When we make an electric circuit, we want it to be super efficient. This means we need low resistance. That's why we use materials like copper instead of wood or rubber in our circuits. Why? Because wood and rubber have high resistivities compared to copper.

So, what's resistivity? It's basically how much a material gets in the way of the flow of electricity. The higher the resistivity, the more it gets in the way. And that's related to the concept of electrical resistance.

If you want your circuit to work like a charm, use materials with low resistivity like copper.

What is electrical resistance?

When we're exploring electric phenomena in circuits, we use materials to guide electric charges for different purposes. There are three basic quantities we use to characterise circuits: resistance, voltage, and current.

Resistance is a measure of how much a material gets in the way of the movement of charges inside it. It's measured in ohms (Ω). Voltage, or potential difference, is the amount of energy per unit of charge needed to move charges between two points of a circuit. It's usually supplied by batteries and is measured in volts (V). Electric current, or just current, is the number of charges that pass through a conductor per unit of time. It's measured in amperes (A).

Resistance plays a big role in Ohm's law, which governs the behaviour of ohmic conductors and certain ranges of non-ohmic conductors. According to Ohm's law, if a circuit has high resistance, less current will be produced (and vice-versa). This means that the bigger the resistance, the bigger the opposition to the movement of charges, and the smaller the current.

But resistance isn't just a simple concept. In fact, the relationship between voltage, current, and resistance can be pretty complex, and not all conductors behave as Ohm's law predicts. Those that do are called ohmic conductors. To understand resistance more deeply, we need to look at the concept of resistivity and how it's generated due to microscopic phenomena.

The definition of resistivity

By studying the relationship between resistivity and resistance, we can understand why resistivity is a characteristic property of materials while resistance is not.

Resistivity measures the resistance of a conductor per unit of length and cross-section. It's a unique value for each material and depends on physical conditions like temperature. Resistivity is measured in ohm-meters (Ωm) and is denoted by the Greek letter ρ.

In simpler terms, resistivity tells us how much a material resists the flow of electricity based on its length and thickness. And because it's a property of the material itself, it doesn't change no matter how the material is shaped or used. This means that resistivity is a characteristic property of that material.

In contrast, resistance depends on both the material and the specific shape and use of the material. So it's not a unique property of the material itself. Understanding the difference between resistivity and resistance is important when designing circuits and choosing materials.

Factors affecting resistivity

Temperature is a key factor in determining resistivity because it's a measure of the average kinetic energy of the particles in a material. As temperature increases, the particles in the conductor move faster on average, which makes them more likely to interfere with the movement of charges. This results in a higher resistivity value.

Another factor that affects resistivity is the metallic nature of the material. Metals are known to be good conductors of electricity, which means they have a lower resistivity than non-metallic materials like wood or rubber. The atomic structure and microscopic spatial disposition of the metal will determine how easy it is for charges to move, and this will ultimately determine the exact value of resistivity.

The characteristic resistivity values of different materials vary widely. For example, copper has a resistivity of 1.68×10−8 Ωm, while rubber has a resistivity of 1×10^13 Ωm. This means that copper is a much better conductor of electricity than rubber.

It's important to note that resistivity is a characteristic property of materials that doesn't depend on their length or cross-section. This means that no matter how a material is shaped or used, its resistivity will always be the same. Understanding the concept of resistivity is crucial for designing circuits and choosing the right materials for specific applications.

The resistivity equation

The equation R = ρL/A captures the relationship between resistance, resistivity, length, and cross-section.

To understand this equation, it's important to remember that current the number of charges that pass through a cross-section of a conductor per unit of time. The cross-section of a conductor is the surface area perpendicular to the direction of the current at each point.

While resistance measures the opposition of a material to the flow of current, we must also consider the material's length because it directly affects the resistance. The longer a conductor is, the greater its resistance will be. This means that resistance and length are directly proportional to each other. On the other hand, the resistance of a conductor is inversely proportional to its cross-sectional area.

Therefore, if we know the resistivity of a material, we can calculate the resistance of a conductor made from that material by multiplying the resistivity by the length and dividing it by the cross-sectional area. This equation allows us to predict how a material will behave in a circuit and choose materials accordingly. 

Length

That is a great analogy! In a crowded street, the street itself can be considered the conductor, and people in the street can be thought of as obstacles that a charge (like yourself) must avoid to reach the other end. By walking just one block instead of three, you encounter fewer people, which means there is less resistance to your movement. This is similar to how a shorter length of a conductor has less resistance to the flow of current compared to a longer length.

