## 30-second summary

## Resistivity

**Resistivity** is a property of materials that describes their ability to resist the flow of electric current. It is defined as the resistance of a unit length of a material with a unit cross-sectional area. Resistivity is usually denoted by the Greek letter rho (ρ) and has units of **ohm-meters** (Ω·m).

Resistivity and resistance are related but distinct concepts in electrical circuits.

Resistance is a measure of how difficult it is for electrical current to flow through a material, and it is measured in ohms (Ω). The resistance of a material depends on its geometry (length, cross-sectional area, etc.) and its resistivity (ρ), which is a fundamental property of the material.

Materials can be classified into different categories based on their electrical resistivity. Here are some common categories:

## Resistivity

**Resistivity** is a property of materials that describes their ability to resist the flow of electric current. It is defined as the resistance of a unit length of a material with a unit cross-sectional area. Resistivity is usually denoted by the Greek letter rho (ρ) and has units of **ohm-meters** (Ω·m).

Resistivity is an intrinsic property of a material and depends on factors such as its chemical composition, temperature, and crystal structure. Materials with high resistivity are poor conductors of electricity, while materials with low resistivity are good conductors.

The resistivity of a material can be calculated using the following formula:

ρ = RA/L

where ρ is the resistivity, R is the resistance of a sample of the material, A is the cross-sectional area of the sample, and L is the length of the sample.

The resistivity of a material can also be measured experimentally using techniques such as four-point probe measurements, which involve passing a known current through a sample of the material and measuring the voltage drop across it.

Resistivity is an important property of materials used in electrical engineering, as it determines their suitability for use in various applications. For example, materials with low resistivity, such as copper and aluminum, are used for wiring and electrical transmission lines, while materials with high resistivity, such as nichrome, are used for heating elements in appliances.

## Resistivity and Resistance

Resistivity and resistance are related but distinct concepts in electrical circuits.

Resistance is a measure of how difficult it is for electrical current to flow through a material, and it is measured in ohms (Ω). The resistance of a material depends on its geometry (length, cross-sectional area, etc.) and its resistivity (ρ), which is a fundamental property of the material.

Resistivity (ρ) is the intrinsic property of a material that describes how much resistance it offers to the flow of electrical current, and it is measured in ohm-meters (Ω·m). Resistivity is a measure of the material’s ability to conduct electricity and is dependent on factors such as temperature, composition, impurities, and pressure.

The relationship between resistance (R), resistivity (ρ), and geometry (l, A) of a conductor is given by the following equation:

R = ρ (l/A)

where l is the length of the conductor and A is its cross-sectional area. This equation shows that the resistance of a conductor increases with length and decreases with increasing cross-sectional area, while the resistivity of the material remains constant.

In summary, resistance is a measure of how much a material resists electrical current, while resistivity is an intrinsic property of a material that describes its ability to conduct electricity.

## Resistivity and Conductivity

**Electrical conductivity** is closely related to resistivity (which is more commonly used):

σ=1/ρ

where σ is the conductivity (in m/Ohm), and ρ is the resistivity (in Ohm/m). To determine the resistance of a wire (which could be made of almost anything: copper, aluminum), use:

R=ρAl=Aσl

where A is the cross-sectional area of the wire (in m^{2}) and l is its length (in meters).

**Classification of Materials according to Electrical Resistivity**

Materials can be classified into different categories based on their electrical resistivity. Here are some common categories:

- Conductors: Materials with low electrical resistivity, such as metals and some types of solutions, are known as conductors. They are able to carry an electric current with minimal resistance and are commonly used in electrical and electronic applications.
- Insulators: Materials with high electrical resistivity, such as plastics, rubber, and glass, are known as insulators. They are not able to carry an electric current easily and are commonly used to isolate and protect electrical components.
- Semiconductors: Materials that have intermediate levels of electrical resistivity, such as silicon and germanium, are known as semiconductors. They can be used to control and manipulate the flow of electric charge and are used extensively in electronics and computer applications.
- Superconductors: Materials that have zero electrical resistance at very low temperatures are known as superconductors. They are able to carry electric current without any loss of energy and are used in specialized applications such as MRI machines and particle accelerators.

Generally, most metals have high conductivity (which is another way of saying metals tend to be conductors) because the electrons in their outermost shell can move easily. Non-metals tend to have low conductivity.

## Resistivity of various materials

Here are 10 examples of materials with their electrical resistivities:

- Copper – Electrical resistivity: 1.68 × 10
^{-8}Ω·m - Aluminum – Electrical resistivity: 2.65 × 10
^{-8}Ω·m - Silver – Electrical resistivity: 1.59 × 10
^{-8}Ω·m - Gold – Electrical resistivity: 2.44 × 10
^{-8}Ω·m - Brass – Electrical resistivity: 6.9 × 10
^{-8}Ω·m

Insulators:

- Glass – Electrical resistivity: 10
^{10}-10^{14}Ω·m - Rubber – Electrical resistivity: 10
^{13}-10^{15}Ω·m - Air – Electrical resistivity: 10
^{16}-10^{19}Ω·m

Semiconductors:

- Silicon – Electrical resistivity: 2.3 × 10
^{3}Ω·cm - Germanium – Electrical resistivity: 4.6 × 10
^{2}Ω·cm

Note: The resistivity values given are approximate and can vary depending on the specific material and conditions. Conductors have low resistivity, insulators have high resistivity, and semiconductors.

