Connection between V vs I graphs and Ohm's Law

[LonelyElectron]

Hi all! Wanted to know if this is along the right lines, or if I'm missing anything... Any help is greatly appreciated :)

1. Homework Statement
Explain the connection between a graph of potential difference versus current, and Ohm’s law.

Homework Equations

slope = rise/run
V=IR
R=V/I

The Attempt at a Solution

Ohm’s law states that potential difference is equal to current multiplied by resistance. If this formula were to be rearranged, it could also state that resistance is equal to potential difference over current; a simple manipulation of the formula. Further, potential difference and current are directly related to each other – meaning if one were to increase so would the other, and if one were to decrease the other would also decrease. Additionally, when graphed, their relationship is linear. A potential difference vs current graph will show this, but based on Ohm’s law, it will automatically showcase the resistance as well. Since the current in a circuit is dependent upon the potential difference, when the slope of the line is calculated, the rise is actually the voltage, and the run is really the current. This makes the slope equal to potential difference over current; the exact same variation of Ohm’s law stated earlier. Therefore, a potential difference versus current graph and Ohm’s law are connected both mathematically and graphically, through formulas and their manipulation.

 

[haruspex]

their relationship is linear

Not always. In particular, as a wire gets hotter from the dissipated energy its resistance increases.
This means the local slope is not necessarily the same as V/I. Which of the two represents resistance?

 

[LonelyElectron]

Not always. In particular, as a wire gets hotter from the dissipated energy its resistance increases.
This means the local slope is not necessarily the same as V/I. Which of the two represents resistance?

Not entirely sure which two you are referring to. Resistance is measured in Ohms and represented by the constant R. The formula for calculating resistance is Ohm's law; V=IR, right?

 

[haruspex]

Not entirely sure which two you are referring to. Resistance is measured in Ohms and represented by the constant R. The formula for calculating resistance is Ohm's law; V=IR, right?

Suppose you plot a graph of y=V against x=I from some experiment, and it turns out not to be a straight line.
At some point, (x, y)= (I, V), is the resistance y/x or is it the slope of the graph at that point?
Are you saying it is y/x?

 

[LonelyElectron]

Suppose you plot a graph of y=V against x=I from some experiment, and it turns out not to be a straight line.
At some point, (x, y)= (I, V), is the resistance y/x or is it the slope of the graph at that point?

I am assuming (similarly to the acceleration in those distance time graphs I think it was) that if the resistance were to vary somewhere along the wires, resulting in a non-linear graph, that the resistance would be the slope between any two points, identifiable using tangents. Right? That means that the slope is still the resistance, and will always be the resistance, meaning (x,y)=(i,v) will continue to be the slope, but some variation has been known to occur... Completely guessing here as all the textbook said was that the relationship is linear, and the slope is the resistance.

 

[LonelyElectron]

WAIT WAIT WAIT. It can be done either way. Ohm's law just states that V=IR, the V and the I can be a slope or a point. Both are possible!

 

[haruspex]

the textbook said was that the relationship is linear, and the slope is the resistance

Ok, if the textbook says it is linear then my answer may be beyond the required level, but...

WAIT WAIT WAIT. It can be done either way. Ohm's law just states that V=IR, the V and the I can be a slope or a point. Both are possible!

I have seen questions on this forum where it is nonlinear, and for that you need to know what the real definition is: it is V/I, so not necessarily equal to the slope.
You might sometimes see reference to, maybe, "marginal resistance"; that would mean the slope, so dV/dI, not V/I.

 

[LonelyElectron]

Ok, if the textbook says it is linear then my answer may be beyond the required level, but...

I have seen questions on this forum where it is nonlinear, and for that you need to know what the real definition is: it is V/I, so not necessarily equal to the slope.
You might sometimes see reference to, maybe, "marginal resistance"; that would mean the slope, so dV/dI, not V/I.

Okay! So my answer is good for this year, and I'll be better prepared for next year! Thanks so much!