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Why do some conductors follow Ohm's law and some do not? Isn't there any universal law that can explain the flow of current?

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If you could be more specific, you would be more likely to get a helpful answer. The reason that a battery doesn't follow Ohm's law is very different than the reason a diode doesn't follow Ohm's law, which in turn is very different than the reason that copper under an extraordinarily high electric field doesn't follow ohm's law, etc. etc. –  Steve B Feb 22 '13 at 15:09
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Physics does not contain any truly 'universal' laws yet. Ohm's law like any other within physics relies on context for validity. $V=IR$ is implicitly only valid in simple circuit theory, in field theory it has the form $J=\sigma E$ as seen in the wiki article.

Variations from Ohm's law are due to electrical properties of the conductor/medium under inspection. The 'law' or equation to be used will be determined by pragmatic concerns of usefulness.

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Downvoted ... First, under almost all circumstances, $J = \sigma E$ if and only if $V=IR$. So neither is more or less "valid". Second, there are plenty of universal laws in physics. Third, "due to electrical properties" is not an explanation of anything. –  Steve B Feb 22 '13 at 15:15
    
to respond, J=sigmaE is clearly more valid as it applies to both field and circuit theory, while V=IR only applies to discrete components in circuit theory. It is a simplification for electrical engineering and requires context for understanding. No there are no universal laws, the Theory of Everything and Grand Unified Theories are still pending. 'Electrical properties' is a short hand for those physical interactions mediated by/explained through electrodynamics. –  Nic Feb 27 '13 at 12:11
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Ohm's law is an assumption that the value of resistance for a resistor is independent of the magnitude and polarity of an applied potential difference. Many conductors, like a diode, depend on the polarity of the applied potential difference-- e.g. you must apply a positive or negative voltage to get a current to move across the conductor.

An ohmic resistor on the other hand, will have a current that is linearly dependent on potential-- so as a larger positive voltage is applied to the resistor, there will be a correspondingly larger positive current through the resistor. Since R = V/I in a circuit, if both V and I are scaling up or down at a constant, linearly proportional rate, R will be constant since the factors that scale the numerator and denominator cancel each other out... and this is what it is meant by a resistor to be ohmic.

Most conductors change their resistance as their temperature change, and like a diodes, some conductors have completely non-linear relationships between current and voltage. Because Ohm's law isn't really a law, a more consistent way to look at a material is to consider it's "instantaneous" resistance, or it's resistivity (a property of a material as opposed to a geometry of a material):

$\rho$=$\frac{E}{J}$

$\rho$ is defined as the resistivity -- this is given by the electric field, $E$, divided by the current density $J$, (the current per cross-sectional area). So intuitively, a material with a high resistivity has a low current density motivated by a large electric field (remember, e-fields are proportional to the force on a particle), and a material with a low resistivity has a large current density motivated by a low electric field...

In any case, the salient thing to take away from this is that Ohm's law is only a "law" because of historical reasons-- in reality Ohm's law describes a special subgroup of conductors that have a linear relationship between current and voltage, but there are many conductors which don't fit this constraint.

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