9.2 Model of Conduction in Metals

Electrons in metals play a crucial role in electrical conduction. Their behavior, described by drift velocity and current density, explains how electric currents flow through conductors like copper wires and power lines.

The conduction model applies to real-world devices like incandescent lamps. It shows how electrons moving through a tungsten filament create light and heat, illustrating the practical applications of electron behavior in metals.

Model of Conduction in Metals

Drift velocity of electrons

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• Average velocity of charge carriers in a conductor caused by an applied electric field
• Proportional to electric field strength $E$ and inversely proportional to resistivity $\rho$ of the material
  • Calculated using the formula $v_d = \frac{E}{\rho n e}$, where $n$ is the number of charge carriers per unit volume and $e$ is the elementary charge
• Typically very small compared to random thermal motion of electrons in a conductor
  • In a copper wire with a current density of $10^6 \text{ A/m}^2$, drift velocity is approximately $10^{-4} \text{ m/s}$
• Net flow of charge carriers in the direction of the electric field results in an electric current
• Drift velocity is essential for understanding the movement of electrons in conductors (copper wires, aluminum power lines)
• Influenced by electron mobility, which describes how easily electrons move through the material in response to an electric field

Current density and electric current

• Vector quantity describing the flow of electric charge per unit cross-sectional area
  • Defined as $J = \frac{I}{A}$, where $I$ is the electric current and $A$ is the cross-sectional area
• Points in the direction of the net flow of positive charge carriers
• Magnitude proportional to electric field strength $E$ and conductivity $\sigma$ of the material
  • Related by the equation $J = \sigma E$, where $\sigma = \frac{1}{\rho}$ is the conductivity and $\rho$ is the resistivity
• Total electric current through a conductor is the integral of current density over the cross-sectional area
  • Expressed as $I = \int_A J \cdot dA$, where $dA$ is the differential area element
• Current density helps analyze the distribution of current in conductors (power transmission lines, printed circuit boards)

Conduction model in incandescent lamps

• Incandescent lamps have a thin tungsten filament heated to high temperature by an electric current
• Conduction model explains the flow of electrons through the tungsten filament
  • Applied voltage creates an electric field within the material
  • Electric field causes free electrons in tungsten to drift, resulting in an electric current
• Electrons collide with tungsten atoms, transferring energy to the lattice
  • Energy transfer heats the filament, causing it to emit light through incandescence
• Filament resistance increases with temperature due to increased lattice vibrations and electron scattering
  • Temperature-dependent resistance described by the temperature coefficient of resistivity $\alpha$
    • Resistance at temperature $T$ is $R = R_0[1 + \alpha(T - T_0)]$, where $R_0$ is the resistance at reference temperature $T_0$
• High operating temperature of filament (around 2500 K) results in significant energy radiated as visible light and infrared
  • Also leads to evaporation of tungsten atoms, limiting the lamp's lifetime
• Conduction model helps understand the operation of incandescent lamps (light bulbs, halogen lamps)

Electron behavior in metals

• Fermi energy represents the highest occupied energy level of electrons in a metal at absolute zero temperature
• Mean free path is the average distance an electron travels between collisions with lattice atoms or impurities
• Lorentz number relates thermal conductivity to electrical conductivity in metals, demonstrating the connection between heat and charge transport