College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 18

Electric Current and Circuits - all with Video Answers

Educators

Chapter Questions

Problem 1

A battery charger delivers a current of $3.0 \mathrm{~A}$ for $4.0 \mathrm{~h}$ to a 12-V storage battery. What is the total charge that passes through the battery in that time?

Sarah Chapman

Sarah Chapman

Numerade Educator

Problem 2

The current in a wire is $0.500$ A. (a) How much charge flows through a cross section of the wire in $10.0 \mathrm{~s}$ ?
(b) How many electrons move through the same cross section in $10.0 \mathrm{~s}$ ?

TH

Thaxter Hensley

Numerade Educator

Problem 3

(a) What is the direction of the current in the vacuum tube shown in the figure? (b) Electrons hit the anode at a rate of $6.0 \times 10^{12}$ per second. What is the current in the tube?

Sarah Chapman

Numerade Educator

Problem 4

In an ion accelerator, $3.0 \times 10^{13}$ helium-4 nuclei (charge $+2 e$ ) per second strike a target. What is the beam current?

TH

Thaxter Hensley

Numerade Educator

Problem 5

The current in the electron beam of a computer monitor is $320 \mu \mathrm{A}$. How many electrons per second hit the screen?

Sarah Chapman

Numerade Educator

Problem 6

A potential difference is applied between the electrodes in a gas discharge tube. In $1.0 \mathrm{~s}, 3.8 \times 10^{16}$ electrons and $1.2 \times 10^{16}$ singly charged positive ions move in opposite directions through a surface perpendicular to the length of the tube. What is the current in the tube?

TH

Thaxter Hensley

Numerade Educator

Problem 7

Two electrodes are placed in a calcium chloride solution and a potential difference is maintained between them. If $3.8 \times 10^{16} \mathrm{Ca}^{2+}$ ions and $6.2 \times 10^{16} \mathrm{Cl}^{-}$ ions per second move in opposite directions through an imaginary area between the electrodes, what is the current in the solution?

Sarah Chapman

Numerade Educator

Problem 8

A Vespa scooter and a Toyota automobile might both use a 12-V battery, but the two batteries are of different sizes and can pump different amounts of charge. Suppose the scooter battery can pump $4.0 \mathrm{kC}$ of charge and the automobile battery can pump $30.0 \mathrm{kC}$ of charge. How much energy can each battery deliver, assuming the batteries are ideal?

TH

Thaxter Hensley

Numerade Educator

Problem 9

What is the energy stored in a small battery if it can move 675 C through a potential difference of $1.20 \mathrm{~V} ?$

Sarah Chapman

Numerade Educator

Problem 10

The label on a $12.0-\mathrm{V}$ truck battery states that it is rated at $180.0 \mathrm{~A} \cdot \mathrm{h}$ (ampere-hours). Treat the battery as ideal.
(a) How much charge in coulombs can be pumped by the battery? [Hint: Convert $\mathrm{A} \cdot \mathrm{h}$ to $\mathrm{A} \cdot \mathrm{s} .]$ (b) How much electric energy can the battery supply? (c) Suppose the radio in the truck is left on when the engine is not running. The radio draws a current of $3.30 \mathrm{~A}$. How long does it take to drain the battery if it starts out fully charged?

Km Neeraj

Numerade Educator

Problem 11

The starter motor in a car draws $220.0 \mathrm{~A}$ of current from the $12.0$ -V battery for $1.20 \mathrm{~s}$. (a) How much charge is pumped by the battery? (b) How much electric energy is supplied by the battery?

Sarah Chapman

Numerade Educator

Problem 12

A solar cell provides an emf of $0.45 \mathrm{~V}$. (a) If the cell supplies a constant current of $18.0 \mathrm{~mA}$ for $9.0 \mathrm{~h}$, how much electric energy does it supply? (b) What is the power-the rate at which it supplies electric energy?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 13

Two copper wires, one double the diameter of the other, have the same current flowing through them. If the thinner wire has a drift speed $v_{1}$, and the thicker wire has a drift speed $v_{2}$, how do the drift speeds of the charge carriers compare?

Mayukh Banik

Numerade Educator

Problem 14

A current of $2.50 \mathrm{~A}$ is carried by a copper wire of radius $1.00 \mathrm{~mm}$. If the density of the conduction electrons is $8.47 \times 10^{28} \mathrm{~m}^{-3}$, what is the drift speed of the conduction electrons?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 15

A current of $10.0 \mathrm{~A}$ is carried by a copper wire of diameter $1.00 \mathrm{~mm}$. If the density of the conduction electrons is $8.47 \times 10^{28} \mathrm{~m}^{-3}$, how long does it take for a conduction electron to move $1.00 \mathrm{~m}$ along the wire?

Mayukh Banik

Numerade Educator

Problem 16

A silver wire of diameter $1.0 \mathrm{~mm}$ carries a current of $150 \mathrm{~mA}$. The density of conduction electrons in silver is $5.8 \times 10^{28} \mathrm{~m}^{-3}$. How long (on average) does it take for a conduction electron to move $1.0 \mathrm{~cm}$ along the wire?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 17

A strip of doped silicon $260 \mu \mathrm{m}$ wide contains $8.8 \times 10^{22}$ conduction electrons per cubic meter and an insignificant number of holes. When the strip carries a current of $130 \mu \mathrm{A}$, the drift speed of the electrons is $44 \mathrm{~cm} / \mathrm{s}$. What is the thickness of the strip?

Mayukh Banik

Numerade Educator

Problem 18

A gold wire of $0.50 \mathrm{~mm}$ diameter has $5.90 \times 10^{28}$ conduction electrons $/ \mathrm{m}^{3}$. If the drift speed is $6.5 \mu \mathrm{m} / \mathrm{s}$, what is the current in the wire?

Km Neeraj

Numerade Educator

Problem 19

A copper wire of cross-sectional area $1.00 \mathrm{~mm}^{2}$ has a current of $2.0$ A flowing along its length. What is the drift speed of the conduction electrons? Assume $1.3$ conduction electrons per copper atom. The mass density of copper is $9.0 \mathrm{~g} / \mathrm{cm}^{3}$ and its atomic mass is $64 \mathrm{~g} / \mathrm{mol}$.

