A brain teaser

[RoveyDoveyGrovey]

I was just wondering something about a simple circuit, that is when you short circuit a battery. You can show that the battery or short circuiting wire will fail by using the equation for power.

p = v^2 / R

So for an AA battery,

p = 1.5^2 / 0

So

p = infinite watts and therefore lots of heat and probably an exploding battery.

but what if you use this form of the power equation,

p = i^2 x r

i = v/r

so,

i = 1.5 / 0

so,
i = infinity

so,

p = infinity x 0

What is infinity x 0??? who wins?? Im guessing infinity because the battery will explode and it won't explode if there is 0 watts but how do you prove this? I was always told that anything multiplied by zero = zero and anything multiplied by infinity = infinity but I was never told what infinity x zero = equals.

 

[alec_t]

p = infinity x 0

No. p= infinity * 1.5
(Actually the current will never reach infinity, because both the shorting wire and the internal structure of the battery have finite resistance).

 

[RoveyDoveyGrovey]

Yep i'm aware of the real world situation, but erm why p = inf x 1.5?? the resistance is zero not 1.5

 

[alec_t]

Did you notice the smiley?
P = i * v

 

[RoveyDoveyGrovey]

Aha I did not, so you are avoiding the question then???

I'm just wondering how is this reconciled, I know you can just use one of the other forms of the Power equation, but this doesn't explain why you end up with infinity x zero.

 

[BrownOut]

Nobody wins. Power is zero in both equations. Remember, if R=0, then V must be equal to zero.

 

[RoveyDoveyGrovey]

AHA, thank you BrownOut, of course you are right. I guess using ideal situations doesn't always work.

 

[BrownOut]

On second thought, I have to think about this a little more. In the equation P=I^2*R, there is infinite current, though V=0. That implys zero power, but wait! The right hand side is: ∞*0, which is undefined. But think about how power tends as the resistance tends towards zero. Just off the top of my head, I would think power would be going up. Maybe someone can verify that. Also, l'hopital's rule might apply and give us some insight.

 

[RoveyDoveyGrovey]

Haha, I keep going round in circles thinking about this.

 

[unclejed613]

no matter which way you look at it, if the resistance is zero, voltage is zero, and current is infinite, and power is zero.
e=i*r=0 since r=0
p=i^2*r=0 since r=0
p=e^2/r=0 since both e and r are zero (technically this could be called undefined because of division by zero, but any division where the numerator is zero results in zero)

yes it is circular reasoning in some respects, but to have power, the values of resistance, voltage, AND current must all be non-zero (even negative values can give useful results).

 

[RoveyDoveyGrovey]

so what you're saying is that if I could short circuit my perfect battery of zero internal resistance with a wire of zero resistance, it wouldn't actually blow up because it wouldn't be dissipating any power and therefore not generating any heat?

but there would be infinite current, but the current wouldn't actually have any voltage to drive through the circuit, so whether the current is infinite or zero or somewhere in between is irrelevant because it has no effect in this case.

 

[crutschow]

Superconducting wires can carry large currents (not infinite of course) while dissipating zero energy. Thus the current flow in an inductor wound with superconducting wire will maintain a current in the windings for an infinite period of time (as long as the superconducting wire is kept cold enough to remain superconducting).

 

[carbonzit]

How about approaching this problem in a different way?

Instead of looking at the isolated case where resistance is zero (at least nominally, if not in the real world), try this: graph power in the circuit as the load resistance goes from infinite (open circuit) to zero (short circuit). Even if the result is undefined at the end of the curve (R=0), you'll be able to see where the power curve is headed (my guess is it will asymptotically approach ∞). That would tell you something, and might even answer the original question (who loses, the battery or the wire?).

 

[alec_t]

You're right CZ. Power approaches infinity asymptotically. Think of it as the square of the current increasing faster than the reduction of resistance. BTW, the voltage across the resistance remains at 1.5 since the battery is ideal.
If anyone's in doubt, it's informative to run a simulation.

 

[BrownOut]

p=i^2*r=0 since r=0

You can't make that claim, becuase the result is: p=∞*0. That result isn't enough information to claim the power is zero, because anything times 0 is 0, anything except infinity that is.

 

[carbonzit]

We can reduce the ridiculousness of this discussion somewhat by recognizing that p=∞ is a singularity which cannot exist in reality. We can discuss things in the neighborhood of this point, which should still shed some light on the problem, without devolving into arguments over how many infinities (or zeroes) can dance on the head of a pin.

 

[RoveyDoveyGrovey]

Haha, I guess the essence of my question is, what is infinity x zero. It is indeed purely a hypothetical question because you can't have zero resistance or infinite current, but so much of the theory taught in electronics revolves around ideal components and situations, well the simple theory that i know of anyway. I would have thought there might be a simple answer but there never seems to be when infinity is involved.

Here's a new one for you, if you eat your own foot do you lose weight?

 

[carbonzit]

Here's a new one for you, if you eat your own foot do you lose weight?

That's easy: of course you do. Your body does not retain 100% of what it consumes.

Think of a better analogy.

 

[BrownOut]

Hi Rovery... the question of what is ∞*0 cannot be answered in this kind of vacuum. There need to be information about the particular equation. The answer will be unique for each equation, well not unique, becuase it will be either ∞ or 0. In the case if p=I^2*R, as someone has correctly ( I think ) pointed out, this will result in ∞. I was wrong to say it was zero in my first post.

PS I tried to prove this result with SPICE, but wasn't successful.

It is indeed purely a hypothetical question because you can't have zero resistance or infinite current, but so much of the theory taught in electronics revolves around ideal components and situations, well the simple theory that i know of anyway.

Hypothetical for sure. But I wouldn't say unimportand.

 

[RoveyDoveyGrovey]

it wasn't supposed to be an analogy, but the answer is of course no you don't lose weight, not until you go to the toilet anyway.

But back to the topic, is zero the ultimate mathematical earth that can even sink infinity, or does infinity win through?