9.3 Resistivity and Resistance

Electric current faces resistance as it flows through materials. Resistance depends on a material's dimensions and resistivity, an intrinsic property. These concepts explain how electrical circuits and their components behave.

Resistors control current flow and voltage in circuits. Their behavior is influenced by factors like length, cross-sectional area, and temperature. These properties affect how resistors function in various applications, from protecting components to shaping signals.

Resistivity and Resistance

Resistance vs resistivity

• Resistance ($R$) quantifies the opposition to the flow of electric current in a material measured in ohms ($\Omega$) depends on the material's dimensions (length and cross-sectional area) and its intrinsic resistivity
• Resistivity ($\rho$) is an intrinsic property of a material that quantifies its resistance to electric current measured in ohm-meters ($\Omega \cdot m$) independent of the material's dimensions varies with temperature and composition (impurities, defects) of the material
• Key difference: resistance depends on both the material's properties and its dimensions, while resistivity is solely a property of the material itself

Conductivity and resistivity relationship

• Conductivity ($\sigma$) measures a material's ability to conduct electric current is the inverse of resistivity: $\sigma = \frac{1}{\rho}$ measured in siemens per meter ($S/m$)
• Relationship: materials with high conductivity (metals like copper, silver) have low resistivity and vice versa insulators (rubber, glass) have low conductivity and high resistivity
• Semiconductors (silicon, germanium) have conductivity and resistivity values between those of conductors and insulators their conductivity can be modified by doping with impurities
• The Fermi level in materials influences their conductivity and resistivity properties

Resistors in electrical circuits

• Resistors are passive electronic components that provide resistance in a circuit used to control current flow, divide voltages, and limit current to protect other components available in various resistance values (ohms) and power ratings (watts)
• Functions of resistors in circuits:
  1. Current limiting: protect sensitive components (LEDs, transistors) by limiting current flow
  2. Voltage division: create desired voltage drops across components in a voltage divider circuit
  3. Load balancing: ensure proper current distribution in parallel circuits by equalizing resistance
  4. Signal conditioning: shape and filter electrical signals (low-pass filters, pull-up/pull-down resistors)

Physical properties of resistors

• Resistance formula: $R = \rho \frac{l}{A}$ where $\rho$ is the resistivity of the material, $l$ is the length, and $A$ is the cross-sectional area of the resistor
• Effect of length on resistance: resistance is directly proportional to length longer resistors (wirewound) have higher resistance than shorter ones (surface-mount) for the same material and cross-sectional area
• Effect of cross-sectional area on resistance: resistance is inversely proportional to cross-sectional area thicker resistors (carbon composition) have lower resistance than thinner ones (thin-film) for the same material and length
• Effect of material resistivity on resistance: higher resistivity materials (carbon, nichrome) result in higher resistance than lower resistivity materials (copper, gold) for the same dimensions

Temperature effects on resistivity

• Temperature coefficient of resistivity ($\alpha$) quantifies the change in resistivity with temperature positive $\alpha$ means resistivity increases with increasing temperature (metals) negative $\alpha$ means resistivity decreases with increasing temperature (semiconductors)
• Temperature dependence of resistivity follows a linear approximation: $\rho(T) = \rho_0[1 + \alpha(T - T_0)]$ where $\rho(T)$ is the resistivity at temperature $T$, $\rho_0$ is the resistivity at reference temperature $T_0$ (usually 20°C)
• Impact on resistance: as resistivity changes with temperature, the resistance of a material also changes according to $R(T) = R_0[1 + \alpha(T - T_0)]$ where $R(T)$ is the resistance at temperature $T$ and $R_0$ is the resistance at reference temperature $T_0$
• Examples: the resistance of a copper wire increases by about 0.4% per °C rise in temperature the resistance of a silicon diode decreases by about 2% per °C rise in temperature

Current flow and electric fields in resistive materials

• Electric fields in resistive materials drive the flow of charge carriers (e.g., electrons in metals)
• Current density in a material is related to the drift velocity of charge carriers
• The relationship between electric field, current density, and resistivity is described by Ohm's law in microscopic form