Km Neeraj

Numerade Educator

Problem 20

An aluminum wire of diameter $2.6 \mathrm{~mm}$ carries a current of $12 \mathrm{~A}$. How long on average does it take an electron to move $12 \mathrm{~m}$ along the wire? Assume $3.5$ conduction electrons per aluminum atom. The mass density of aluminum is $2.7 \mathrm{~g} / \mathrm{cm}^{3}$ and its atomic mass is $27 \mathrm{~g} / \mathrm{mol}$.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 21

A $12-\Omega$ resistor has a potential difference of $16 \mathrm{~V}$ across
it. What current flows through the resistor?

Mayukh Banik

Numerade Educator

Problem 22

Current of $83 \mathrm{~mA}$ flows through the resistor in the diagram. (a) What is the resistance of the resistor? (b) In what direction does the current flow through the resistor?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 23

A copper wire and an aluminum wire of the same length have the same resistance. What is the ratio of the diameter of the copper wire to that of the aluminum wire?

Mayukh Banik

Numerade Educator

Problem 24

A bird sits on a high-voltage power line with its feet $2.0 \mathrm{~cm}$ apart. The wire is made from aluminum, is $2.0 \mathrm{~cm}$ in diameter, and carries a current of $150 \mathrm{~A}$. What is the potential difference between the bird's feet?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 25

A person can be killed if a current as small as $50 \mathrm{~mA}$ passes near the heart. An electrician is working on a humid day with hands damp from perspiration. Suppose his resistance from one hand to the other is $1 \mathrm{k} \Omega$ and he is touching two wires, one with each hand. (a) What potential difference between the two wires would cause a $50-\mathrm{mA}$ current from one hand to the other? (b) An electrician working on a "live" circuit keeps one hand behind his or her back. Why?

Mayukh Banik

Numerade Educator

Problem 26

An electric device has the current-voltage $(I-V)$ graph shown. What is its resistance at (a) point 1 and (b) point 2?

Km Neeraj

Numerade Educator

Problem 27

If $46 \mathrm{~m}$ of nichrome wire is to have a resistance of $10.0 \Omega$ at $20^{\circ} \mathrm{C}$, what diameter wire should be used?

Mayukh Banik

Numerade Educator

Problem 28

The resistance of a conductor is $19.8 \Omega$ at $15.0^{\circ} \mathrm{C}$ and $25.0 \Omega$ at $85.0^{\circ} \mathrm{C}$. What is the temperature coefficient of resistance of the material?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 29

A common flashlight bulb is rated at $0.300 \mathrm{~A}$ and $2.90 \mathrm{~V}$ (the values of current and voltage under operating con. ditions). If the resistance of the bulb's tungsten filament at room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ is $1.10 \Omega$, estimate the temperature of the tungsten filament when the bulb is turned on.

Mayukh Banik

Numerade Educator

Problem 30

Find the maximum current that a fully charged D-cell can supply - if only briefly-such that its terminal voltage is at least $1.0 \mathrm{~V}$. Assume an emf of $1.5 \mathrm{~V}$ and an internal resistance of $0.10 \Omega$.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 31

A battery has a terminal voltage of $12.0 \mathrm{~V}$ when no current flows. Its internal resistance is $2.0 \Omega$. If a $1.0-\Omega$ resistor is connected across the battery terminals, what is the terminal voltage and what is the current through the $1.0-\Omega$ resistor?

Mayukh Banik

Numerade Educator

Problem 32

(a) What are the ratios of the resistances of (a) silver and (b) aluminum wire to the resistance of copper wire $\left(R_{\mathrm{Ag}} / R_{\mathrm{Cu}}\right.$ and $\left.R_{\mathrm{A} 1} / R_{\mathrm{Cu}}\right)$ for wires of the same length and the same diameter? (c) Which material is the best conductor, for wires of equal length and diameter?

Narayan Hari

Numerade Educator

Problem 33

. A wire with cross-sectional area $A$ carries a current $I$. Show that the electric field strength $E$ in the wire is proportional to the current per unit area (I/A) and identify the constant of proportionality. [Hint: Assume a length $L$ of wire. How is the potential difference across the wire related to the electric field in the wire? (Which is uniform?) Use $V=I R$ and the connection between resistance and resistivity.]

Mayukh Banik

Numerade Educator

Problem 34

A copper wire has a resistance of $24 \Omega$ at $20^{\circ} \mathrm{C}$. An aluminum wire has three times the length and twice the radius of the copper wire. The resistivity of copper is $0.6$ times that of aluminum. Both Al and Cu have temperature coefficients of resistivity of $0.004^{\circ} \mathrm{C}^{-1}$. (a) What is the resistance of the aluminum wire at $20^{\circ} \mathrm{C} ?$ (b) The graph shows a $V-I$ plot for the copper wire. What is the resistance of the wire when operating steadily at a current of $10 \mathrm{~A} ?$ (c) What must the temperature of the copper wire have been when operating at $10 \mathrm{~A}$ ? Ignore changes in the wire's dimensions.

Km Neeraj

Numerade Educator

Problem 35

Refer to Problem 34 . With the copper wire connected to an ideal battery, the current increases greatly when the wire is immersed in liquid nitrogen. Ignoring changes in the wire's dimensions, state whether each of the following quantities increases, decreases, or stays the same as the wire is cooled: the electric field in the wire, the resistivity, and the drift speed. Explain your answers.

Mayukh Banik

Numerade Educator

Problem 36

Suppose a collection of five batteries is connected as shown.
(a) What is the equivalent emf of the collection? Treat them as ideal sources of emf. (b) What is the current through the resistor if its value is $3.2 \Omega$ ?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 37

Suppose four batteries are connected in series as shown.
(a) What is the equivalent emf of the set of four batteries? Treat them as ideal sources of emf. (b) If the current in the circuit is $0.40 \mathrm{~A}$, what is the value of the resistor $\underline{R}$ ?

