What is more accurate definition of Ohm's law?

Question

What is more accurate definition of Ohm's law and resistance? is it

$$R=\frac {V}{I}$$

or

$$R=\frac{dV}{dI}$$

This is doubt that developed in my mind during a class where professor derived power equation where he used second one for resistance in the derivation. I checked Wikipedia. They showed the first relation as accurate. Of course if the first relation is correct and resistance \$R\$ is constant, then we can use second relation. But what if resistance is not constant?

For a practical problem, suppose my voltage source is current dependent and is given by

$$V=I^2+2I$$

Then how will you find resistance of given circuit at a given value of current \$I\$?

Answer 1

This answer is probably inherently displeasing to the feeling of natural order for some :-) :

A law of nature is simply a statement of observed results under defined conditions.

Ohms law is essentially a statement that the ratio of the two two variables V & I is typically observed to remain approximately constant as the variables vary.

It is arguably saying the opposite of what it may seem - ie not so much
"R is the ratio between ..." but
it is more "if the ratio between V & I is constant then we call this constant resistance" and "this approximates typical behaviour of a significant proportion of real world products".

At any given moment R IS the ratio between V and I. If this ratio has changed then R has changed. So V/I never changes for specific values of V & I with all other conditions held constant, whereas dV/dI typically does change in the real world.

So R = V/I is an accurate statement
R = dV/dI is usually an approximation and
where it all falls apart it just means that the observation does not apply under thos e conditions.

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That's woolier than I'd like but seems to convey what I'm trying to say. I hope :-).

Answer 2

If a circuit element is Ohmic, then the voltage across and current through are proportional

$$V \propto I $$

and the constant of proportionality is the resistance \$R\$

$$V = R \cdot I$$

This relationship, Ohm's Law, is obviously a linear relationship and thus the slope of the associated VI curve is just the constant of proportionality \$R\$

$$\frac{dV}{dI} = \frac{V}{I} = R $$

However, for a non-linear circuit element, e.g., \$V = R \cdot (I + \epsilon I^3)\$, the voltage across is not proportional to the current through and thus

$$\frac{V}{I} \ne \frac{dV}{dI}$$

Now we can define the terms static resistance and dynamic (or differential) resistance:

$$R_{static} = \frac{V}{I} $$

$$r_d = \frac{dV}{dI} $$

The static resistance is useful for DC analysis while the dynamic resistance is useful for small-signal analysis (where we linearize the circuit element about the DC operating point).

For more, see the Wikipedia article section static and differential resistance.

Answer 3

You're entering the field of superposition and small signal analysis with \$\frac{dV}{dI}\$. It simplifies a complex model in such a way that you can work reasonably accurate with it and with reasonably simple equations.

Answer 4

I think the important bit you are missing is that when we use the variable \$V\$ for voltage we are already talking about a voltage difference, not an absolute value. It is implied here that we are talking about the difference between the voltage at some point and the voltage at ground (0V by definition). So, Ohm's law is already in the form

$$ R = \frac{dV}{dI}$$

and the two forms are equivalent.

If you start talking about how real resistors or nonlinear elements behave then you need to stop talking about Ohm's law. Ohm's law applies to ideal resistances only.

Answer 5

Start with \$v=ir\$, and differentiate with respect to \$i\$.

$$ \frac{\mathrm{d}v}{\mathrm{d}i} = i \frac{\mathrm{d}r}{\mathrm{d}i} + r \frac{\mathrm{d}i}{\mathrm{d}i} = r + i \frac{\mathrm{d}r}{\mathrm{d}i} $$

If \$r\$ is constant with respect to \$i\$, you get your equation. If not, you need to include the extra term. Calc is your friend!