Chapter Summary

16.1 Current

Electric current [latex]I[/latex] is the rate at which charge flows, given by
[latex]I = \frac{Δ Q}{Δ t} ,[/latex]

where [latex]Δ Q[/latex] is the amount of charge passing through an area in time [latex]Δ t[/latex].
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 [latex]\text{1 A } = \text{ 1 C}/\text{s}.[/latex]
Current is the flow of free charges, such as electrons and ions.
Drift velocity [latex]v_{\text{d}}[/latex] is the average speed at which these charges move.
Current [latex]I[/latex] is proportional to drift velocity [latex]v_{\text{d}}[/latex], as expressed in the relationship [latex]I = \text{nqAv}_{\text{d}}[/latex]. Here, [latex]I[/latex] is the current through a wire of cross-sectional area [latex]A[/latex]. The wire’s material has a free-charge density [latex]n[/latex], and each carrier has charge [latex]q[/latex] and a drift velocity [latex]v_{\text{d}}[/latex].
Electrical signals travel at speeds about [latex]\text{10}^{\text{12}}[/latex] times greater than the drift velocity of free electrons.

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

The resistance [latex]R[/latex] of a cylinder of length [latex]L[/latex] and cross-sectional area [latex]A[/latex] is [latex]R = \frac{ρL}{A}[/latex], where [latex]\rho[/latex] is the resistivity of the material.
Values of [latex]\rho[/latex] in Table 16.1 show that materials fall into three groups—conductors, semiconductors, and insulators.
Temperature affects resistivity; for relatively small temperature changes [latex]Δ T[/latex], resistivity is [latex]\rho = \rho_{0} \left(\right. \text{1 } + \alpha Δ T \left.\right)[/latex], where [latex]\rho_{0}[/latex] is the original resistivity and [latex]\alpha[/latex] is the temperature coefficient of resistivity.
Table 16 .2 gives values for [latex]\alpha[/latex], the temperature coefficient of resistivity.
The resistance [latex]R[/latex] of an object also varies with temperature: [latex]R = R_{0} \left(\right. \text{1 } + \alpha Δ T \left.\right)[/latex], where [latex]R_{0}[/latex] is the original resistance, and [latex]R[/latex] is the resistance after the temperature change.

Electric power [latex]P[/latex] is the rate (in watts) that energy is supplied by a source or dissipated by a device.
Three expressions for electrical power are [latex]P = \text{IV},[/latex] [latex]P = \frac{V^{2}}{R} ,[/latex] and [latex]P = I^{2} R .[/latex]
The energy used by a device with a power [latex]P[/latex] over a time [latex]t[/latex] is [latex]E = \text{Pt}[/latex].

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 [latex]V = V_{0} \text{sin 2} \pi \text{ft}[/latex], where [latex]V[/latex] is the voltage at time [latex]t[/latex], [latex]V_{0}[/latex] is the peak voltage, and [latex]f[/latex] is the frequency in hertz.
In a simple circuit, [latex]I = \text{V}/\text{R}[/latex] and AC current is [latex]I = I_{0} \text{sin 2} \pi \text{ft}[/latex], where [latex]I[/latex] is the current at time [latex]t[/latex], and [latex]I_{0} = V_{0} /\text{R}[/latex] is the peak current.
The average AC power is [latex]P_{\text{ave}} = \frac{1}{2} I_{0} V_{0}[/latex].
Average (rms) current [latex]I_{\text{rms}}[/latex] and average (rms) voltage [latex]V_{\text{rms}}[/latex] are [latex]I_{\text{rms}} = \frac{I_{0}}{\sqrt{2}}[/latex] and [latex]V_{\text{rms}} = \frac{V_{0}}{\sqrt{2}}[/latex], where rms stands for root mean square.
Thus, [latex]P_{\text{ave}} = I_{\text{rms}} V_{\text{rms}}[/latex].
Ohm’s law for AC is [latex]I_{\text{rms}} = \frac{V_{\text{rms}}}{R}[/latex].
Expressions for the average power of an AC circuit are [latex]P_{\text{ave}} = I_{\text{rms}} V_{\text{rms}}[/latex],[latex]P_{\text{ave}} = \frac{V_{\text{rms}}^{ 2}}{R}[/latex], and[latex]P_{\text{ave}} = I_{\text{rms}}^{ 2} R[/latex], analogous to the expressions for DC circuits.

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 16 .3 lists shock hazards as a function of current.
Figure 16 .22 graphs the threshold current for two hazards as a function of frequency.

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).

The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: [latex]R_{\text{s}} = R_{1} + R_{2} + R_{3} + . . . .[/latex]
Each resistor in a series circuit has the same amount of current flowing through it.
The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula: [latex]\frac{1}{R_{\text{p}}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . .[/latex]
Each resistor in a parallel circuit has the same full voltage of the source applied to it.
The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

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