Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 27

Circuits - all with Video Answers

Educators

Chapter Questions

Problem 1

In Fig. 27-14, take $R_{1}=R_{2}=R_{3}=10 \Omega$. If the potential difference across the ideal battery is $\Delta V_{\mathrm{B}}=12 \mathrm{~V}$, find: (a) the equivalent resistance of the circuit and (b) the direction the current flows in the circuit. (c) Which point, $A$ or $B$, is at higher potential?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 2

Batteries Figure 27-15 shows two ideal batteries with $\Delta V_{\mathrm{B} 1}=$ $12 \mathrm{~V}$ and $\Delta V_{\mathrm{B} 2}=8 \mathrm{~V}$. (a) What is the direction of the current in the resistor? (b) Which battery is doing positive work?
(c) Which point, $A$ or $B$, is at the higher potential?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 3

In Fig. $27=14$, take $R_{1}=$ $10 \Omega, R_{2}=15 \Omega$, and $R_{3}=20 \Omega$. If the
potential difference across the ideal battery is $\Delta V_{\mathrm{B}}=15 \mathrm{~V}$, find: $($ a) the equivalent resistance of the circuit,
(b) the current through each of the resistors, and (c) the total current in the circuit.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 4

In Fig. $27-16$, if the potential at point $P$ is $100 \mathrm{~V}$, what is the potential at point $Q ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 5

In Fig. 27-14, take $R_{1}=$ $12 \Omega, R_{2}=15 \Omega$, and $R_{3}=25 \Omega .$ If
the potential difference across the ideal battery is $\Delta V_{\mathrm{B}}=15 \mathrm{~V}$, find the potential differences across each of the resistors

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 6

Figure $27-17$ shows a $6.00 \Omega$ resistor connected to a $12.0 \mathrm{~V}$ battery by means of two copper wires. The wires each have length $20.0 \mathrm{~cm}$ and radius $1.00 \mathrm{~mm}$. In such circuits we generally neglect the potential differences along wires and the transfer of energy to thermal energy in them. Check the validity of this neglect for the circuit of Fig. $27-17:$ What are the potential differences across (a) the resistor and (b) each of the two sections of wire? At what rate is energy lost to thermal energy in
(c) the resistor and (d) each of the two sections of wire?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 7

The current in a single-loop circuit with one resistance $R$ is $5.0 \mathrm{~A}$. When an additional resistance of $2.0 \Omega$ is inserted in series with $R$, the current drops to $4.0 \mathrm{~A}$. What is $R ?$

Salamat Ali

Numerade Educator

Problem 8

A simple ohmmeter is made by connecting an ideal 1.50 V flashlight battery in series with a resistance $R$ and an ammeter that reads from 0 to $1.00 \mathrm{~mA}$, as shown in Fig. 27-18. Resistance $R$ is adjusted so that when the clip leads are shorted together, the meter deflects to its full-scale value of $1.00 \mathrm{~mA}$. What external resistance across the leads results in a deflection of (a) $10 \%$, (b) $50 \%$, and (c) $90 \%$ of full scale?
(d) If the ammeter has a resistance of $20.0 \Omega$ and the internal resistance of the battery is negligible, what is the value of $R ?$

Shelby Mohamed

Numerade Educator

Problem 9

What are the sizes and directions of the currents through resistors (a) $R_{2}$ and (b) $R_{3}$ in Fig. 27-19, where each of the three resistances is $4.0 \Omega ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 10

The resistances in Figs. $27-20 a$ and $b$ are all $6.0 \Omega$, and the batteries are ideal $12 \mathrm{~V}$ batteries (a) When switch $\mathrm{S}$ in Fig. $27=20 a$ is closed, what is the change in the electric potential difference $\Delta V_{B,}$ across resistor 1 , or does $\Delta V_{R_{1}}$ remain the same? (b) When switch $\mathrm{S}$ in Fig. $27-20 b$ is closed what is the change in the electric potential difference $\Delta V_{R_{1}}$ across resistor 1, or does $\Delta V_{R_{1}}$ remain the same?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 11

(a) In Fig. 27-21, what is the equivalent resistance of the network shown? (b) What is the current in each resistor? Put $R_{1}=100 \Omega, R_{2}=R_{3}=50 \Omega, R_{4}=$
$75 \Omega$, and $\Delta V_{\mathrm{B}}=6.0 \mathrm{~V}$; assume the battery is ideal.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 12

Plot 1 in Fig. $27-22 a$ gives the electric potential difference $\Delta V_{R_{1}}$ set up across $R_{1}$ versus the current $i$ that can appear in resistor $1 .$ Plots 2 and 3 are similar plots for resistors 2 and 3 , respectively. Figure $27-22 b$ shows a circuit with those three resistors and a $6.0 \mathrm{~V}$ battery. What is the current in resistor 2 in that circuit?