The resistivity of a material depends on various factors, including:

- Temperature: The resistivity of most materials increases with temperature. This is because, at higher temperatures, there is more thermal energy available to dislodge electrons from their atoms and increase the resistance to the flow of electrical current.
- Composition: The resistivity of a material is largely dependent on its chemical composition. Materials with more free electrons, such as metals, generally have lower resistivity compared to materials with fewer free electrons, such as insulators.
- Impurities: The presence of impurities in a material can increase its resistivity. This is because impurities can cause defects in the crystal structure of the material, which can disrupt the flow of electrons and increase resistance.
- Pressure: The resistivity of a material can change with pressure, particularly in materials that are semiconductors. For example, increasing pressure on silicon can decrease its resistivity.
- Magnetic fields: In some materials, the resistivity can change in the presence of a magnetic field. This phenomenon is known as the magneto-resistive effect and is often utilized in the construction of electronic devices such as magnetic sensors and hard disk drives.

## How to measure resistivity

The resistivity of a material can be measured using a variety of techniques, depending on the nature of the material and the accuracy required. Here are some common methods for measuring resistivity:

- Four-point probe method: This is a widely used and accurate technique for measuring resistivity. It involves placing four electrical contacts onto the surface of the material in a square configuration and measuring the voltage difference between the inner two contacts while passing a known current through the outer two contacts. The resistivity can then be calculated using the dimensions of the probe and the measured values.
- Van der Pauw method: This method involves placing four electrical contacts on the surface of a material in a circular or elliptical configuration and measuring the voltage difference between pairs of opposite contacts while passing a known current through the remaining two. By rotating the sample and taking measurements at multiple angles, the resistivity can be calculated.
- Hall effect measurement: This technique involves measuring the voltage induced by an applied magnetic field perpendicular to the direction of current flow in the material. By measuring the Hall voltage and the applied magnetic field, the resistivity can be calculated.
- Transmission line method: This method involves sending a high-frequency electrical signal through the material and measuring its attenuation over a known distance. The resistivity can then be calculated using the dimensions of the sample and the measured values.

These are just a few examples of the methods used to measure resistivity. Other techniques include impedance spectroscopy, ac and dc methods, and more specialized methods for specific types of materials.

## How does electric current flow

Ohm’s law can be explained at a microscopic level by understanding the behavior of electrons in a conductor.

In a conductor, such as a metal wire, there are free electrons that are able to move through the material. These electrons collide with the atoms of the conductor as they move, which creates a resistance to their motion. The resistance of a conductor is related to the number of collisions that occur as electrons move through it.

When a voltage is applied across a conductor, it creates an electric field that causes the free electrons to move in a particular direction. The electrons experience a force due to this electric field, which causes them to accelerate and move through the conductor. However, the electrons do not move in a straight line but rather undergo a random motion due to collisions with the atoms of the conductor, losing energy and scattering in random directions. This creates resistance to the flow of electrons and causes some of the energy of the electric field to be converted into heat.

Ohm’s law can be understood in terms of this electron behavior. The current through a conductor is directly proportional to the voltage applied across it, because a higher voltage creates a stronger electric field that causes the electrons to move faster, resulting in a higher current. However, the current is inversely proportional to the resistance of the conductor, because a higher resistance means that there are more collisions and, therefore fewer free electrons available to carry the current.

Thus, Ohm’s law can be understood as a balance between the forces driving the electrons (the electric field) and the forces resisting their motion (collisions with atoms), resulting in a relationship between the current, voltage, and resistance of a conductor.

The drift velocity of electrons in a conductor is typically quite slow, on the order of a few millimeters per second, even though the current in the conductor may be quite high. This is because the electrons are constantly colliding with the atoms of the conductor, which slows down their overall motion. Drift velocity is proportional to current. In a resistive material, it is also proportional to the magnitude of an external electric field.

While the drift velocity is relatively slow, it is still an important concept in understanding the behavior of electric currents in conductors. The overall flow of electric charge in a conductor is determined by the combination of the drift velocity and the number of charge carriers moving through the conductor.

For example, when a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field (over a distance of a few micrometers).

## Resistance – **Hydraulic Analogy**

The hydraulic analogy, or the electric-fluid analogy, is a widely used analogy between hydraulics and electricity, which is a useful tool for teaching and for those who are struggling to understand how circuits work. it can also be applied to heat transfer problems.

Since electric current is invisible and the processes in play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. The relationship between voltage and current is defined (in ohmic devices like resistors) by Ohm’s law. Ohm’s Law is analogous to the Hagen–Poiseuille equation, as both are linear models relating flux and potential in their respective systems.

Electricity (as well as heat) was originally understood to be a kind of fluid, and the names of certain electric quantities (such as current) are derived from hydraulic equivalents.

**Voltage**is like the pressure difference that pushes water through the hose. It is measured in volts (V). This model assumes that the water is flowing horizontally so that the force of gravity can be ignored.**Current**is equivalent to a hydraulic volume flow rate; that is, the volumetric quantity of flowing water over time. Usually measured in amperes. The wider pipe is, the more water will flow through. It is measured in amps (I or A).**Resistance**is like pipe diameter or obstacles in the hose that slow down the water flow. It is measured in ohms (Ω). In hydraulics, resistance is associated with the pressure loss coefficient.**Resistors**are comparable to a section of the pipe network where the radius of the pipe is constricted, restricting the rate of fluid flow in that region, the same way that a resistor limits current.