Mayukh Banik

Numerade Educator

Problem 38

(a) Find the equivalent capacitance between points $A$ and $B$ for the three capacitors. (b) What is the charge on the $6.0-\mu \mathrm{F}$ capacitor if a $44.0-\mathrm{V} \mathrm{emf}$ is connected to the terminals $A$ and $\bar{B}$ for a long time?

Narayan Hari

Numerade Educator

Problem 39

(a) Find the equivalent capacitance between points $A$ and $B$ for the five capacitors. (b) If a 16.0-V emf is connected to the terminals $A$ and $B$, what is the charge on a single equivalent capacitor that replaces all five?
(c) What is the charge on the $3.0-\mu \mathrm{F}$ capacitor?

Narayan Hari

Numerade Educator

Problem 40

(a) What is the equivalent resistance between points $A$ and $B ?$ (b) A 276-V emf is connected to the terminals $A$ and $B$. What is the current in the $12-\Omega$ resistor?

Ivan Kochetkov

Numerade Educator

Problem 41

(a) What is the equivalent resistance between points $A$ and $B$ if $R=1.0 \Omega ?(\mathrm{~b})$ If a $20-\mathrm{V} \mathrm{emf}$ is connected to the terminals $A$ and $B$, what is the current in the $2.0-\Omega$ resistor?

Ivan Kochetkov

Numerade Educator

Problem 42

If a $93.5-\mathrm{V}$ emf is connected to the terminals $A$ and $B$ and the current in the $4.0-\Omega$ resistor is $17 \mathrm{~A}$, what is the value of the unknown resistor $R ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 43

(a) What is the equivalent capacitance between points $A$ and $B$ if $C=1.0 \mu \mathrm{F}$ ? (b) What is
the charge on the $4.0-\mu \mathrm{F}$ capacitor when it is fully charged?

Ivan Kochetkov

Numerade Educator

Problem 44

The equivalent capacitance between points $A$ and $B$ is $1.63 \mu \mathrm{F}$. (a) What is the capacitance of the unknown capacitor $C ?$ (b) What is the charge on the $4.0-\mu \mathrm{F}$ capacitor when it is fully charged?

Ivan Kochetkov

Numerade Educator

Problem 45

A $24-V$ emf is connected to the terminals $A$ and $B$.
(a) What is the current in one of the $2.0-\Omega$ resistors?
(b) What is the current in the $6.0-\Omega$ resistor? (c) What is the current in the leftmost $4.0-\Omega$ resistor?

Mayukh Banik

Numerade Educator

Problem 46

(a) Find the equivalent resistance between points $A$ and $B$ for the combination of resistors shown. (b) An $18-\mathrm{V}$ $\mathrm{emf}$ is connected to the terminals $A$ and $B$. What is the current through the $1.0-\Omega$ resistor connected directly to point $A$ ? (c) What is the current in the $8.0-\Omega$ resistor?

Shahab Ullah

Numerade Educator

Problem 47

(a) What is the resistance between points $A$ and $B$ ? Each resistor has the same resistance $R$. [Hint: Redraw the circuit.] (b) What is the resistance between points $\underline{B}$ and $C ?(\mathrm{c})$ If a $32-\mathrm{V} \mathrm{emf}$ is connected to terminals $\bar{A}$ and $B$ and if each $R=2.0 \Omega$, what is the current in one of the resistors?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 48

(a) Find the equivalent resistance between points $A$ and $\bar{B}$ for the combination of resistors shown. (b) What is the potential difference across each of the $4.0-\Omega$ resistors? (c) What is the current in the $3.0-\Omega$ resistor?

Vishal Gupta

Numerade Educator

Problem 49

(a) Find the value of $\mathrm{a}$ single capacitor to replace the three capacitors in the diagram. (b) What is the potential difference across the 12 left side of the diagram? (c) What is the charge on the $12-\mu \mathrm{F}$ capacitor to the far right side of the circuit?

Shahab Ullah

Numerade Educator

Problem 50

A $6.0$ -pF capacitor is needed to construct a circuit. The only capacitors available are rated as $9.0 \mathrm{pF}$. How can a combination of three $9.0-\mathrm{pF}$ capacitors be assembled so that the equivalent capacitance of the combination is $60 \mathrm{nF}^{\circ}$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 51

(a) Find the equivalent resistance between terminals $A$ and $B$ to replace all of the resistors in the diagram. (b) What current flows through the emf?
(c) What is the current through the $4.00-\Omega$ resistor at the bottom?

Mayukh Banik

Numerade Educator

Problem 52

Find the current in each branch of the circuit. Specify the direction of each.

Cal Wilkens

Numerade Educator

Problem 53

Find the current in each branch of the circuit. Specify the direction of each.

Cal Wilkens

Numerade Educator

Problem 54

Find the unknown emf and the unknown currents in the circuit.

Km Neeraj

Numerade Educator

Problem 55

Find the unknown emf and the unknown resistor in the circuit.

Vishal Gupta

Numerade Educator

Problem 56

The figure shows a simplified circuit diagram for an automobile. The equivalent resistor $R$ represents the total electrical load due to spark plugs, lights, radio, fans, starter, rear window defroster, and the like in par-allel. If $R=0.850 \Omega$, find the current in
cach branch. What is the terminal voltage of the battery? Is the battery charging or discharging?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 57

What is the power dissipated by the resistor in the circuit if the $\operatorname{cmf}$ is $2.00 \mathrm{~V} ?$

Ivan Kochetkov

Numerade Educator

Problem 58

Refer to the figure with Problem 57. What is the power dissipated by the resistor in the circuit if $R=5.00 \Omega$ ?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 59

What is the current in a $60.0-\mathrm{W}$ bulb when connected to a 120 -V emf?

Mayukh Banik

Numerade Educator

Problem 60

What is the resistance of a $40.0-\mathrm{W}, 120-\mathrm{V}$ lightbulb?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 61

If a chandelier has a label stating $120 \mathrm{~V}, 5.0 \mathrm{~A}$, can its power rating be determined? If so, what is it?

