# Relationship between conductivity and resistivity

### What Is the Difference Between Resistivity & Conductivity? | Sciencing Electrical resistivity is a fundamental property of a material that quantifies how strongly that .. so the relation between resistivity and conductivity simplifies to. Heaney, Michael B. "Electrical Conductivity and Resistivity. . The quantitative relationship between the resistivity # and the simple microscopic. Resistivity and conductivity are both properties of conductors. What Is the Difference Between Resistivity & Conductivity Relationship.

And remember, Ohm's law says that the voltage across a resistor equals the current through that resistor times the resistance of that resistor. So, this pretty much gives you a way to define resistance. The resistance of the resistor is defined to be the amount of voltage applied across it divided by the amount of current through it.

And this is good, we like definitions, because we want to be sure that we know what we're talking about. That's the definition of resistance. Remember, it has units of ohms. But be careful, don't fall into the trap of thinking about this the way some people do. Some people think, "Oh okay, if I want a bigger resistance, "I'll just increase the voltage, 'cause that'll give me "a bigger number up top.

If you increase the voltage, you're gonna increase the current. And this ratio is gonna stay the same. The resistance is a constant. And this resistor, if you're not changing the material makeup or size or dimensions of this resistor, this number that is the resistance is a constant, if it's truly an Ohmic material. So, Ohmic materials maintain a constant resistance, regardless of what voltage or current you throw at them.

It'll just be constant. Yeah, if you throw too much current or voltage, the thing'll burn up. I don't suggest you do that. So, there's an operating range here, but if you're within that range, this resistance, this number, this number of ohms is a constant.

It stays the same no matter what voltage or current you put through it. So, we define it by talking about voltage and current, but it doesn't even really depend on that.

If you really want to change this ratio, this number that comes out here for the resistance, you need to change something about the resistor itself. Its size, what it's made out of, its length, its shape. So let's figure out how to do that if we take this resistor. Imagine taking this resistor, bringing it into the shop.

What is it gonna look like? Well, for simplicity's sake, let's say we just have a perfectly cylindrical resistor. So, this is the wire going into one end.

This is your resistor. It's a cylinder, let's say. And this is the wire going out of the other end, so this is the blown up version of this resistor. One thing you could depend on is the length. So, the length of this resistor could affect the resistance of this resistor.

Another thing it could depend on is the area of this front part here, this cross sectional area. It's called the cross sectional area, because that's the direction that the current's heading into. This current's heading into that area there, like a tunnel, and it comes out over here.

Now this is full, this isn't hollow. This is made up of some material. Maybe it's a metal or some sort of carbon compound or a semiconductor, but it's a solid material right here that the current flows into and then flows out of.

So, what would happen if we made this resistor longer? Let's say we start changing some of these variables, and we increase the length of this resistor. Well, now this current's gotta flow through a longer resistor. It's gotta flow through this resistor for more of this path. And it makes sense to me to think that the resistance is going to increase. If I increase the length of this resistor, then the resistance is gonna go up.

How about the area, this cross sectional area? Let's say I increase this area, I make it a wider, larger diameter cylinder. Well, it makes sense to me to think that now that current's got more room to flow through, essentially. There's a bigger area through which this current can flow. It's not as restricted. That means the resistance should go down.

And if we try to put this in a mathematical formula. What that means is, if I increase the length, R should depend on the length. It turns out it's directly proportional to the length. If I double the length of a resistor, I get twice the resistance. But area, if I increase the area, I should get less resistance. So, over here in this formula, my area has go to go on the bottom. The resistance of the resistor is inversely proportional to this cross sectional area.

But there's one more quantity that this resistance could depend on, and that's what the material is actually made of. So, the geometry determines the resistance as well as what the material is made of.

## Electric Resistance

Some materials just naturally offer more resistance than others. Metals offer very little resistance, and non-metals typically offer more resistance. So, we need a way to quantify the natural resistance a material offers, and that's called the resistivity. And it's represented with the greek letter rho. And the bigger the resistivity of a material, the more it naturally resists the flow of current through it. To give you an idea of the numbers here, the resistivity of copper. Well, that's a metal, it's going to be small.

We'll talk more about the units in a second. But the resistivity of something like rubber, an insulator, is huge. It can be on the order of 10 to the 13th. So, there's a huge range of possible values as you go from metal conductor to semiconductor to insulator, huge range of possible resistivities. And this is the last key here. This is the last element in this equation. The resistivity goes right here.

So, the bigger the resistivity, the bigger the resistance. And then it also depends on these geometrical factors of length and area. So, here's a formula to determine what factors actually change the resistance of a resistor. The resistivity, the length, and the area. So, what are the units of resistivity? Well, I can rearrange this formula, and I can get that the resistivity equals the resistance times the area of the resistor divided by the length. And so that gives me units of ohms times meters squared, 'cause that's area, divided by meters.

And so I end up getting ohms. One of these meters cancels out. Those are the units of these resistivities. But how do you remember this formula? It's kind of complicated. I mean, is area on top, is length on bottom? Hopefully you can remember why those factors affected it. But sometimes students have a hard time remembering this formula. One of my previous students from a few years ago figured out a way to remember it. He thought this looked like "Replay". So, this R is like R, and the equal sign kind of looks like an E. And the rho kinda looks like a P. In this case, the Fermi level falls within a band gap.

Since there are no available states near the Fermi level, and the electrons are not freely movable, the electronic conductivity is very low. Free electron model Like balls in a Newton's cradleelectrons in a metal quickly transfer energy from one terminal to another, despite their own negligible movement. A metal consists of a lattice of atomseach with an outer shell of electrons that freely dissociate from their parent atoms and travel through the lattice.

This is also known as a positive ionic lattice. When an electrical potential difference a voltage is applied across the metal, the resulting electric field causes electrons to drift towards the positive terminal. The actual drift velocity of electrons is typically small, on the order of magnitude of meters per hour. However, due to the sheer number of moving electrons, even a slow drift velocity results in a large current density.

Most metals have electrical resistance. In simpler models non quantum mechanical models this can be explained by replacing electrons and the crystal lattice by a wave-like structure. When the electron wave travels through the lattice, the waves interferewhich causes resistance. The more regular the lattice is, the less disturbance happens and thus the less resistance.

The amount of resistance is thus mainly caused by two factors. First, it is caused by the temperature and thus amount of vibration of the crystal lattice.

The temperature causes bigger vibrations, which act as irregularities in the lattice. Second, the purity of the metal is relevant as a mixture of different ions is also an irregularity.

Semiconductor and Insulator electricity In metals, the Fermi level lies in the conduction band see Band Theory, above giving rise to free conduction electrons. However, in semiconductors the position of the Fermi level is within the band gap, about halfway between the conduction band minimum the bottom of the first band of unfilled electron energy levels and the valence band maximum the top of the band below the conduction band, of filled electron energy levels.

That applies for intrinsic undoped semiconductors. This means that at absolute zero temperature, there would be no free conduction electrons, and the resistance is infinite. However, the resistance decreases as the charge carrier density i. In extrinsic doped semiconductors, dopant atoms increase the majority charge carrier concentration by donating electrons to the conduction band or producing holes in the valence band.

A "hole" is a position where an electron is missing; such holes can behave in a similar way to electrons.

### Electrical resistivity and conductivity - Wikipedia

For both types of donor or acceptor atoms, increasing dopant density reduces resistance. Hence, highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers dominates over the contribution from dopant atoms, and the resistance decreases exponentially with temperature. Conductivity electrolytic In electrolyteselectrical conduction happens not by band electrons or holes, but by full atomic species ions traveling, each carrying an electrical charge.