In a conductor, the electrons that carry the current also encounter resistance as they move through the material. The longer the path they must travel, the more obstacles they encounter, and the more resistance they face. This is why the length of a conductor is a crucial factor in determining its resistance. By choosing a shorter length of a conductor, we can reduce the obstacles that electrons encounter and, therefore, reduce the resistance of the material.

Cross-section

The cross-sectional area of a conductor plays a crucial role in determining its resistance. As you mentioned, resistance measures the opposition to the flow of current, but the amount of current that flows through conductor is directly proportional to its cross-sectional area. This means that if we double the size of the cross-section, we also double the amount of current that can flow through it. While the resistance still exists, the larger cross-section allows more current to flow through the material, reducing the overall resistance.

In your analogy of friends uniformly separated at the other end of the street, if you were to count how many friends reach your end of the street per unit of time, you would count double that amount if you were in a street that was two times wider with double the number of friends. This is because, in a uniform density of charges in the material, the larger cross-section allows for a greater flow of charges (like friends in your analogy), resulting in less resistance and more current flowing through the material.

To summarize, resistance increases with the length of a conductor as moving charges encounter more particles that obstruct them. Resistance decreases with the cross-sectional area of a conductor as a larger cross-section allows for more charges to flow through the material per unit of time.

How to calculate resistance usingresistivity

Thank you for providing an excellent example to illustrate the concepts we discussed earlier!

In your example, we are comparing the efficiency of a silver wire, with a cross-sectional area of 1², wire. To determine the cross-sectional area of the carbon wire that would transfer current as efficiently as the silver wire, we can use the formula for resistance, which is:

Resistance = (Resistivity x Length) / Cross-sectional area

We know the length of the wire (1m) and resistivity values for both silver and carbon, which we can find in a table of material properties. By rearranging the formula, we can solve for the cross-sectional area of the carbon wire that would have the same resistance as the silver wire.

When we plug in the values for silver, we get:

Resistance of silver wire = (1.59 x 10^-8 Ωm x 1m) / 0.0001m² = 1.59 x 10^-4 Ω

To find the cross-sectional area of the carbon wire needed for the same resistance, we can rearrange the formula:

Cross-sectional area of carbon wire = (Resistivity x Length) / Resistance

When we plug in the values for carbon and the resistance we just found for silver, we get:

Cross-sectional area of carbon wire = (5.7 x 10^-8 Ωm x 1m) / (1.59 x 10^-4 Ω) = 2.27 x 10^-4 m²

This means that if we were to use a carbon wire with a diameter of approximately 0.5m (assuming a cylindrical shape), it would have the same resistance and transfer current as efficiently as the silver wire with a cross-sectional area of 1cm².

However, as you pointed out, this diameter is quite large compared to the silver wire. If we were to use copper instead of carbon, the cross-sectional area needed for the same resistance would be smaller, allowing us to use a wire with a diameter closer to that of the silver wire. This is why we typically use copper to make electrical cables rather than carbon.

Resistivity - key takeaways

Resistance measures the opposition of a medium to the flow of charges through it, while resistivity measures the intrinsic opposition of a material to the flow of charges per unit of length and cross-sectional area. Resistivity is a more fundamental quantity than resistance because it is determined solely by the microscopic characteristics of the material, and does not depend on the size or width of the conductor.

Resistivity is a characteristic of each material at certain external conditions, such as temperature, and can vary depending on the material and its properties. For example, as the temperature of a material increases, the resistivity typically increases as well.

As you mentioned, resistance increases as the length of the conductor increases because moving charges encounter more particles that obstruct their flow. Conversely, resistance decreases with an increase in cross-sectional area because a larger cross-section allows more charges to flow through the material per unit of time.

Overall, understanding the relationship between resistance and resistivity, as well as their dependence on length and cross-sectional area, is crucial in designing and optimizing electrical circuits and systems.

Resistivity

What is resistivity?
Resistivity is a quantity that measures the characteristic opposition of amaterial to the movement of charges inside of it per unit of length andcross-section.
How do you calculate resistance with resistivity?
If we know the resistivity of a material, we can calculate the resistance of aconductor made out of it by multiplying it by the length and dividing it by thecross-section.
What is meant by the electrical resistivity of a material?
Resistivity is a quantity that measures the resistance of a conductor per unitof length and cross-section. It is different for each material and depends oncertain physical conditions, such as temperature. 
What is the equation for resistivity?
The equation for resistivity is ρ=R·A/L

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