AP E&M Chapter 13.5 Other things about Circuits

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.

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\[\begin{aligned} R &= \frac V I \\ &= V \frac 1 {JA} \\ &= V \frac \rho {EA} \\ &= V \frac \rho {\frac V L A} \\ &= \frac {\rho L} A \end{aligned}\]

Therefore, the resistance of a resistor with length \(L\), cross-sectional area \(A\), and resistivity \(\rho\) is

\[\boxed{ R = \frac {\rho L} A }\]