Microscopic Ohm's Law: Point-Dependent Resistance?

[RaduAndrei]

The microscopic Ohm's law is:
J = sigma * E

So the density in every point is proportional to the electric field in that point.

My question. This sigma is also point-dependent? So the resistor could be made of different materials and ohm's law would hold for every point differently.

Or is this law only for resistors made of a single material?

I am thinking is the first case.

 

[RaduAndrei]

And a second question.

If we apply a voltage across a uniform cross-sectional area wire, is the electric field also uniform?
And if the cross-sectional area is not uniform, then the electric field is not uniform.

Is that so? Is the uniformity of the electric field given by the uniformity of the area?

 

[ZapperZ]

First things first. Are you aware of the Drude model? Have you looked it up if you haven't?

Zz.

 

[RaduAndrei]

I've heard of it, but did not read it. I don't want to enter into much physics. I just want to understand something quickly for my electronics: deriving the resistance as a function of length, area, resistivity.

And then occupy my time with electronics and not physics.

But I can't find anywhere something acceptable. They all say "assuming the electric field is uniform". Why "assuming"?

And I don't see where the uniform cross sectional area condition comes into play.

 

[Shinji83]

You should study the classical theory of conduction and the Drude model first.
Trust me that you can't understand electronics if you don't get this elementary physics, things are going to get much much more complex once you study transistors.

Btw to sum it up.

For electrons J = -n*e*v_d
where n is the carrier density in the material (number of electrons/cm³)
e is the charge of the electron
v_d is the drift velocity, that is the mean velocity that charge carriers get in the material under a constant electric field.

v_d = - (e*τ/m_e)*E
τ is a characteristic time which represents the mean value between two successive collisions of the electrons with the lattice of the material.

So J=σE

with σ= n*e²*τ/m_e

This answers to your question, σ is a function of the space because it depends on n which is a function of x,y,z in the material.
Second answer, if you apply a constant voltage across a wire what stays constand is the current in every section.
That means that if the cross section varies, J must vary as well. If n is costant in the material hence σ is constant in the material, then the only possibility is to have a lower drift velocity where the cross section is larger, which means that ,yes, electric field is not constant over the length of the wire, it's lower where the resistance is lower, but its integral over the length is still equal to the voltage applied.

 

[RaduAndrei]

Thanks for answer.

So when deriving the resistance, they say:
v = integral of E*dl = E*L
i = integral of J*ds = J*S
Thus ro = E/J

So a full demonstration would look like this.

Consider a resistor made up of one single, ohmic material.

Thus:
E(x,y,z) = ro(x,y,z) * J(x,y,z) because the material is ohmic

and

ro(x,y,z) = ro = ct because we have one single material.

Thus: E(x,y,z) = ro * J(x,y,z)

Now we apply a potential difference across it that creates an electric field:
v = integral of E*dl

Assuming the electric field to be uniform and pointing along the length of the resistor (which imposes the condition that the cross-sectional area must be uniform), we have:
v = E*L

Because E and ro are uniform, then J is also uniform and pointing along the length of the resistor. Thus the current spreads out uniformly along the conductor:
i = integral of J*ds = J*S

Thus R = v/i = (E/J)*L/S = ro*L/S

Is this okay?

 

[Shinji83]

Thanks for answer.

So when deriving the resistance, they say:
v = integral of E*dl = E*L
i = integral of J*ds = J*S
Thus ro = E/J

So a full demonstration would look like this.

Consider a resistor made up of one single, ohmic material.

Thus:
E(x,y,z) = ro(x,y,z) * J(x,y,z) because the material is ohmic

and

ro(x,y,z) = ro = ct because we have one single material.

Thus: E(x,y,z) = ro * J(x,y,z)

Now we apply a potential difference across it that creates an electric field:
v = integral of E*dl

Assuming the electric field to be uniform and pointing along the length of the resistor (which imposes the condition that the cross-sectional area must be uniform), we have:
v = E*L

Because E and ro are uniform, then J is also uniform and pointing along the length of the resistor. Thus the current spreads out uniformly along the conductor:
i = integral of J*ds = J*S

Thus R = v/i = (E/J)*L/S = ro*L/S

Is this okay?

Yes.
But ro(x,y,z) or sigma(x,y,z) is not constant because you have a single material.
But because we assume the material is homogeneous hence has a perfect structure with no fluctuations of parameters in space.

 

[RaduAndrei]

Yes.
But ro(x,y,z) or sigma(x,y,z) is not constant because you have a single material.
But because we assume the material is homogeneous hence has a perfect structure with no fluctuations of parameters in space.

Aa, yes. You're right.