Shelby Mohamed

Numerade Educator

Problem 13

Two In Fig. $27-23, R=10 \Omega$. What is the equivalent resistance between points $A$ and $B ?$ (Hint: This circuit section might look simpler if you first assume that points $A$ and $B$ are connected to a battery.) tween points $D$ and $E .$ Hint:
Imagine that a battery is connected between points $D$ and $E$.)

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 14

Figure $27=24$ shows a circuit containing three switches, labeled $\mathrm{S}_{1}, \mathrm{~S}_{2}$, and $\mathrm{S}_{3} .$ Find the current at $a$ for all possible combinations of switch settings. Put $\Delta V_{\mathrm{B}}=120 \mathrm{~V}, R_{1}=20.0 \Omega$, and $R_{2}$
$=10.0 \Omega$. Assume that the battery has no resistance.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 15

Two $\quad$ lightbulbs, one of resistance $R_{1}$ and the other of resistance $R_{2}$, are connected to a battery (a) in parallel and (b) in series. Which bulb is brighter in each case if $R_{1}=R_{2}$ ? How is your answer different if $R_{1}>R_{2} ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 16

In Fig. $27=5$ calculate the potential difference be tween points $c$ and $d$ by as many path as possible. Assume that $\Delta V_{\mathrm{B} 1}=4.0 \mathrm{~V}$ $\Delta V_{\mathrm{B} 2}=1.0 \mathrm{~V}, \quad R_{1}=R_{2}=10 \Omega$, an
$R_{3}=5.0 \Omega$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 17

(a) In Fig. $27-25$, deter mine what the ammeter will reac assuming $\Delta V_{\mathrm{B}}=5.0 \mathrm{~V} \quad$ (for $\mathrm{th}$
$\begin{array}{lll}\text { ideal battery), } & R_{1}=2.0 & \Omega,\end{array}$
$R_{2}=4.0 \Omega$, and $R_{3}=6.0 \Omega$. (b) The ammeter and the source of emf are now physically interchanged. Show that the ammeter reading remains unchanged.

Shelby Mohamed

Numerade Educator

Problem 18

In Fig. 27 26, find the equivalent resistance between points (a) $F$ and $H$ and $($ b) $F$ and $G$. (Hint: for each pair of points, imagine that a battery is connected across the pair).

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 19

Each In Fig. $27-27$ find the current in each resistor and the potential difference between points $a$ and $b$. Put $\Delta V_{\mathrm{B} 1}=6.0 \mathrm{~V}$, $\Delta V_{\mathrm{B} 2}=5.0 \mathrm{~V}, \Delta V_{\mathrm{B} 3}=4.0 \mathrm{~V}, R_{1}=$
$100 \Omega$, and $R_{2}=50 \Omega$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 20

By using only two resistors - singly, in series, or in parallel-you are able to obtain resistances of 3.0, 4.0, 12, and $16 \Omega$. What are the two resistances?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 21

A copper wire of radius $a=0.250 \mathrm{~mm}$ has an aluminum jacket of outer radius $b=0.380 \mathrm{~mm} .$ (a) There is a current $i=2.00 \mathrm{~A}$ in the composite wire. Using Table $26-2$, calculate the current in each material. (b) If a potential difference $V=12.0 \mathrm{~V}$ between the ends maintains the current, what is the length of the composite wire?

Shelby Mohamed

Numerade Educator

Problem 22

Between $D$ and $E$ In Fig. $27=28$, find the equivalent resistance between points $D$ and $E$. (Hint:
Imagine that a battery is connected between points $D$ and $E$.)