Mayukh Banik

Numerade Educator

Problem 62

A portable CD player does not have a power rating listed. but it has a label stating that it draws a maximum current of $2.50 .0 \mathrm{~mA}$. The player uses three $1.50-\mathrm{V}$ batteries connected in series. What is the maximum power consumed?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 63

How much work are the batteries in the circuit doing in every $10.0$ -s time interval?

Ivan Kochetkov

Numerade Educator

Problem 64

Show that $\mathrm{A}^{2} \times \Omega=\mathrm{W}$ (amperes squared times ohms $=$ watts).

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 65

Consider the circuit in the diagram. (a) Draw the simplest equivalent circuit and label the values of the resistor(s).
(b) What current flows from the battery? (c) What is the potential difference between points $A$ and $B ?$ (d) What current flows through each branch between points $A$ and $B$ ? (e) Determine the power dissipated in the $50.0-\Omega$ resistor, the $70.0-\Omega$ resistor, and the $40.0$ $\Omega$ resistor.

Mayukh Banik

Numerade Educator

Problem 66

(a) What is the equivalent resistance of this circuit if $R_{1}=10.0 \Omega$ and $R_{2}=15.0 \Omega ?$ (b) What current flows through $R_{1} ?$ (c) What is the voltage drop across $R_{2}$ ?
(d) What current flows through $R_{2}$ ?
(e) How much power is dissipated in $R_{2}$ ?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 67

In her bathroom, Mindy has an overhead heater that consists of a coiled wire made of nichrome that gets hot when turned on. The wire has a length of $3.0 \mathrm{~m}$ when it is uncoiled. The heating element is attached to the normal $120-V$ wiring and when the wire is glowing red hot it has a temperature of about $420^{\circ} \mathrm{C}$ and dissipates $2200 \mathrm{~W}$ of power. Nichrome has a resistivity of $108 \times 10^{-8} \Omega \cdot \mathrm{m}$ at $20^{\circ} \mathrm{C}$ and a temperature coefficient of resistivity of $0.00040^{\circ} \mathrm{C}^{-1}$. (a) What is the resistance of the heater when it is turned on?
(b) What current does the wire carry? (c) If the wire has a circular cross section, what is its diameter? Ignore the small changes in the wire's diameter and length due to changes in temperature. (d) When the heater is first turned on, it has not yet heated up, so it is operating at $20^{\circ} \mathrm{C}$. What is the current through the wire when it is first turned on?

Mayukh Banik

Numerade Educator

Problem 68

At what rate is electric energy converted to internal energy in the $4.00-\Omega$ and $5.00-\Omega$ resistors in the figure?

Mayukh Banik

Numerade Educator

Problem 69

A battery has a $6.00-\mathrm{V} \mathrm{emf}$ and an internal resistance of $0.600 \Omega .$ (a) What is the voltage across its terminals when the current drawn from the battery is $1.20 \mathrm{~A} ?$
(b) What is the power supplied by the battery?

Mayukh Banik

Numerade Educator

Problem 70

During a "brownout," which occurs when the power companies cannot keep up with high demand, the voltage of the household circuits drops below its normal $120 \mathrm{~V}$. (a) If the voltage drops to $108 \mathrm{~V}$, what would be the power consumed by a "100-W" lightbulb (that is, a lightbulb that consumes $100.0 \mathrm{~W}$ when connected to $120 \mathrm{~V}$ )? Ignore (for now) changes in the resistance of the lightbulb filament. (b) More realistically, the lightbulb filament will not be as hot as usual during the brownout. Does this make the power drop more or less than that you calculated in part (a)? Explain.

Km Neeraj

Numerade Educator

Problem 71

A source of emf $\mathscr{E}$ has internal resistance $r$. (a) What is the terminal voltage when the source supplies a current $I ?(\mathrm{~b})$ The net power supplied is the terminal voltage times the current. Starting with $P=I \Delta V$, derive Eq. (18-22) for the net power supplied by the source. Interpret each of the two terms, (c) Suppose that a battery of emf $\mathscr{8}$ and internal resistance $r$ is being recharged:
another emf sends a current $I$ through the battery in the reverse direction (from positive terminal to negative). At what rate is electric energy converted to chemical energy in the recharging battery? (d) What is the power supplied by the recharging circuit to the battery?

Mayukh Banik

Numerade Educator

Problem 72

Redraw the circuit in Problem 40 to show how an ammeter would be connected to measure (a) the current through the $15-\Omega$ resistor and (b) the current through the $24-\Omega$ resistor.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 73

Redraw the circuit in Problem 40 to show how a voltmeter would be connected to measure (a) the potential drop across the $15-\Omega$ resistor and (b) the potential drop across the $24-\Omega$ resistor.

Mayukh Banik

Numerade Educator

Problem 74

a) Redraw the circuit to show how an ammeter would be connected to measure the current through the $1.40-\mathrm{k} \Omega$ resistor. (b) Assuming the ammeter to be ideal, what is its reading? (c) If the ammeter has a resistance of $120 \Omega$, what is its reading?

Mayukh Banik

Numerade Educator

Problem 75

(a) Redraw the circuit to show how a voltmeter would be connected to measure the voltage across the $83.0-\mathrm{k} \Omega$ resistor. (b) Assuming the voltmeter to be ideal, what is its reading? (c) If the voltmeter has a resistance of $1.00 \mathrm{M} \Omega$, what is its reading?

Ivan Kochetkov

Numerade Educator

Problem 76

A galvanometer has a coil resistance of $50.0 \Omega$. It is to be made into an ammeter with a full-scale deflection equal to $10.0 \mathrm{~A}$. If the galvanometer deflects full scale for a current of $0.250 \mathrm{~mA}$, what size shunt resistor should be used?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 77

A galvanometer has a coil resistance of $34.0 \Omega$. It is to be made into a voltmeter with a full-scale deflection equal to $100.0 \mathrm{~V}$. If the galvanometer deflects full scale for a current of $0.120 \mathrm{~mA}$, what size resistor should be placed in series with the galvanometer?

