infinite" depends in large part on whether you are a mathematician or an engineer.

Ya, sir...I just misinterpreted some threads....now all the working till we reach 6/0 is clear to me.

Just that "final answer" or maybe "answers" sort of thing is troubling my poor 15-year-old brain.

 

[Darshit Sharma]

My subjective and somewhat inconsistent take is that "the circuit" under consideration does not include the battery if it is not connected.

It's more about communication than about physics.

$\$

Okay, so after some re-reading-threads exercise...I feel I have understood.

However, I would like to ask if that's the common and more apt consideration of an "open circuit" because I saw another thread here in this same post which said 0.6 ohms, which amounts to the battery's internal resistance to be the total resistance in case of an open circuit, unlike your interpretation.

And why are we not adding the infinite resistance of the open switch even if it is in series?

My friends are saying 0.6 and 2.4, the answer key says 2.4 and you all are also saying 0.6 and 2.4, so which one should be correct and why? And if this is an incorrect-framing-of-the-question issue, please instruct me about the different resistances we can get if we consider different interpretations and different "points."

 

[BvU]

It's more about communication than about physics.

Just that "final answer" or maybe "answers" sort of thing is troubling my poor 15-year-old brain.

Mine is 70 and I still have trouble. Final answers hardly exist outside textbooks, even in the 'exact' sciences. Still very useful as models in philosophical context. black and white isn't Black and White

For someone living and breathing with an ohmmeter the 0.6 Ohm is preferably ignored. And an open switch is the same as switching off the meter to get a cup of coffee.

For someone in the context of being taught the first steps: bow to the inevitable (post #24) and move on !

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[Steve4Physics]

My friends are saying 0.6 and 2.4, the answer key says 2.4 and you all are also saying 0.6 and 2.4,

I'm not! See Post #9.

so which one should be correct and why? And if this is an incorrect-framing-of-the-question issue, please instruct me about the different resistances we can get if we consider different interpretations and different "points."

I can’’t help adding this.

In real life, an open switch (or ‘key’) is not ideal. It has a very large - but finite - resistance. Let’s say around $10^9 \Omega$.

And a closed switch is not ideal. It has a very small – but non zero – resistance. Let’s say around $10^{-3} \Omega$.

A = circuit with switch closed.
B = circuit with switch open.

Note that B is the same circuit as A – except that the $10^{-3} \Omega$ ‘resistor’ has been replaced by a $10^9 \Omega$ ‘resistor’.

We calculate the circuit resistances in exactly the same way for both A and B.
For A we get (near enough) $3.0 \Omega$.
For B we get (near enough) $10^9 \Omega$ (not $2.4 \Omega, 0.6 \Omega$ or $3.0 \Omega$).

The same argument applies with ideal components - the limiting case. So the circuit resistance of B is infinite.

If the question is asking about some other resistance, the question must define the required resistance clearly – it does not.

IMO.