Section Summary

20.1 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 $\mathrm{\Delta }t$.
• 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 $v_{\text{d}}$, as expressed in the relationship $I={\text{nqAv}}_{\text{d}}$. Here, $I$ is the current through a wire of cross-sectional area $A$. The wire’s material has a free-charge density $n$, and each carrier has charge $q$ and a drift velocity $v_{\text{d}}$.
• Electrical signals travel at speeds about ${\text{10}}^{\text{12}}$ times greater than the drift velocity of free electrons.

20.2 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 $I$, voltage $V$, and resistance $R$ in a simple circuit to be $I=\frac{V}{R}.$
• Resistance has units of ohms ($\text{Ω}$), related to volts and amperes by $\mathrm{1\; \Omega }=\text{1 V/A}$.
• There is a voltage or $\text{IR}$ drop across a resistor, caused by the current flowing through it, given by $V=\text{IR}$.

20.3 Resistance and Resistivity

• The resistance $R$ of a cylinder of length $L$ and cross-sectional area $A$ is $R=\frac{\mathrm{\rho L}}{A}$, where $\rho$ is the resistivity of the material.
• Values of $\rho$ in Table 20.1 show that materials fall into three groups—conductors, semiconductors, and insulators.
• Temperature affects resistivity; for relatively small temperature changes $\mathrm{\Delta }T$, resistivity is $\rho ={\rho }_0(\text{1}+\alpha \mathrm{\Delta }T)$, where ${\rho }_0$ is the original resistivity and $\text{α}$ is the temperature coefficient of resistivity.
• Table 20.2 gives values for $\alpha$, the temperature coefficient of resistivity.
• The resistance $R$ of an object also varies with temperature: $R=R_0(\text{1}+\alpha \mathrm{\Delta }T)$, where $R_0$ is the original resistance, and $R$ is the resistance after the temperature change.

20.4 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 $E=\text{Pt}$.

20.5 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 $V=V_0\text{sin 2}\pi \text{ft}$, where $V$ is the voltage at time $t$, $V_0$ is the peak voltage, and $f$ is the frequency in hertz.
• In a simple circuit, $I=\text{V/R}$ and AC current is $I=I_0\text{sin 2}\pi \text{ft}$, where $I$ is the current at time $t$, and $I_0=V_0\text{/R}$ is the peak current.
• The average AC power is $P_{\text{ave}}=\frac{1}{2}{I}_0{V}_0$.
• 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, $P_{\text{ave}}=I_{\text{rms}}{V}_{\text{rms}}$.
• Ohm’s law for AC is $I_{\text{rms}}=\frac{V_{\text{rms}}}{R}$.
• Expressions for the average power of an AC circuit are $P_{\text{ave}}=I_{\text{rms}}{V}_{\text{rms}}$, $P_{\text{ave}}=\frac{V_{\text{rms}}^2}{R}$, and $P_{\text{ave}}=I_{\text{rms}}^{2}R$, analogous to the expressions for DC circuits.

20.6 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 20.3 lists shock hazards as a function of current.
• Figure 20.25 graphs the threshold current for two hazards as a function of frequency.

20.7 Nerve Conduction–Electrocardiograms

• Electric potentials in neurons and other cells are created by ionic concentration differences across semipermeable membranes.
• Stimuli change the permeability and create action potentials that propagate along neurons.
• Myelin sheaths speed this process and reduce the needed energy input.
• This process in the heart can be measured with an electrocardiogram (ECG).