Resistors and Resistivity

Introduction:

Resistors are components that impede the flow of electric current in a circuit, offering resistance.

Resistance and the Resistor Equation:

• Definition: Resistance (\( R \)) determines how much a resistor opposes the flow of current. It is influenced by the resistor’s material, length, and cross-sectional area.
• Resistor Equation: \[ R = \rho \times \frac{L}{A} \] Where:
  • \( R \) = Resistance (in ohms)
  • \( \rho \) = Resistivity of the material (in ohm-meter)
  • \( L \) = Length of the resistor (in meters)
  • \( A \) = Cross-sectional area of the resistor (in square meters)

Analogy with the Circulatory System:

Arteries clogged with plaque constrict the space blood can flow in, escalating the vessel’s resistance. The longer the blood has to flow in this restricted space the greater the resistance. Similarly, long, thin resistors have more resistance than short, wide ones.

This means the smaller the resistors area (i.e the more plaque) the greater the resistance and the longer the resistors length (i.e the more extensive the plaque) the greater the resistance as well.

The material of blood vessels and resistors is also important because some materials naturally resist flow more. Think of it like how thick a liquid is, or its viscosity. For example, if our blood was as thick as honey, it would barely move. Similarly, if a material in a resistor has a high resistivity it doesn’t let electrons flow easily and the resistance is higher, making it harder for electricity to pass through.

Resistivity:

• Definition: Resistivity (\( \rho \)) is an inherent property of materials indicating how much they resist electric current. The higher the resistivity, the greater the resistance provided by a given volume of that material.
  • \( \rho \) = Resistivity (in ohm-meter)

Conductivity:

• Definition: Conductivity (\( \sigma \)) is the inverse of resistivity. It measures a material’s ability to conduct electric current. \[ \sigma = \frac{1}{\rho} \] Where:
  • \( \sigma \) = Conductivity (in siemens per meter, S/m)

How It’s Tested:

Questions often examine your understanding of how changing one variable affects overall resistance. For instance, you might encounter scenarios asking you to predict the effect on resistance if the material’s resistivity increases, or if the resistor’s length is extended. Here is a guide to how those changes play out:

1. Resistance (R) is directly proportional to the resistivity (ρ) of the material. This means if the resistivity increases, resistance increases as well.
2. Resistance (R) is directly proportional to the length (l) of the resistor. So, a longer resistor will have a higher resistance.
3. Resistance (R) is inversely proportional to the cross-sectional area (A) of the resistor. This means that as the area increases, resistance decreases.

The MCAT may not always ask about the resistance formula directly. Instead, you could be asked to apply it alongside Ohm’s Law. This means you have to first understand how a change in a particular variable affects resistance. Then, consider what that change in resistance implies according to Ohm’s Law.

Rather than trying to memorize all possible variations of equations that could combine with the resistivity equation, focus on identifying the common variable shared between the equations at hand. From there, you can deduce how changes to this variable influence other aspects of the circuit’s behavior.