AP E&M Chapter 13.5 Other things about Circuits
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Current Density and Drift Velocity
Current Density
The current density (电流密度) $\vec{J}$ is a vector field within a wire.
\[\boxed{ I = \int \vec{J} \cdot d\vec{A} }\]Suppose a uniform current flows through a straight wire, $\vec{J}$ is constant and parallel to $d\vec{A}$. Therefore, the current density for the case of a straight wire is
\[\boxed{ J = \frac I A }\]Drift Velocity
The drift velocity (漂移速度) is defined as the average electron velocity:
\[\boxed{ \vec{v}_D = \frac 1 N \sum\limits_{i=1}^N \vec{v}_i }\]Relationship between Current Density & Drift Velocity
The charge carrier density $n$ (载流子密度) denotes the number of charge carriers per volume. The SI unit for charge carrier density is $\ce{m^{-3}}$.
\[\begin{aligned} J &= \frac I A \\ &= \frac 1 A \frac {dQ} {dt} \\ &= \frac 1 A \frac {e dN} {dt} \\ &= \frac 1 A \frac {n e A v_D dt} {dt} \\ &= n e v_D \end{aligned}\]Therefore, the relationship between current density and drift velocity is
\[\boxed{ \vec{J} = n e \vec{v}_D }\]$e$ is the elementary charge (元电荷), which is electric charge carried by a single proton.
\[e = 1.602176634 \times 10^{-19} \ \ce{C}\]Resistivity & Conductivity
Resistivity (电阻率) is defined as the ratio of the electric field to the current density:
\[\boxed{ \vec{E} = \rho \vec{J} }\] \[\boxed{ \rho = \frac E J }\]This is known as a reformulation of Ohm’s law.
Conductivity (电导率) is defined as the reciprocal of resistivity.
\[\boxed{ \sigma = \frac 1 \rho = \frac J E }\]Resistivity and conductivity are properties of materials. Resistance depends on the material and the geometry of the object.
\[\begin{aligned} R &= \frac V I \\ &= V \frac 1 {JA} \\ &= V \frac \rho {EA} \\ &= V \frac \rho {\frac V L A} \\ &= \frac {\rho L} A \end{aligned}\]
Therefore, the resistance of a resistor with length $L$, cross-sectional area $A$, and resistivity $\rho$ is
\[\boxed{ R = \frac {\rho L} A }\]