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 23

Four $18.0 \Omega$ resistors are connected in parallel across a $25.0 \mathrm{~V}$ battery. What is the current through the battery?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 24

(a) In Fig. 2729 , what is the equivalent resistance of the network shown? (b) What is the current in each resistor? Put $R_{1}=100 \Omega, R_{2}=R_{3}=50 \Omega, R_{4}=75 \Omega$, and $\Delta \mathrm{V}_{\mathrm{B}}=6.0 \mathrm{~V} ;$ assume
the battery is ideal.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 25

Nine copper wires of length $l$ and diameter d are connected in parallel to form a single composite conductor of resistance $R$. What must be the diameter $D$ of a single copper wire of length $l$ if it is to have the same resistance?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 26

A voltmeter (of resistance $\left.R_{\Delta \mathrm{V}}\right)$ and an ammeter (of resistance $\left.R_{\mathrm{A}}\right)$ are connected to measure a resistance $R$, as in Fig. $27-30 a$. The resistance is given by $R=\Delta V / i$, where $\Delta V$ is the voltmeter reading and $i$ is the current in the resistance $R$. Some of the current $i^{\prime}$ registered by the ammeter goes through the voltmeter, so that the ratio of the meter readings $\left(=\Delta V / i^{\prime}\right)$ gives only an apparent resistance reading $R^{\prime}$. Show that $R$ and $R^{\prime}$ are related by
$$
\frac{1}{R}=\frac{1}{R^{\prime}}-\frac{1}{R_{\Delta \mathrm{V}}}
$$
Note that as $R_{\Delta \mathrm{V}} \rightarrow \infty, R^{\prime} \rightarrow R$. Ignore $R_{0}$ for now.

Shelby Mohamed

Numerade Educator

Problem 27

If an ammeter and a voltmeter are used to measure resistance, they may also be connected as in Fig. 27-30b. Again the ratio of the meter readings gives only an apparent resistance $R^{\prime}$. Show that now $R^{\prime}$ is related to $R$ by
$$
R=R^{\prime}-R_{\mathrm{A}}
$$
in which $R_{\Lambda}$ is the ammeter resistance. Note that as $R_{\mathrm{A}} \rightarrow 0 \Omega$, $R^{\prime} \rightarrow R$. Ignore $R_{0}$ for now.

Shelby Mohamed

Numerade Educator

Problem 28

In Fig. 27-30, the ammeter and voltmeter resistances are $3.00 \Omega$ and $3.00 \Omega$, respectively. Take $\Delta V_{\mathrm{B}}=12.0 \mathrm{~V}$ for the ideal battery and $R_{0}=100 \Omega$. If $R=85.0 \Omega$, (a) what will the meters read for the two different connections (Figs. $27-30 a$ and $b$ )? (b) What apparent resistance $R^{\prime}$ will be computed in each case?

Shelby Mohamed

Numerade Educator

Problem 29

You are given a number of $10 \Omega$ resistors, each capable of dissipating only $1.0 \mathrm{~W}$ without being destroyed. What is the minimum number of such resistors that you need to combine in series or in parallel to make a $10 \Omega$ resistance that is capable of dissipating at least $5.0 \mathrm{~W}$ ?

Salamat Ali

Numerade Educator

Problem 30

In Fig. 27-31a, resistor 3 is a variable resistor and the battery is an ideal $12 \mathrm{~V}$ battery. Figure $27-31 b$ gives the current $i$ through the battery as a function of $R_{3} .$ The curve has an asymptote of $2.0 \mathrm{~mA}$ as $R_{3} \rightarrow \infty$. What are (a) resistance $R_{1}$ and (b) resistance $R_{2} ?$

Shelby Mohamed

Numerade Educator

Problem 31

Figure $27-32$ shows a section of a circuit. The electric potential difference between points $A$ and $B$ that connect the section to the rest of the circuit is $V_{A}-V_{B}$ $=78 \mathrm{~V}$, and the current through the $6.0 \Omega$ resistor is $6.0 \mathrm{~A}$. Is the device represented by "Box" absorbing or providing energy to the circuit and at what rate?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 32

In Fig. $27-33$, a resistor and an arrangement of $n$ resistors in parallel are connected in series with an ideal battery. All the resistors have the same resistance. If one more identical resistor were added in parallel to the $n$ resistors already in parallel. the current through the battery would change by $1.25 \%$. What is the value of $n$ ?

