Concept Summary

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Topic | Discussion | Learning Aids
Electromotive force | 20.1 ELECTROMOTIVE FORCE AND CURRENT There must be at least one source or generator of electrical energy in an electric circuit. The electromotive force (emf) of a generator, such as a battery, is the maximum potential difference (in volts) that exists between the terminals of the generator. |
Current The ampere | The rate of flow of charge is called the electric current. If the rate is constant, the current I is given by

| (20.1)

where Δq is the magnitude of the charge crossing a surface in a time Δt, the surface being perpendicular to the motion of the charge. The SI unit for current is the coulomb per second (C/s), which is referred to as an ampere (A).
Direct current Alternating current | When the charges flow only in one direction around a circuit, the current is called direct current (dc). When the direction of charge flow changes from moment to moment, the current is known as alternating current (ac). |
Conventional current | Conventional current is the hypothetical flow of positive charges that would have the same effect in a circuit as the movement of negative charges that actually does occur. |
Resistance The ohm | 20.2 OHM’S LAW The definition of electrical resistance R is R = V/I, where V (in volts) is the voltage applied across a piece of material and I (in amperes) is the current through the material. Resistance is measured in volts per ampere, a unit called an ohm ().
If the ratio of the voltage to the current is constant for all values of voltage and current, the resistance is constant. In this event, the definition of resistance becomes Ohm’s law as follows:
Ohm’s law |

| (20.2)

| 20.3 RESISTANCE AND RESISTIVITY The resistance of a piece of material of length L and cross-sectional area A is |
Resistance in terms of length and area Resistivity |

| (20.3)

where is the resistivity of the material.
Temperature dependence of resistivity Temperature coefficient of resistivity | The resistivity of a material depends on the temperature. For many materials and limited temperature ranges, the temperature dependence is given by

| (20.4)

where and are the resistivities at temperatures T and T0, respectively, and is the temperature coefficient of resistivity.
Temperature dependence of resistance | The temperature dependence of the resistance R is given by

| (20.5)

where R and R0 are the resistances at temperatures T and T0, respectively.
| 20.4 ELECTRIC POWER When electric charge flows from point A to point B in a circuit, leading to a current I, and the voltage between the points is V, the electric power associated with this current and voltage is |
Electric power |

| (20.6a)

Power delivered to a resistor | For a resistor, Ohm’s law applies, and it follows that the power delivered to the resistor is also given by either of the following two equations:

| (20.6b)

| (20.6c)

Alternating voltage | 20.5 ALTERNATING CURRENT The alternating voltage between the terminals of an ac generator can be represented by

| (20.7)

where V0 is the peak value of the voltage, t is the time, and f is the frequency (in Hertz) at which the voltage oscillates.
Alternating current | Correspondingly, in a circuit containing only resistance, the ac current is

| (20.8)

where I0 is the peak value of the current and is related to the peak voltage via I0 = V0/R.
Root mean square current Root mean square voltage | For sinusoidal current and voltage, the root mean square (rms) current and voltage are related to the peak values according to the following equations:

| (20.12)

| (20.13)

Average ac power | The power in an ac circuit is the product of the current and the voltage and oscillates in time. The average power is

| (20.15a)

Average ac power delivered to a resistor | For a resistor, Ohm’s law applies, so that and the average power delivered to the resistor is also given by the following two equations:

| (20.15b)

| (20.15c)

Equivalent series resistance | 20.6 SERIES WIRING When devices are connected in series, there is the same current through each device. The equivalent resistance RS of a series combination of resistances (R1, R2, R3, etc.) is

| (20.16)

| The power delivered to the equivalent resistance is equal to the total power delivered to any number of resistors in series. |
Equivalent parallel resistance | 20.7 PARALLEL WIRING When devices are connected in parallel, the same voltage is applied across each device. In general, devices wired in parallel carry different currents. The reciprocal of the equivalent resistance RP of a parallel combination of resistances (R1, R2, R3, etc.) is

| (20.17)

| The power delivered to the equivalent resistance is equal to the total power delivered to any number of resistors in parallel. |

Use Self-Assessment Test 20.1 to evaluate your understanding of Sections 20.1, 20.2, 20.3, 20.4, 20.5, 20.6 and 20.7.

| 20.8 CIRCUITS WIRED PARTIALLY IN SERIES AND PARTIALLY IN PARALLEL Sometimes, one section of a circuit is wired in series, while another is wired in parallel. In such cases the circuit can be analyzed in parts, according to the respective series and parallel equivalent resistances of the various sections. |
Internal resistance Terminal voltage | 20.9 INTERNAL RESISTANCE The internal resistance of a battery or generator is the resistance within the battery or generator. The terminal voltage is the voltage between the terminals of a battery or generator and is equal to the emf only when there is no current through the device. When there is a current I, the internal resistance r causes the terminal voltage to be less than the emf by an amount Ir. |
The junction rule | 20.10 KIRCHHOFF’S RULES Kirchhoff’s junction rule states that the sum of the magnitudes of the currents directed into a junction equals the sum of the magnitudes of the currents directed out of the junction. |
The loop rule | Kirchhoff’s loop rule states that, around any closed-circuit loop, the sum of the potential drops equals the sum of the potential rises. The Reasoning Strategy given at the end of Section 20.10 explains how these two rules are applied to analyze any circuit. |
Galvanometer Ammeter Voltmeter | 20.11 THE MEASUREMENT OF CURRENT AND VOLTAGE A galvanometer is a device that responds to electric current and is used in nondigital ammeters and voltmeters. An ammeter is an instrument that measures current and must be inserted into a circuit in such a way that the current passes directly through the ammeter. A voltmeter is an instrument for measuring the voltage between two points in a circuit. A voltmeter must be connected between the two points and is not inserted into a circuit as an ammeter is. |
Equivalent parallel capacitance | 20.12 CAPACITORS IN SERIES AND IN PARALLEL The equivalent capacitance CP for a parallel combination of capacitances (C1, C2, C3, etc.) is

| (20.18)

In general, each capacitor in a parallel combination carries a different amount of charge. The equivalent capacitor carries the same total charge and stores the same total energy as the parallel combination.
Equivalent series capacitance | The reciprocal of the equivalent capacitance CS for a series combination (C1, C2, C3, etc.) of capacitances is

| (20.19)

The equivalent capacitor carries the same amount of charge as any one of the capacitors in the combination and stores the same total energy as the series combination.
Charging a capacitor | 20.13 RC CIRCUITS The charging or discharging of a capacitor in a dc series circuit (resistance R, capacitance C) does not occur instantaneously. The charge on a capacitor builds up gradually, as described by the following equation:

| (20.20)

where q is the charge on the capacitor at time t and q0 is the equilibrium value of the charge. The time constant of the circuit is
Time constant |

| (20.21)

Discharging a capacitor | The discharging of a capacitor through a resistor is described as follows:

| (20.22)

where q0 is the charge on the capacitor at time t = 0 s.

Use Self-Assessment Test 20.2 to evaluate your understanding of Sections 20.8, 20.9, 20.10, 20.11, 20.12 and 20.13.

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