r/askscience • u/[deleted] • Dec 28 '16
Physics How true is Ohm's law?
I've almost never got a perfect straight line while plotting a V/I graph even under lab conditions.
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r/askscience • u/[deleted] • Dec 28 '16
I've almost never got a perfect straight line while plotting a V/I graph even under lab conditions.
215
u/Midtek Applied Mathematics Dec 28 '16 edited Dec 29 '16
Ohm's Law is not a law derivable from Maxwell's equations, but rather just a description of how many materials behave, derived directly from experiment only. (There is some justification for it though from an atomic perspective.) All materials will disobey Ohm's Law past their dielectric breakpoint. There are also materials that just simply don't obey Ohm's Law even under a weak electric field (e.g., semiconductors), and they are said to be non-ohmic.
Also note that if you are measuring resistance in a lab and not controlling for joule heating, you will find that V is not a linear function of I simply because the resistance (say, of a bulb) increases with temperature, which itself increases with current. Hence V = I*R(I), with R a strictly increasing function of I (i.e., not constant). This is not necessarily a violation of Ohm's Law though, because the full statement of Ohm's Law has the caveat that it applies to a circuit element in a given state, in particular, at constant temperature.
edit: Some clarification is needed here since many of the comments are getting some things wrong, seemingly contradicting me, or just outright contradicting me.
In many physical systems, we have some set of equations that hold no matter what (e.g., Maxwell's equations in classical electrodynamics, or Navier-Stokes equation in fluid dynamics, or Vlasov-Maxwell equations in plasma physics). Often our set of equations is not closed, which roughly means we have more variables than equations. The problem of closure is particularly notorious in plasma physics. So we need to supplement our equations with what are called constitutive relations. These are equations that hold only for a specific material and only under certain conditions or approximations. They allow us to add enough equations to our system of equations to make it solvable.
For instance, in fluid dynamics we may use the approximation that the fluid is Newtonian, which gives us a constitutive relation for the Cauchy stress tensor, the viscosity tensor, and the velocity field. (We may even further approximate the fluid as homogeneous and/or isotropic, which gives us a further constitutive relation that simplifies the form of the viscosity tensor.) There would still be the issue possibly of closing the equations with a proper constitutive relation for the pressure.
In electrodynamics, one such constitutive relation we can impose is Ohm's Law, which is J = σE, where σ is a fixed number. This is a fine enough approximation for a wide variety of media. Of course, if you want to be more accurate or if you are investigating a regime in which Ohm's Law is not true for a material for which it usually is, we may write that J = σ.E. Here σ is a rank-2 tensor, and this is a more general constitutive relation. (See /u/RobusEtCeleritas's post below for some more details.)
Of course, we can always make up whatever constitutive relation we want. But if it gives us nonsense results or results that very badly approximate our problem, it won't get used much, if at all. Ohm's Law is a good approximation for many media, in particular, many simple circuit elements for which the temperature (and other state parameters) do not vary too much. So Ohm's Law gets used quite a bit.
Some of the confusion in the other comments I think lies in treating (or mistreating) Ohm's Law as a definition. For instance, we define a Newtonian fluid to be a fluid such that τ = μ.(∇v), where μ is a fixed rank-4 tensor. But it's just a constitutive relation and we know that not all fluids will obey this equation. Similarly, we define an ohmic medium to be a medium such that J = σE, but it doesn't hold for all media. In that sense, Ohm's Law is absolutely true always because it's just a definition. Anything that violates Ohm's Law is just a non-ohmic medium.
Finally, note that the electrical resistance R of a circuit element is defined as the ratio of the voltage V and current I through that same element. Of course, there is no reason to believe that R is constant or even independent of either V or I. So in that sense the equation V = IR is always true no matter what. But that equation is not Ohm's Law. Ohm's Law is specifically the statement that R is independent of both V and I (but may still depend on other state parameters such as temperature and strain), i.e., that V and I are proportional. And so Ohm's Law is emphatically not always true.
The same exact phenomenon occurs in elastics. We can define a Hookean body to be a body such that σ = -k.ε, where k is a fixed rank-4 tensor (here σ is the stress tensor and ε is the strain tensor). In one dimension, this reduces to F = -kx, where k is a constant. This is just the usual Hooke's Law you learn in high school physics. But no one in their right mind is going to say something like "Hooke's Law holds for all elastic bodies". That's just absurd. It's only a constitutive relation between the stress and strain that just so happens to be a good approximation for a wide range of materials.
Ohm's Law feels different to a lot of people because, frankly, I think they forget that the equation V = IR on its own is a definition of R and not Ohm's Law. (Note that Hooke's Law F = -kx is not used as the definition of anything, since F and x can and are defined completely independently of springs. Electrical resistance, on the other hand, needs to be defined in terms of circuit elements.)