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So I know there are several versions of the equation for Power in circuits, based on how you interconvert Ohm's law V=IR. I got the reference here: However I'm a bit confused about these 2 versions. In one, power is directly proportional to resistance, and in the other, it's inversely proportional. How is this possible? P=I^2*R vs. P=V^2/R

Most importantly, what kind of clue can I use when reading a question on the MCAT to know when to use which equation? For example, in Question 4 of this Khan Academy passage, is there a specific reason why they use P=V^2/R?

Here is a screen shot of it. Thank you so much!
 

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May 30, 2020
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Basically
P=IV
V=IR, I=V/R
so therefore substituting V/R into I in P=IV we get P=V^2/R
The Khan Academy passage uses P=V^2/R to relate power to resistance. Furthermore looking at the equation we know R=pL/A (resistance increases as area(or diameter) decreases and resistance increases as length increases (as seen in a long wire). Thus plugging R=pL/A into P=V^2/R we can determine that resistivity will increases as area (or diameter of axon) decreases and as resistivity increases the power required to maintain a constant voltage would increase.
I would suggest thinking of electricity as water motion and flow, it really helps.
 
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Your question about why one form of the equation has R in the numerator while the other form has R in the denominator is interesting—if I'm understanding right, it sounds like you're trying to go beyond the algebra to understand why that's the case. So, in general, it's a good idea to develop an intuitive understanding of what equations mean as you study them, rather than just treating them as abstract things to memorize or as mathematical tricks, but that's easier to do with some equations than others. This is partly because you have to take the equation as a whole: if you want to understand intuitively why R is in the denominator in P=V^2/R, you would also have to have an intuitive sense of what it means to say that P is proportional to the square of voltage...and at least for me, that's not easy to do, so I prefer to start with P=VI (which makes more intuitive sense) and then just do the algebra for the alternative forms.

Here's a similar example: work (in joules) can be defined as having units of N·m, corresponding to the familiar concept of work = force times distance. However, in P/V curves, work (in joules) is also defined as Pa/m^3 (pressure divided by volume). In a similar spirit to your question, one might ask: how can work have a direct relationship with meters, but also an inverse relationship with meters cubed? The answer relates to how pressure is defined (as N/m^2). So the units for pressure divided by volume can be broken down to (N/m^2)/m^3, which simplifies to N·m. So, work is still the same underlying idea, but we're just expressing it differently based on the units we're using. It's a similar deal with the different ways of writing P=VI.

Re: the passage question, in a nutshell, the reason why you rewrite it that way is that of the three quantities we can mess around with (V, I, and R), R is the only one that is related to radius (through resistivity). In other words, you pick the version of the power equation that answers the question you're interested in (or aligns with the answer choices you have to choose among).

Hope this helps!!
 
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Your question about why one form of the equation has R in the numerator while the other form has R in the denominator is interesting—if I'm understanding right, it sounds like you're trying to go beyond the algebra to understand why that's the case. So, in general, it's a good idea to develop an intuitive understanding of what equations mean as you study them, rather than just treating them as abstract things to memorize or as mathematical tricks, but that's easier to do with some equations than others. This is partly because you have to take the equation as a whole: if you want to understand intuitively why R is in the denominator in P=V^2/R, you would also have to have an intuitive sense of what it means to say that P is proportional to the square of voltage...and at least for me, that's not easy to do, so I prefer to start with P=VI (which makes more intuitive sense) and then just do the algebra for the alternative forms.

Here's a similar example: work (in joules) can be defined as having units of N·m, corresponding to the familiar concept of work = force times distance. However, in P/V curves, work (in joules) is also defined as Pa/m^3 (pressure divided by volume). In a similar spirit to your question, one might ask: how can work have a direct relationship with meters, but also an inverse relationship with meters cubed? The answer relates to how pressure is defined (as N/m^2). So the units for pressure divided by volume can be broken down to (N/m^2)/m^3, which simplifies to N·m. So, work is still the same underlying idea, but we're just expressing it differently based on the units we're using. It's a similar deal with the different ways of writing P=VI.

Re: the passage question, in a nutshell, the reason why you rewrite it that way is that of the three quantities we can mess around with (V, I, and R), R is the only one that is related to radius (through resistivity). In other words, you pick the version of the power equation that answers the question you're interested in (or aligns with the answer choices you have to choose among).

Hope this helps!!
Thank you so much for your detailed answer. I think you hit the point where I was confused about. It makes more sense now! :)
 

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