Shelby Mohamed

Numerade Educator

Problem 33

In Fig. $27-34$, where each resistance is $4.00 \Omega$, what are the sizes and directions of currents (a) $i_{1}$ and (b) $i_{2}$ ? At what rates is energy being transferred at (c) the $4.00 \mathrm{~V}$ battery and (d) the $12.0 \mathrm{~V}$ battery, and for each, is the battery supplying or absorbing energy?

Shelby Mohamed

Numerade Educator

Problem 34

Fig. $27-35 a$ are ideal. $\Delta V_{\mathrm{B} 1}$ of battery 1 has a fixed value but $\Delta V_{\mathrm{B} 2}$ of battery $2 \mathrm{can}$ be varied between $1.0 \mathrm{~V}$ and $10 \mathrm{~V}$. The plots in Fig. $27-35 b$ give the currents through the two batteries as a function of $\Delta V_{\mathrm{B} 2}$. You must decide which plot corresponds to which battery, but for both plots, a negative current occurs when the direction of the current through the battery is opposite the direction of that battery's potential difference. What are (a) $\Delta V_{\mathrm{B} 1}$ (b) resistance $R_{1}$, and (c) resistance $R_{2}$ ?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 35

(a) How much work does an ideal battery with $\Delta V_{\mathrm{B}}=12.0 \mathrm{~V}$ do on an electron that passes through the battery from the positive to the negative terminal? (b) If $3.4 \times 10^{18}$ electrons pass through each second, what is the power of the battery?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 36

Figure $27-36$ shows a portion of a circuit. The rest of the circuit draws current $i$ at the connections $A$ and $B$, as indicated. Take $\Delta V_{B 1}=10 \mathrm{~V}, \Delta V_{\mathrm{B} 2}=15 \mathrm{~V}, R_{1}=R_{2}=5.0 \Omega, R_{3}=$
$R_{4}=8.0 \Omega$, and $R_{5}=12 \Omega$. For each of four values of $i-0,4.0,8.0$, and $12 \mathrm{~A}-$ find the current through each ideal battery and state whether the battery is charging or discharging. Also find the potential difference $\Delta V_{A B}$ between points $A$ and $B$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 37

In Fig. 27-37, $R_{s}$ is to be adjusted in value by moving the sliding contact across it until points $a$ and $b$ are brought to the same potential. (One tests for this condition by momentarily connecting a sensitive ammeter between $a$ and $b$; if these points are at the same potential, the ammeter will not deflect.) Show that when this adjustment is made, the following relation holds:
$$
R_{x}=R_{s}\left(\frac{R_{2}}{R_{1}}\right)
$$
An unknown resistance $\left(R_{x}\right)$ can be measured in terms of a standard $\left(R_{s}\right)$ using this device, which is called a Wheatstone bridge.

Salamat Ali

Numerade Educator

Problem 38

In Fig. 27-38, what are currents (a) $i_{2}$, (b) $i_{4}$, (c) $i_{1}$,
(d) $i_{3}$, and $(\mathrm{e})$ $i_{5} ?$

Manish Haldankar

Manish Haldankar

Numerade Educator

Problem 39

What are the sizes and directions of (a) current $i_{1}$ and (b) current $i_{2}$ in Fig. $27-39$, where each resistance is $2.00 \Omega ?$ (Can you answer this making only mental calculations?) (c) At what rate is energy being transferred in the $5.00 \mathrm{~V}$ battery at the left, and is the energy being supplied or absorbed by the battery?

Shelby Mohamed

Numerade Educator

Problem 40

Size and Direction Three (a) What are the size and direction of
current $i_{1}$ in Fig. $27-40$, where each resistance is $2.0 \Omega$ ? What are the powers of (b) the $20 \mathrm{~V}$ battery, (c) the $10 \mathrm{~V}$ battery, and (d) the $5.0 \mathrm{~V}$ battery, and for each, is energy being supplied or absorbed?