Mayukh Banik

Numerade Educator

Problem 78

A galvanometer is to be turned into a voltmeter that deflects full scale for a potential difference of $100.0 \mathrm{~V}$. What size resistor should be placed in series with the galvanometer if it has an internal resistance of $75 \Omega$ and deflects full scale for a current of $2.0 \mathrm{~mA}$ ?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 79

Many voltmeters have a switch by which one of several series resistors can be selected. Thus, the same meter can be used with different full-scale voltages. What size series resistors should be used in the voltmeter of Problem 78 to give it full-scale voltages of (a) $50.0 \mathrm{~V}$ and (b) $500.0 \mathrm{~V} ?$

Mayukh Banik

Numerade Educator

Problem 80

An ammeter with a full scale deflection for $I=10.0 \mathrm{~A}$ has an internal resistance of $24 \Omega$. We need to use this ammeter to measure currents up to $12.0 \mathrm{~A}$. The lab instructor advises that we get a resistor and use it to protect the ammeter. (a) What size resistor do we need and how should it be connected to the ammeter, in series or in parallel? (b) How do we interpret the ammeter readings?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 81

A voltmeter has a switch that enables voltages to be measured with a maximum of $25.0 \mathrm{~V}$ or $10.0 \mathrm{~V}$. For a range of voltages to $25.0 \mathrm{~V}$, the switch connects a resistor of magnitude $9850 \Omega$ in series with the galvanometer; for a range of voltages to $10.0 \mathrm{~V}$, the switch connects a resistor of magnitude $3850 \Omega$ in series with the galvanometer. Find the coil resistance of the galvanometer and the galvanometer current that causes a full-scale deflection.

Mayukh Banik

Numerade Educator

Problem 82

In the circuit, $R=30.0 \mathrm{k} \Omega$ and $C=0.10 \mu \mathrm{F}$. The capacitor is allowed to charge fully and then the switch is changed from position $a$ to position $b$. What will the voltage across the resistor be $8.4 \mathrm{~ms}$ later?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 83

In the circuit shown, assume the battery emf is $20.0 \mathrm{~V}$, $R=1.00 \mathrm{M} \Omega$, and $C=2.00 \mu \mathrm{F}$.
The switch is closed at $t=0 .$ At what time $t$ will the voltage across the capacitor be $15.0 \mathrm{~V}$ ?

Ivan Kochetkov

Numerade Educator

Problem 84

A charging $R C$ circuit controls the intermittent windshield wipers in a car. The emf is $12.0 \mathrm{~V}$. The wipers are triggered when the voltage across the $125-\mu \mathrm{F}$ capacitor reaches $10.0 \mathrm{~V}$; then the capacitor is quickly discharged (through a much smaller resistor) and the cycle repeats. What resistance should be used in the charging circuit if the wipers are to operate once every $1.80 \mathrm{~s} ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 85

A capacitor is charged to an initial voltage $V_{0}=9.0 \mathrm{~V}$. The capacitor is then discharged by connecting its terminals through a resistor. The current $I(t)$ through this resistor, determined by measuring the voltage $V_{\mathrm{R}}(t)=I(t) R$ with an oscilloscope, is shown in the graph. (a) Find the capacitance $C$, the resistance $R$, and the total energy dissipated in the resistor. (b) At what time is the energy in the capacitor half its initial value? (c) Graph the voltage across the capacitor, $V_{C}(t)$, as a function of time.

Mayukh Banik

Numerade Educator

Problem 86

A defibrillator passes a brief burst of current through the heart to restore normal beating. In one such defibrillator, a $50.0-\mu \mathrm{F}$ capacitor is charged to $6.0 \mathrm{kV}$. Paddles are used to make an electric connection to the patient's chest. A pulse of current lasting $1.0 \mathrm{~ms}$ partially discharges the capacitor through the patient. The electrical resistance of the patient (from paddle to paddle) is $240 \Omega$. (a) What is the initial energy stored in the capacitor? (b) What is the initial current through the patient?
(c) How much energy is dissipated in the patient during the $1.0 \mathrm{~ms} ?$ (d) If it takes $2.0 \mathrm{~s}$ to recharge the capacitor, compare the average power supplied by the power source with the average power delivered to the patient.
(c) Referring to your answer to part (d), explain one reason a capacitor is used in a defibrillator.

Mayukh Banik

Numerade Educator

Problem 87

Capacitors are used in many applications where one needs to supply a short burst of relatively large current. A $100.0-\mu \mathrm{F}$ capacitor in an electronic flash lamp supplies a burst of current that dissipates $20.0 \mathrm{~J}$ of energy (as light and heat) in the lamp. (a) To what potential difference must the capacitor initially be charged?
(b) What is its initial charge? (c) Approximately what is the resistance of the lamp if the current reaches $5.0 \%$ of its original value in $2.0 \mathrm{~ms}$ ?

Mayukh Banik

Numerade Educator

Problem 88

Consider the circuit shown with $R_{1}=25 \Omega, R_{2}=33 \Omega$, $C_{1}=12 \mu \mathrm{F}, C_{2}=23 \mu \mathrm{F}, C_{3}=46 \mu \mathrm{F}, \quad$ and $\quad V=6.0 \mathrm{~V}$
(a) Draw an equivalent circuit with one resistor and one capacitor and label it with the values of the equivalent resistor and capacitor. (b) A long time after switch $S$ is closed, what are the charge on capacitor $C_{1}$ and the current in resistor $R_{1} ?$ (c) What is the time constant of the circuit? (d) At what time after switch $S$ is closed is the voltage across the combination of three capacitors $50 \%$ of its final value?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 89

In the circuit, the capacitor is initially uncharged. At $t=0$ switch $S$ is closed. Find the currents $I_{1}$ and $I_{2}$ and voltages $V_{1}$ and $V_{2}$ (assuming $V_{3}=0$ ) at points 1 and 2 at the following times:
(a) $t=0$ (i.c., just after the switch is closed), (b) $t=1.0 \mathrm{~ms}$, and (c) $t=5.0 \mathrm{~ms}$.

