20.3 Resistance and Resistivity

Electrical resistance and resistivity are key concepts in understanding how materials oppose the flow of electric current. Resistance depends on an object's dimensions, while resistivity is an intrinsic property of the material itself.

Temperature plays a crucial role in resistance, typically causing it to increase as temperature rises. This relationship is important in various applications, from designing electrical systems to creating temperature sensors.

Electrical Resistance and Resistivity

Resistivity vs resistance

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• Resistivity ($\rho$) is an intrinsic property of a material that quantifies its ability to resist the flow of electric current
  • Depends on the material's composition and temperature (copper, silicon)
  • Measured in ohm-meters ($\Omega \cdot m$)
  • Remains constant for a given material at a specific temperature
  • Inversely related to conductivity
• Resistance ($R$) is the opposition to the flow of electric current in a specific object or component
  • Depends on the material's resistivity, length, and cross-sectional area (wire, resistor)
  • Measured in ohms ($\Omega$)
  • Can vary depending on the object's dimensions and temperature

Resistance calculations using resistivity

• The resistance of a uniform object can be calculated using the formula: $R = \rho \frac{L}{A}$
  • $R$ is the resistance in ohms ($\Omega$)
  • $\rho$ is the material's resistivity in ohm-meters ($\Omega \cdot m$)
  • $L$ is the length of the object in meters ($m$)
  • $A$ is the cross-sectional area of the object in square meters ($m^2$)
• For a cylindrical wire, the cross-sectional area is $A = \pi r^2$, where $r$ is the radius of the wire
  • Example: electrical wiring in homes and buildings
• For a rectangular prism, the cross-sectional area is $A = w \cdot h$, where $w$ is the width and $h$ is the height
  • Example: thin film resistors in electronic circuits
• When objects are connected in series, their resistances add: $R_{total} = R_1 + R_2 + ... + R_n$
  • Example: multiple resistors connected end-to-end in a circuit
• When objects are connected in parallel, their reciprocal resistances add: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
  • Example: multiple resistors connected side-by-side in a circuit

Temperature effects on resistance

• The thermal coefficient of resistivity ($\alpha$) describes how a material's resistivity changes with temperature
  • Measured in inverse kelvins ($K^{-1}$)
  • For most materials, resistivity increases with increasing temperature (metals, semiconductors)
• The change in resistivity due to temperature can be calculated using the formula: $\rho_T = \rho_0[1 + \alpha(T - T_0)]$
  • $\rho_T$ is the resistivity at temperature $T$
  • $\rho_0$ is the resistivity at a reference temperature $T_0$ (usually 20℃)
  • $\alpha$ is the thermal coefficient of resistivity
  • $T$ is the new temperature in kelvins ($K$) or degrees Celsius (℃)
• The change in resistance due to temperature can be calculated using the formula: $R_T = R_0[1 + \alpha(T - T_0)]$
  • $R_T$ is the resistance at temperature $T$
  • $R_0$ is the resistance at a reference temperature $T_0$
  • Example: temperature sensors (thermistors) and heating elements

Electric Current and Voltage

• Electric current is the flow of charge carriers through a conductor
  • Measured in amperes (A)
  • Related to resistance by Ohm's Law: $V = IR$
• Voltage is the electric potential difference between two points
  • Measured in volts (V)
  • Creates an electric field that drives the current
• The drift velocity of charge carriers is influenced by the electric field and the material's properties