Shelby Mohamed

Numerade Educator

Problem 41

Size and Direction Four (a) What are the size and direction of $\begin{array}{lll}\text { current } i_{1} \text { in Fig. } 27-41 ? & \text { (b) } \text { How }\end{array}$ much energy is dissipated by all four resistors in $1.0 \mathrm{~min} ?$

Shelby Mohamed

Numerade Educator

Problem 42

A $5.0$ A current is set up in a circuit for $6.0$ min by a rechargeable battery with a $6.0 \mathrm{~V}$ emf. By how much is the chemical energy of the battery reduced?

Salamat Ali

Numerade Educator

Problem 43

A standard flashlight battery can deliver about $2.0 \mathrm{~W} \cdot \mathrm{h}$ of energy before it runs down. (a) If a battery costs $80 \mathrm{c}$, what is the cost of operating a $100 \mathrm{~W}$ lamp for $8.0 \mathrm{~h}$ using batteries? (b) What is the cost if energy is provided at $12 \phi$ per kilowatt-hour?

Shelby Mohamed

Numerade Educator

Problem 44

Power is supplied by a device of emf $\mathscr{8}$ to a transmission line with resistance $R$. Find the ratio of the power dissipated in the line for $\mathscr{E}=110000 \mathrm{~V}$ to that dissipated for $\mathscr{E}=$ $110 \mathrm{~V}$, assuming the power supplied is the same for the two cases.

Shelby Mohamed

Numerade Educator

Problem 45

A certain car battery with a $12 \mathrm{~V}$ emf has an initial charge of $120 \mathrm{~A} \cdot \mathrm{h}$. Assuming that the potential across the terminals stays constant until the battery is completely discharged, for how long can it deliver energy at the rate of $100 \mathrm{~W}$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 46

A wire of resistance $5.0 \Omega$ is connected to a battery whose emf $\mathscr{8}$ is $2.0 \mathrm{~V}$ and whose internal resistance is $1.0 \Omega$. In $2.0 \mathrm{~min}$, (a) how much energy is transferred from chemical to electrical form? (b) How much energy appears in the wire as thermal energy? (c) Account for the difference between (a) and (b).

Shelby Mohamed

Numerade Educator

Problem 47

Assume that the batteries in Fig. $27-42$ have negligible internal resistance. Find
(a) the current in the circuit, (b) the power dissipated in each resistor, and (c) the power of each battery, stating whether energy is supplied by or absorbed by it.

Shelby Mohamed

Numerade Educator

Problem 48

In Fig. $27-43 a$, both batteries have emf $\mathscr{B}=1.20 \mathrm{~V}$ and the external resistance $R$ is a variable resistor. Figure $27-43 b$ gives the electric potentials $\Delta V_{T}$ between the terminals of each battery as functions of $R$ : Curve 1 corresponds to battery 1 and curve 2 corresponds to battery $2 .$ What are the internal resistances of (a) battery 1 and (b) battery $2 ?$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 49

The following table gives the electric potential difference $\Delta V_{T}$ across the terminals of a battery as a function of current $i$ being drawn from the battery. (a) Write an equation that represents the relationship between the terminal potential difference $\Delta V_{T}$ and the current $i$. Enter the data into your graphing calculator and perform a linear regression fit of $\Delta V_{T}$ versus $i$. From the parameters of the fit, find (b) the battery's emf and (c) its internal resistance. $\begin{array}{lccccccc}i(\mathrm{~A}): & 50 & 75 & 100 & 125 & 150 & 175 & 200 \\ \Delta V_{T}(\mathrm{~V}): & 10.7 & 9.0 & 7.7 & 6.0 & 4.8 & 3.0 & 1.7\end{array}$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 50

In Fig. 27-11a, put $\mathscr{8}=2.0 \mathrm{~V}$ and $r=100 \Omega$. Plot
(a) the current and (b) the potential difference across $R$, as functions of $R$ over the range 0 to $500 \Omega$. Make both plots on the same graph. (c) Make a third plot by multiplying together, for various values of $R$, the corresponding values on the two plotted curves. What is the physical significance of this third plot?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 51

A car battery with a $12 \mathrm{~V}$ emf and an internal resistance of $0.040 \Omega$ is being charged with a current of $50 \mathrm{~A}$.
(a) What is the potential difference across its terminals? (b) At what rate is energy being dissipated as thermal energy in the battery? (c) At what rate is electric energy being converted to chemical energy? (d) What are the answers to (a) and (b) when the battery is used to supply $50 \mathrm{~A}$ to the starter motor?