Km Neeraj

Numerade Educator

Problem 90

In the circuit, the initial energy stored in the capacitor is $25 \mathrm{~J}$. At $t=0$ the switch is closed. (a) Sketch a graph of the voltage across the resistor $\left(V_{R}\right)$ as a function of $t$. Label the vertical axis with key numerical value(s) and units. (b) At what time is the energy stored in the capacitor $1.25 \mathrm{~J} ?$

Mayukh Banik

Numerade Educator

Problem 91

A $20-\mu \mathrm{F}$ capacitor is discharged through a $5-\mathrm{k} \Omega$ resis tor. The initial charge on the capacitor is $200 \mu \mathrm{C}$.
(a) Sketch a graph of the current through the resistor as a function of time. Label both axes with numbers and units. (b) What is the initial power dissipated in the resistor? (c) What is the total energy dissipated?

Mayukh Banik

Numerade Educator

Problem 92

a) In a charging $R C$ circuit, how many time constants have elapsed when the capacitor has $99.0 \%$ of its fina charge? (b) How many time constants have clapsed when the capacitor has $99.90 \%$ of its final charge?
(c) How many time constants have elapsed when the current has $1.0 \%$ of its initial value?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 93

A capacitor is charged by a $9.0-\mathrm{V}$ battery. The charging current $I(t)$ is shown. (a) What, approximately, is the total charge on the capacitor in the end?

Km Neeraj

Numerade Educator

Problem 94

A charged capacitor is discharged through a resistor. The current $I(t)$ through this resistor, determined by measuring the voltage $V_{\mathrm{R}}(t)=I(t) R$ with an oscilloscope, is shown in the graph. The total energy dissipated in the resistor is $2.0 \times 10^{-4} \mathrm{~J}$. (a) Find the capacitance $C$, the resistance $R$, and the initial charge on the capacitor.

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator

Problem 95

In the physics laboratory, Oscar measured the resistance between his hands to be $2.0 \mathrm{k} \Omega$. Being curious by nature, he then took hold of two conducting wires that were connected to the terminals of an emf with a terminal voltage of $100.0 \mathrm{~V}$, (a) What current passes through Oscar? (b) If one of the conducting wires is grounded and the other has an alternate path to ground through a $15-\Omega$ resistor (so that Oscar and the resistor are in parallel), how much current would pass through Oscar if the maximum current that can be drawn from the emf is $1.00 \mathrm{~A}$ ?

Mayukh Banik

Numerade Educator

Problem 96

Chelsea inadvertently bumps into a set of batterics with an emf of $100.0 \mathrm{~V}$ that can supply a maximum power of 5.0 W. If the resistance between the points where she contacts the batteries is $1.0 \mathrm{k} \Omega$, how much current passes through her?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 97

The wiring circuit for a typical room is shown schematically. (a) Of the six locations for a circuit breaker indicated by $A, B, C, D, E$, and $F$, which one would best protect the household against a short circuit in any one of the three appliances? Explain. (b) The room circuit is supplied with $120 \mathrm{~V}$. Suppose the heater draws $1500 \mathrm{~W}$, the lamp draws $300 \mathrm{~W}$, and the microwave draws $1200 \mathrm{~W}$. The circuit breaker is rated at $20.0 \mathrm{~A}$. Can all three devices be operated simultaneously without tripping the breaker? Explain.

Mayukh Banik

Numerade Educator

Problem 98

Several possibilities are listed for what might or might not happen if the insulation in the current-carrying wires of the figure breaks down and point $b$ makes electric contact with point $c$. Discuss each possibility. (a) The person touching the microwave oven gets a shock;
(b) the cord begins to smoke; (c) a fuse blows out;
(d) an electrical fire breaks out inside the kitchen wall.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 99

A $1.5-V$ flashlight battery can maintain a current of $0.30 \mathrm{~A}$ for $4.0 \mathrm{~h}$ before it is exhausted. How much chemical energy is converted to electrical energy in this process? (Assume zero internal resistance of the battery.)

Mayukh Banik

Numerade Educator

Problem 100

In the diagram, the positive terminal of the $12-V$ battery is grounded-it is at zero potential. At what potential is point $X$ ?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 101

$\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ represent ammeters with negligible resistance. What are the values of the currents (a) in $\mathrm{A}_{1}$ and
(b) in $\bar{A}_{2}$ ?

Vishal Gupta

Numerade Educator

Problem 102

Repeat Problem 101 if each of the ammeters has resistance $0.200 \Omega$.

Mayukh Banik

Numerade Educator

Problem 103

In a pacemaker used by a heart patient, a capacitor with a capacitance of $25 \mu \mathrm{F}$ is charged to $1.0 \mathrm{~V}$ and then discharged through the heart every $0.80 \mathrm{~s}$. What is the average discharge current?

Mayukh Banik

Numerade Educator

Problem 104

A certain electric device has the current-voltage $(I-V)$ graph shown with Problem 26. What is the power dissipated at points 1 and $2 ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 105

A $1.5$ -horsepower motor operates on $120 \mathrm{~V}$. Ignoring $I^{2} R$ losses, how much current does it draw?

Mayukh Banik

Numerade Educator

Problem 106

(a) Given two identical, ideal batteries $(\mathrm{emf}=\mathscr{E})$ and two identical lightbulbs (resistance $=R$ assumed constant), design a circuit to make both bulbs glow as brightly as possible. (b) What is the power dissipated by each bulb? (c) Design a circuit to make both bulbs glow, but one more brightly than the other. Identify the brighter bulb.

Km Neeraj

Numerade Educator

Problem 107

Two circuits are constructed using identical, ideal batteries $(\mathrm{emf}=\mathscr{B})$ and identical lightbulbs (resistance $=R$ ). If each bulb in circuit 1 dissipates $5.0 \mathrm{~W}$ of power, how much power does each bulb in circuit 2 dissipate? Ignore changes in the resistance of the bulbs due to temperature changes.

