What is meant by Electrical resistance and Electrical resistivity? Some different thoughts needed

Question

The Electrical resistance is defined as the oppose of flow of charge by a material(wire for example). Also $R = \rho l/A$

What i think is that "charge" here actually means the magnitude of charge only. It doesn't mean charge particles. And the reason to say that is, when we increase the area of cross section of wire carrying current the amount of charge flowing is affected while the flow of charge particles remains uneffected. Double amount of charge starts flowing when area of cross-section is doubled because the number of electrons flowing gets doubled while the drift velocity remains same as before.

Means electrical resistance opposes the amount of charge passing through a cross section of wire.

Electrical Resistivity is also sometimes given the same definition, the oppose of flow of charge by a material. Here the "charge" should mean the charge particles.

If you replace a copper wire with tungsten of identical dimension, the electron flow reduces due to the reduction in speed of electrons or the electrons drift velocity gets reduced. If you change the cross section of wire, the amount of charge flowing changes even the material and voltage provided remains same.

It means that Electrical resistivity is the property of material(let say wire) to resist flow of charge particles (not magnitude of charge).

In short: The Resitance 'always' opposes the current whereas the Resitivity 'always' oppose the velocity of electrons contributing to current in wire. "Resistance doesn't always oppose the velocity of electrons"

I want to know your thoughts on how correct or wrong it is.

Answer 1

No, you have got it wrong. Firstly, you seem to be trying to say that you can have more or less charge without having more or fewer charged particles. The current flowing through a wire is electrons- the moving charge is the movement of electrons. You cannot have charge without charged particles, as they are the basic building blocks of it.

The difference between resistance and resistivity is like the difference between density and moment of inertia. All gold has the same density- it is a property of the material; but different amounts of gold arranged in different ways will have different moments of inertia. Likewise, resistivity is a fundamental property of the material, so all copper wires will have the same resistivity; but different amounts of copper arranged in different ways can have different values of resistance. An amount of copper made into a short thick wire will have lower resistance than the same amount of copper formed into a very long thin wire.

Note that you can increase a current in two ways- one is to increase the speed at which the electrons are moving and the other is to have more electrons moving in parallel.

Answer 2

In this answer, the charge of electron is assumed to be $e$ $(e\gt0)$ for simplicity. The result is effectively the same.

$m$ is the mass of an electron.

In any (linear or ohmic) resisting conductor, resistance or resistivity arises due to the collisions of moving electrons with the nuclei.

It is assumed that the electrons come momentarily to rest after a collision with a nucleus and the average time between two collisions(average collision time) is $\tau$($\tau$ is a characteristic property of a material).

Hence the average acceleration on a moving electron is $-\frac {m\vec v}{\tau}$.

The equation of motion of an electron in such a wire under an electric field $\vec E$ along the length of the wire is

$m\frac{d\vec v}{dt}=e\vec E - m\frac {\vec v}{\tau}$

After a long time i.e., after a steady state is reached, $\vec v = \frac{e\tau\vec E}m$

The current density $\vec j = ne\vec v$ (the amount charge flowing though a unit cross-section in a unit time i.e., no. of electrons crossing in a unit time, times the charge of an electron)(where n is the electron density or no. of electrons present in a unit volume)

$\vec j=\frac{ne^2\tau}m \vec E$

Here $\frac{ne^2\tau}m$ is defined as a constant conductivity(or specific conductance) which represents the ability of a conductor to conduct, denoted by $\sigma$. It is independent of the dimensions of the conductor(as $\tau$ is unique for a material). Resistivity($\rho$) is defined as the reciprocal of conductivity and the ability of a $\mathbf {material}$(note that it is material and not wire) to resist the flow of charge.

Taking a linear conductor $|\vec E|=\frac V\ell$ (where $V$ is the voltage across the ends of the wire and $\ell$ is the length of the wire) and $|\vec j|=\frac iA$ (where $i$ is the amount of current passing through the wire and $A$ is the area of cross-section of the wire).

$i = \frac{ne^2\tau}m \frac A\ell V$

$i = \frac {\sigma A}\ell V$

$V = \frac {\rho\ell}A i$

This constant was replaced resistance($R$) known as the ability of a wire to resist the flow of charges.

If we change the cross section of a wire, the drift velocity is almost same and the current density is same, but the current changes due to the change in cross-section.

Resistivity is the characteristic property of a material and Resistance is the effective property of a wire. Both seem like resisting the flow of charges, but they are just constants that tell us about how materials behave.

Answer 3

Given a material of uniform cross section $A$ and length $L$ its resistance is $R=\frac{\rho L}{A}$ where $\rho$ is the resistivity of the material.

If $A$ and $L$ are constant then $R \propto \rho$, so any mention of difference in microscopic properties, ie the mechanism of the "opposition" to the flow of charges, is misguided.

Resistivity is an intrinsic property of the material, ie resistivity is independent of how much of a material is present and is independent of the form of the material and resistance is an extrinsic property of an object as resistance depends of the size and shape of an object.