Shelby Mohamed

Numerade Educator

Problem 52

What Value of $\boldsymbol{R}$ (a) In Fig. 27 -
44 , what value must $R$ have if the current in the circuit is to be $1.0 \mathrm{~mA} ?$ Take $\mathscr{E}_{1}=2.0 \mathrm{~V}, \mathscr{B}_{2}=$
$3.0 \mathrm{~V}$, and $r_{1}=r_{2}=3.0 \Omega .$ (b) What is the rate at which thermal energy appears in $R ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 53

In Fig. $27-45$, circuit section $A B$ absorbs energy at a rate of $50 \mathrm{~W}$ when a current $i=1.0$ A passes through it in the indicated direction. (a) What is the potential difference between $A$ and $B ?$ (b) emf device $X$ does not have internal resistance. What is its emf? (c) What is its polarity (the orientation of its positive and negative terminals)?

Shelby Mohamed

Numerade Educator

Problem 54

When the lights of an automobile are switched on, an ammeter in series with them reads $10 \mathrm{~A}$ and a voltthem reads $12 \mathrm{~V}$. See Fig. $27-46$. When the electric starting motor is turned on, the ammeter reading drops to $8.0 \mathrm{~A}$ and the lights $\mathrm{dim}$ somewhat. If the internal resistance of the battery is $0.050 \Omega$ and that of the ammeter is negligible, what are (a) the emf of the battery and (b) the current through the starting motor when the lights are on?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 55

Two batteries having the same emf 8 but different internal resistances $r_{1}$ and $r_{2}\left(r_{1}>r_{2}\right)$ are connected in series to an external resistance $R$. (a) Find the value of $R$ that makes the potential difference zero between the terminals of one battery. (b) Which battery is it?

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 56

The starting motor of an automobile is turning too slowly, and the mechanic has to decide whether to replace the motor, the cable, or the battery. The manufacturer's manual says that the $12 \mathrm{~V}$ battery should have no more than $0.020 \Omega$ internal resistance, the motor no more than $0.200 \Omega$ resistance, and the cable no more than $0.040 \Omega$ resistance. The mechanic turns on the motor and measures $11.4 \mathrm{~V}$ across the battery, $3.0 \mathrm{~V}$ across the cable, and a current of $50 \mathrm{~A}$. Which part is defective?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 57

(a) In Fig. $27-11 a$, show that the rate at which energy is dissipated in $R$ as thermal energy is a maximum when $R=r .$ (b) Show that this maximum power is $P=\mathscr{E}^{2} / 4 r$.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 58

A solar cell generates a potential difference of $0.10$ $\mathrm{V}$ when a $500 \Omega$ resistor is connected across it, and a potential difference of $0.15 \mathrm{~V}$ when a $1000 \Omega$ resistor is substituted. What are
(a) the internal resistance and (b) the emf of the solar cell? (c) The area of the cell is $5.0 \mathrm{~cm}^{2}$, and the rate per unit area at which it receives energy from light is $2.0 \mathrm{~mW} / \mathrm{cm}^{2} .$ What is the efficiency of the cell for converting light energy to thermal energy in the $1000 \Omega$ external resistor?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 59

Two batteries of emf $\mathscr{E}$ and internal resistance $r$ are connected in parallel across a resistor $R$, as in Fig. 27 $47 a$. (a) For what value of $R$ is the rate of electrical energy dissipation by the resistor a maximum? (b) What is the maximum energy dissipation rate?

Konstantin Pavlovskii

Konstantin Pavlovskii

Numerade Educator

Problem 60

You are given two batteries of emf $\mathscr{E}$ and internal resistance $r$. They may be connected either in parallel (Fig. $27-47 a$ ) or in series (Fig. $27-47 b$ ) and are to be used to establish a current in a resistor $R$. (a) Derive expressions for the current in $R$ for both arrangements. Which will yield the larger current (b) when $R>r$ and $(\mathrm{c})$ when $R<r ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 61