Mayukh Banik

Numerade Educator

Problem 108

Given two identical, ideal batteries of emf $\mathscr{E}$ and two identical lightbulbs of resistance $R$ (assumed constant), find the total power dissipated in the circuit in terms of $\mathscr{E}$ and $R$.

Km Neeraj

Numerade Educator

Problem 109

Consider a $60.0$ -W lightbulb and a $100.0$ -W lightbulb designed for use in a household lamp socket at $120 \mathrm{~V}$.
(a) What are the resistances of these two bulbs? (b) If they are wired together in a series circuit, which bulb shines brighter (dissipates more power)? Explain.
(c) If they are connected in parallel in a circuit, which bulb shines brighter? Explain.

Mayukh Banik

Numerade Educator

Problem 110

A 500-W electric heater unit is designed to operate with an applied potential difference of $120 \mathrm{~V}$. (a) If the local power company imposes a voltage reduction to lighten its load, dropping the voltage to $110 \mathrm{~V}$, by what percentage does the heat output of the heater drop? (Assume the resistance does not change.) (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than calculated in part (a)?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 111

The Wheatstone bridge is a circuit used to measure unknown resistances. The bridge in the figure is balanced-no current flows through the galvanometer. (a) What is the unknown resistance $R_{x} ?$ [Hint: What is the potential difference between points $A$ and $B ?]$
(b) Does the resistance of the galvanometer affect the measurement? Explain.

Mayukh Banik

Numerade Educator

Problem 112

In the circuit shown, an emf of $150 \mathrm{~V}$ is connected across a
resistance network. What is the current through $R_{2}$ ? Each of the resistors has a value of $10 \Omega$.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 113

(a) What is the resistance of the heater element in a $1500-W$ hair dryer that plugs into a $120-V$ outlet?
(b) What is the current through the hair dryer when it is turned on? (c) At a cost of $$\$ 0.10$$ per $\mathrm{kW} \cdot \mathrm{h}$, how much does it cost to run the hair dryer for $5.00 \mathrm{~min} ?$ (d) If you were to take the hair dryer to Europe where the voltage is $240 \mathrm{~V}$, how much power would your hair dryer be using in the brief time before it is ruined? (e) What current would be flowing through the hair dryer during this time?

Narayan Hari

Numerade Educator

Problem 114

A string of 25 decorative lights has bulbs rated at $9.0 \mathrm{~W}$ and the bulbs are connected in parallel. The string is connected to a $120-V$ power supply. (a) What is the resistance of each of these lights? (b) What is the current through each bulb? (c) What is the total current coming from the power supply? (d) The string of bulbs has a fuse that will blow if the current is greater than $2.0$ A. How many of the bulbs can you replace with $10.4-\mathrm{W}$ bulbs without blowing the fuse?

Mayukh Banik

Numerade Educator

Problem 115

A $2.00-\mu \mathrm{F}$ capacitor is charged using a $5.00-\mathrm{V}$ battery and a $3.00-\mu \mathrm{F}$ capacitor is charged using a $10.0-\mathrm{V}$ battery. (a) What is the total energy stored in the two capacitors? (b) The batteries are disconnected and the two capacitors are connected together (+ to + and $-$ to $-$ ). Find the charge on each capacitor and the total energy in the two capacitors after they are connected. (c) Explain what happened to the "missing" energy.

Vishal Gupta

Numerade Educator

Problem 116

Three identical lightbulbs are connected with wires to an ideal battery. The two terminals on cach socket connect to the two terminals of its lightbulb. Wires do not connect with one another where they appear to cross in the picture. Ignore the change of the resistances of the filaments due to temperature changes.
(a) Which of the schematic circuit diagrams correctly represent(s) the circuit? (List more than one choice if more than one diagram is correct.) (b) Which bulb(s) is/are the brightest? Which is/are the dimmest? Or are they all the same? Explain. (c) Find the current through each bulb if the filament resistances are each $24.0 \Omega$ and the emf is $6.0 \mathrm{~V}$.

Mayukh Banik

Numerade Educator

Problem 117

A portable radio requires an emf of $4.5 \mathrm{~V}$. Olivia has only two nonrechargeable $1.5-V$ batteries, but she finds ? larger $6.0-\mathrm{V}$ battery. (a) How can she arrange the batteries to produce an emf of $4.5 \mathrm{~V}$ ? Draw a circuit diagram. (b) Is it advisable to use this combination with her radio? Explain.

Ivan Kochetkov

Numerade Educator

Problem 118

We can model some of the electrical properties of an unmyelinated axon as an electric cable covered with defective insulation so that current leaks out of the axon to the surrounding fluid. We assume the axon consists of a cylindrical membrane filled with conducting fluid. A current of ions can travel along the axon in this fluid and can also leak out through the membrane. The inner radius of the cylinder is $5.0 \mu \mathrm{m} ;$ the membrane thickness is $8.0 \mathrm{~nm}$. (a) If the resistivity of the axon fluid is 2.0 $\Omega \cdot \mathrm{m}$, calculate the resistance of a $1.0-\mathrm{cm}$ length of axon to current flow along its length. (b) If the resistivity of the porous membrane is $2.5 \times 10^{7} \Omega \cdot \mathrm{m}$, calculate the resistance of the wall of a $1.0-\mathrm{cm}$ length of axon to current flow across the membrane. (c) Find the length of axon for which the two resistances are equal. This length is a rough measure of the distance a signal can travel without amplification.

Mayukh Banik

Numerade Educator

Problem 119

A piece of gold wire of length $L$ has a resistance $R_{0}$. Suppose the wire is drawn out so that its length increases by a factor of three. What is the new resistance $R$ in terms of the original resistance?

Mayukh Banik

Numerade Educator

Problem 120

A voltmeter with a resistance of $670 \mathrm{k} \Omega$ is used to measure the voltage across the $83.0-\mathrm{k} \Omega$ resistor in the figure with Problems 74 and 75 . What is the voltmeter reading?

