Current

Current is the rate of flow of charge; the amount of charge that passes per unit time.

Macroscopic View of Current

\[I = \frac{\Delta Q}{\Delta t}\]

Its units are amperes $1A = \frac{C}{s}$.

• The current points from high to low voltage; that is, it points in the same direction as the $\vec E$ field

Microscopic View of Current

\[I = \eta Av_de\]

where $\eta$ is charge carrier density, $A$ is a cross sectional area, $v_d$ is electron drift velocity, and e is the elementary charge $1.6\times 10^{-19}C$

Charge carrier density $\eta$ has units $\frac{\mathrm{Number\ of\ Charges}}{m^3}$.

Current Density

Macroscopic View

Current $J$ is equal to

\[J = \frac{I}{A}\]

where $I$ is current and $A$ is a cross sectional area.

Microscopic View

Current $J$ is equal to

\[J = \eta v_de\]

where $\eta$ is charge carrier density, $v_d$ is electron drift velocity, and $e$ the elementary charge $1.6\times 10^{-19}C$.

Kirchoff’s Junction Rule (Node Equation)

\[I_{in}=I_{out}\]

Material Properties of Conductors

Conductivity

Conductivity may be defined as $\sigma$ (sigma) where

\[J = \sigma E\]

The units of conductivity are $\frac{A}{V\cdot m}$.

Resistivity

Resistivity $\rho$ (rho) is the inverse of conductivity

\[\rho = \frac{1}{\sigma}\]

whose units are $\frac{V\cdot m}{A}$.

Ohm’s Law

From the equation for conductivity may be derived the general form of Ohm’s Law

\[V = IR\]

where $R=\frac{\rho L}{A}$. The units of $R$ are $\frac{V}{A}\equiv \Omega$ or an Ohm.

Power

Power is measured in Watts. The power supplied by a battery is equal to

\[P = IV\]

where $I$ is current and $V$ voltage.

Power Dissipated in a Resistor

\[P = I^2R\]

Drift Speed

Drift speed is equal to

\[v_d = \frac{e\tau}{m}E\]

where $\tau$ is the mean time between collisions.