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. Understanding these concepts is crucial for analyzing electrical circuits and their components.

Resistors play a vital role in controlling 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

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• 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