Mayukh Banik

Numerade Educator

Problem 121

A gold wire and an aluminum wire have the same dimensions and carry the same current. The electron density (in electrons/cm') in aluminum is three times larger than the density in gold. How do the drift specds of the electrons in the two wires, $v_{\mathrm{Au}}$ and $v_{\mathrm{A}}$, compare?

Mayukh Banik

Numerade Educator

Problem 122

Copper and aluminum are being considered for the cables in a high-voltage transmission line where each must carry a current of $50 \mathrm{~A}$. The resistance of each cable is to be $0.15 \Omega$ per kilometer. (a) If this line carries power from Niagara Falls to New York City (approximately $500 \mathrm{~km}$ ), how much power is lost along the way in the cable? Compute for each choice of cable material (b) the necessary cable diameter and
(c) the mass per meter of the cable. The electrical resistivities for copper and aluminum are given in Table $18.1$; the mass density of copper is $8920 \mathrm{~kg} / \mathrm{m}^{3}$ and that of aluminum is $2702 \mathrm{~kg} / \mathrm{m}^{3}$.

Mayukh Banik

Numerade Educator

Problem 123

The circuit is used to study the charging of a capacitor.
(a) At $t=0$, the switch is closed. What initial charging current is measured by the ammeter? (b) After the current has decayed to zero, what are the voltages at points $A, B$, and $C ?$

Mayukh Banik

Numerade Educator

Problem 124

A parallel plate capacitor is constructed from two square conducting plates of length $L=0.10 \mathrm{~m}$ on a side. There is air between the plates, which are separated by a distance $d=89 \mu \mathrm{m} .$ The capacitor is connected to a $10.0-\mathrm{V}$ battery. (a) After the capacitor is fully charged, what is the charge on the upper plate?
(b) The battery is disconnected from the plates and the capacitor is discharged through a resistor $R=0.100 \mathrm{M} \Omega .$ Sketch the current through the resistor as a function of time $t$ ( $t=0$ corresponds to the time when $R$ is connected to the capacitor). (c) How much energy is dissipated in $R$ over the whole discharging process?

Mayukh Banik

Numerade Educator

Problem 125

About $5.0 \times 10^{4} \mathrm{~m}$ above Earth's surface, the atmosphere is sufficiently ionized that it behaves as a conductor. The Earth and the ionosphere form a giant spherical capacitor, with the lower atmosphere acting as a leaky dielectric. (a) Find the capacitance $C$ of the Earth-ionosphere system by treating it as a parallel plate capacitor. Why is it OK to do that? [Hint: Compare Earth's radius to the distance between the "plates."] (b) The fair-weather electric ficld is about $150 \mathrm{~V} / \mathrm{m}$, downward. How much energy is stored in this capacitor? (c) Due to radioactivity and cosmic rays, some air molecules are ionized even in fair weather. The resistivity of air is roughly $3.0 \times 10^{14} \Omega \cdot \mathrm{m}$. Find the resistance of the lower atmosphere and the total current that flows between Earth's surface and the ionosphere. [Hint: Since we treat the system as a parallel plate capacitor, treat the atmosphere as a dielectric of uniform thickness between the plates.]
(d) If there were no lightning, the capacitor would discharge. In this model, how much time would elapse before Earth's charge were reduced to $1 \%$ of its normal value? (Thunderstorms are the sources of emf that
maintain the charge on this leaky capacitor.)

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 126

Near Earth's surface the air contains both negative and positive ions, due to radioactivity in the soil and cosmic rays from space. As a simplified model, assume there are $600.0$ singly charged positive ions per $\mathrm{cm}^{3}$ and $500.0$ singly charged negative ions per $\mathrm{cm}^{3}$; ignore the presence of multiply charged ions. The electric field is $100.0 \mathrm{~V} / \mathrm{m}$, directed downward.
(a) In which direction do the positive ions move? The negative ions? (b) What is the direction of the current due to these ions? (c) The measured resistivity of the air in the region is $4.0 \times 10^{13} \Omega \cdot \mathrm{m} .$ Calculate the drift speed of the ions, assuming it to be the same for positive and negative ions. [Hint: Consider a vertical tube of air of length $L$ and cross-sectional area $A$. How is the potential difference across the tube related to the electric field strength?] (d) If these conditions existed over the entire surface of the Earth, what is the total current due to the movement of ions in the air?

Mayukh Banik

Numerade Educator

Problem 127

A battery with an emf of $1.0 \mathrm{~V}$ is connected to a $1.0-\mathrm{k} \Omega$ resistor and a diode (a nonohmic device) as shown in part (a) of the figure. The current that flows through the diode for a given voltage drop is shown in part
(b) of the figure. (a) What is the current through the diode? (b) What is the current through the battery?
(c) What is the total power dissipated in the diode and resistor? (d) Suppose the battery emf were increased so that the power dissipated in the $1.0-\mathrm{k} \Omega$ resistor doubled. Would you expect the power dissipated in the diode to double? If not, would it increase by a factor greater than 2 or less than 2 ? Explain briefly.

Mayukh Banik

Numerade Educator

Problem 128

Poiseuille's law [Eq. $(9-15)]$ gives the volume flow rate of a viscous fluid through a pipe. (a) Show that Poiseuille's law can be written in the form $\Delta P=I R$, where $I=\Delta V / \Delta t$ represents the volume flow rate and $R$ is a constant of proportionality called the fluid flow resistance. (b) Find $R$ in terms of the viscosity of the fluid and the length and radius of the pipe. (c) If two or more pipes are connected in series so that the volume flow rate through them is the same, do the resistances of the pipes add as for electrical resistors $\left(R_{\mathrm{eq}}=R_{1}+R_{2}+\cdots\right) ?$ Explain.
(d) If two or more pipes are connected in parallel, so the pressure drop across them is the same, do the reciprocals of the resistances add as for electrical resistors $\left(1 / R_{\mathrm{eq}}=\right.$ $\left.1 / R_{1}+1 / R_{2}+\cdots\right) ?$ Explain.

Km Neeraj

Numerade Educator