Chapter 28 - Current and Resistance

• A circuit is a closed network through which charged particles can flow.
• Various devices (e.g., bulb, battery) in a circuit are known as circuit elements.
• Current is the rate at which charged particles pass through a cross-section of filament.
• Current equation: $I =$$\frac{dq}{dt} =$$\frac{\Delta q}{\Delta t}$
• Current density is given by: $J \equiv \frac{I}{A}$ and is a vector that points in the same direction as the electric field ($\vec{J} \propto \vec{E}$).
• Conductivity: the degree to which a specified material conducts electricity.
• Conductivity equation: $\sigma \equiv \frac{ne^2}{m_e}t_{mf}$
• Conductivity in terms of the electric field: $\vec{J} =$$\sigma \vec{E}$
• Resistivity (Ω • m) is the measure of a material's ability to resist conducting electricity.
• Resistivity equation: $\rho \equiv \frac{1}{\sigma} =$$\frac{m_e}{ne^2t_{mf}}$
• Resistivity in terms of the electric field: $\vec{E} =$$\rho \vec{J}$
• Resistance is the measure of opposition to the flow of current in an electrical circuit.
• Resistance depends on only filament properties: (1) the resistivity, (2) the length, and (3) the cross-sectional area.
• Ohm's law: $\Delta V =$$IR$
• For Ohm's law, the potential difference (voltage) is proportional to the current. Ohm's law is found from the slope of an I-V graph.
• The temperature dependence of resistivity is given by: $\rho =$$\rho_0(1+\alpha T-\alpha T_0)$, where $\rho _0$, is the resistivity at temperature $T_0$, and $\alpha$ is the temperature component of resistivity.
• For a conductor, the electric field $\vec{E}$ is proportional to its current density $\vec{J}$.
• A circuit element that obeys Ohm's law over a wide range of potential differences is described as an Ohmic conductor.
• Power supplied to a circuit element (default): $P =$$I\Delta V$
• Power supplied by a battery (with terminal potential) $P_{bat} =$$I \epsilon$
• Power supplied by a circuit with resistance: $P =$$I^2R =$$\frac{\Delta V^2}{R}$