Sections

Section Summary
# Section Summary

### 3.1 Current

- Electric current $I$ is the rate at which charge flows, given by
$$I=\frac{\mathrm{\Delta}Q}{\mathrm{\Delta}t}\text{,}$$where $\mathrm{\Delta}Q$ is the amount of charge passing through an area in time $\mathrm{\Delta}\mathrm{t.}$
- The direction of conventional current is taken as the direction in which positive charge moves.
- The SI unit for current is the ampere (A), where $\text{1 A}=\text{1 C/s.}$
- Current is the flow of free charges, such as electrons and ions.
- Drift velocity ${v}_{\text{d}}$ is the average speed at which these charges move.
- Current $I$ is proportional to drift velocity ${v}_{\text{d}}\text{,}$ as expressed in the relationship $I={\text{nqAv}}_{\text{d}}\text{.}$ Here, $I$ is the current through a wire of cross-sectional area $\mathrm{A.}$ The wire's material has a free-charge density $n$, and each carrier has charge $q$ and a drift velocity ${v}_{\text{d}}\text{.}$
- Electrical signals travel at speeds about ${\text{10}}^{\text{12}}$ times greater than the drift velocity of free electrons.

### 3.2 Ohm’s Law: Resistance and Simple Circuits

- A simple circuit
*is*one in which there is a single voltage source and a single resistance. - One statement of Ohm's law gives the relationship between current $\mathrm{I,}$ voltage $\mathrm{V,}$ and resistance $R$ in a simple circuit to be $I=\frac{V}{R}.$
- Resistance has units of ohms ($\text{\Omega}$), related to volts and amperes by $\mathrm{1\; \Omega}=\text{1 V/A}\text{.}$
- There is a voltage or $\text{IR}$ drop across a resistor, caused by the current flowing through it, given by $V=\text{IR}\text{.}$

### 3.3 Resistance and Resistivity

- The resistance $R$ of a cylinder of length $L$ and cross-sectional area $A$ is $R=\frac{\mathrm{\rho L}}{A}\text{,}$ where $\rho $ is the resistivity of the material.
- Values of $\rho $ in Table 3.1 show that materials fall into three groups—
*conductors, semiconductors, and insulators*. - Temperature affects resistivity; for relatively small temperature changes $\mathrm{\Delta}\mathrm{T,}$ resistivity is $\rho ={\rho}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$, where ${\rho}_{0}$ is the original resistivity and $\text{\alpha}$ is the temperature coefficient of resistivity.
- Table 3.2 gives values for $\alpha $, the temperature coefficient of resistivity.
- The resistance $R$ of an object also varies with temperature: $R={R}_{0}(\text{1}+\alpha \mathrm{\Delta}T)\text{,}$ where ${R}_{0}$ is the original resistance, and $R$ is the resistance after the temperature change.

### 3.4 Electric Power and Energy

- Electric power $P$ is the rate (in watts) that energy is supplied by a source or dissipated by a device.
- Three expressions for electrical power are
$$P=\text{IV,}$$$$P=\frac{{V}^{2}}{R}\text{,}$$
and

$$P={I}^{2}R\text{.}$$ - The energy used by a device with a power $P$ over a time $t$ is $E=\text{Pt}\text{.}$

### 3.5 Alternating Current versus Direct Current

- Direct current (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
- The voltage source of an alternating current (AC) system puts out $V={V}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$, where $V$ is the voltage at time $t\text{,}$${V}_{0}$ is the peak voltage, and $f$ is the frequency in hertz.
- In a simple circuit, $I=\text{V/R}$ and AC current is $I={I}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$, where $I$ is the current at time $t$, and ${I}_{0}={V}_{0}\text{/R}$ is the peak current.
- The average AC power is ${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}\text{.}$
- Average (rms) current ${I}_{\text{rms}}$ and average (rms) voltage ${V}_{\text{rms}}$ are ${I}_{\text{rms}}=\frac{{I}_{0}}{\sqrt{2}}$ and ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$, where rms stands for root mean square.
- Thus, ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\text{.}$
- Ohm's law for AC is ${I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{R}$.
- Expressions for the average power of an AC circuit are ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\text{,}$${P}_{\text{ave}}=\frac{{V}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}}{R}$, and ${P}_{\text{ave}}={I}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}R\text{,}$ analogous to the expressions for DC circuits.

### 3.6 Electric Hazards and the Human Body

- The two types of electric hazards are thermal (excessive power) and shock (current through a person).
- Shock severity is determined by current, path, duration, and AC frequency.
- Table 3.3 lists shock hazards as a function of current.
- Figure 3.28 graphs the threshold current for two hazards as a function of frequency.