Today I studied about Ohm's law which states-

The current through a conductor between two points is directly proportional to the voltage across the two points.By introducing constant of proportionality $\text{R}$,which is resistance we get $\text{V=IR}$.

Now,my question is why did we introduce $\text{R}$=resistance?We could have introduced any constant $\text{k}$.Shortly speaking how did we know that the constant of proportionality is the resistance?

How do we know that it is this constant of proportionality that is responsible for opposing the motion of electric current through a circuit?

I know somewhat an experimental proof of this.But is there any rigorous theoretically proof that explains this?

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    $\begingroup$ The proportionality constant given by $\frac{V}{I}$ is defined as the resistance. You can't ask why a definition is true. $\endgroup$ – Prahar Apr 27 '16 at 13:54
  • $\begingroup$ Since $R=V/I$ is a definition, are you asking about a proof that $I$ is actually proportional to $V$? $\endgroup$ – YakovL Apr 27 '16 at 14:02

Georg Ohm's original experiments, 1825, established that for a set temperature, the current through a specific length of a conductor was proportional to the potential difference applied.

Ohm's law is empirical; it cannot be derived directly from Maxwell's equations as it depends upon material properties. It is violated by many materials, and even then depends upon pressure and temperature. The microscopic version can be derived from Drude's quasi-quantum model; this model has a variety of defects which are often studied in a Condensed Matter theory course.

There are several ways to describe this; the conductor has conductivity, $\sigma $, or its reciprocal, resistivity, $\rho =1/\sigma$. Resistance is then the property of a particular piece of that conductor, say $R=\rho × Length/Area$, where we have a uniform cross sectional area for the specified length; e.g., the resistance of a piece of copper wire.

Resistivity and conductivity are tabulated; you can then make components of known resistance.

So the resistance measured in the example given is the particular value that applies to that example.

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  • $\begingroup$ Then why do we believe in Ohm's Law if it does not have a mathematical proof...an experiment cannot be a foolproof used to say that for all generalized cases the law will be true..... $\endgroup$ – tatan Apr 27 '16 at 13:53
  • $\begingroup$ I wrote a paper on this -- follow the link in my profile. The physics that is responsible for these properties is studied in condensed matter theory, classical & quantum, following a course in statistical mechanics. The simplified version is the Drude model, which is quasi-quantum mechanical. $\endgroup$ – Peter Diehr Apr 27 '16 at 13:58
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    $\begingroup$ There's no mathematical proof for physics. We just create equations which are consistent with the world around us. Sometimes these equations can be applied to other situations and create predictions, but there's no formal proof. We can derive Ohm's law from more basic assumptions. A more general version of Ohm's law is $\vec{j} = \sigma \vec{E}$, but we still have just made that equation from assumptions about the universe. $\endgroup$ – Tweej Apr 27 '16 at 14:01

I'm not sure exactly what you're asking for. But let's say there's an electric current flowing through a straight wire segment of length $l$, then the change in $\Delta\phi$, or $V$, would be defined by $$\Delta\phi = \int \mathbf{E}\cdot d\mathbf{l}$$

Because it is a straight wire,

$$\Delta\phi = E\cdot l$$

But we have a definition of current

$$I = \int_A \mathbf{J}\cdot\mathbf{n}da = JA$$

And we have the electric current density $\mathbf{J} = g\mathbf{E}$

If we combine the equations we get

$$I = \frac{gA}{l}\Delta\phi$$ Where the reciprocal of $\frac{gA}{l}$ we define as "Resistance" and denote $R$.

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The definition of resistance of a component is

$\text{resistance of component (R)} = \dfrac{\text{potential difference across component (V)}}{\text{current passing through component (I)}}$

This is not Ohm's law it is the definition of resistance.

It so happens that for some components it is found that $V \propto I$ which is called Ohm's law.
This is an experimental law which also can be derived from theory after making some assumptions about the microscopic nature of matter.

If a component obeys Ohm's law then the resistance of that component is constant and resistance is found to be a useful parameter to know.

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  • $\begingroup$ But see definition of transconductance and transresistance. $\endgroup$ – Peter Diehr Apr 27 '16 at 15:43
  • $\begingroup$ @PeterDiehr Those are useful parameters when the resistance is not constant and enable one to approximate a non-linear current-voltage characteristic into a linear one over a small range of currents and voltages. $\endgroup$ – Farcher Apr 27 '16 at 20:22

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