In Fig. 27-48, $\mathscr{E}_{1}=3.00 \mathrm{~V}, \mathscr{E}_{2}=1.00 \mathrm{~V}$, $R_{1}=5.00 \Omega, R_{2}=2.00 \Omega, R_{3}=4.00$
$\Omega$, and both batteries are ideal. What is the rate at which energy is dissipated in (a) $R_{1}$, (b) $R_{2}$, and (c) $R_{3}$ ? What is the power of (d) battery 1 and (e) battery $2 ?$

Keshav Singh

Numerade Educator

Problem 62

Value of $R$ In the circuit of Fig. $27 \cdot 49$, for what value of $R$ will the ideal battery transfer energy to the resistors (a) at a rate of $60.0$ $\mathrm{W},(\mathrm{b})$ at the maximum possible rate, and (c) at the minimum possible rate? (d) What are those rates?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 63

(a) Calculate the current through each ideal battery in Fig. $27-50 .$ Since the batteries are ideal $\mathscr{B}=\Delta V_{\mathrm{B}}$ in each case. Assume that $R_{1}=1.0 \Omega$, $R_{2}=2.0 \Omega, \mathscr{E}_{1}=2.0 \mathrm{~V}$ and $\mathscr{E}_{2}=$
$\mathscr{S}_{3}=4.0 \mathrm{~V} .$ (b) Calculate $V_{a}-V_{b}$

Keshav Singh

Numerade Educator

Problem 64

In the circuit of Fig. $27-51,8$ has a constant value but $R$ can be varied. Find the value of $R$ that results in the maximum heating in that resistor. The battery is ideal.

Aniket Bajaj

Numerade Educator

Problem 65

For the circuit in Fig. $27-52$, indicate whether the statements are true or false. If a statement is false, give a correct statement.
(a) Some of the current is used up when the bulb is lit; the current in wire $B$ is smaller than the current in wire $\bar{A}$.
(b) A current probe will have the same readings if connected to read the current in wire $A$ or wire $B .$ The current flows from the battery, through wire $A$, through the bulb, and then back to the battery through wire $B$.
(c) The current flows toward the bulb in both wires $A$ and $B$.
(d) The (positive) current flows from the battery, through wire $A$, and then back to the battery through wire $B$.
(e) If wire $A$ is left connected but wire $B$ is disconnected, the bulb will still light.

Mayukh Banik

Numerade Educator

Problem 66

Use the Model (a) Use our model for
electric current to rank the networks shown in Fig. $27-53$ in order by resistance. Explain your reasoning. (b) If a battery were connected to each of the circuits, in which case would the current through the battery be the largest? The smallest? Explain your reasoning.

Vishal Gupta

Numerade Educator

Problem 67

Examine the circuits shown in Fig. 27 . 54 and indicate whether you think each of the following two statements are true or false. Please explain your reasoning.
(a) Circuits 1 and 2 are different. The brightness of the two bulbs in circuit 1 are the same, but in circuit 2 the bulb closest to the battery in brighter than the bulb that is further away.
(b) Circuit diagrams only show electrical connections, so the drawings in circuits 1 and 2 are electrically equivalent and the brightness of the two bulbs is the same in both circuits 1 and 2 .

Chris Johnson

Numerade Educator

Problem 68

(a) Identify which of the nice, neat circuit diagrams $(A, B, C$, or $D)$ in Fig. $27-55 c$ corresponds to the messy circuit drawing in Fig. $27-55 a$. Explain the reasons for your answer.
(b) Which neat circuit diagram corresponds to the messy circuit drawing in Fig. 27-55b. Explain the reasons for your answer.

Sikandar Baig

Numerade Educator

Problem 69

(a) For the circuit in Fig. 27-56, at which point $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ or $\mathrm{E}$ is the voltage the lowest? Explain. (b) At which point is the potential energy of a positive charge the highest? Explain. (c) At which point is the current the largest? Explain.

Andrew Duncan

Numerade Educator

Problem 70

Bulbs 1 Through 6 (a) For the circuit shown in Fig. $27-57$, rank bulbs 1 through 6 in order of descending brightness. Explain the reasoning for your ranking. (b) Now assume that the filament of lightbulb 6 breaks. Again rank the bulbs in order of descending brightness. Explain the reasoning for your ranking.

Sheh Lit Chang

University of Washington

Problem 71

The circuit diagram in Fig. $27-58$ shows two unlabeled resistors attached to identical bulbs. Ex-
plain how you would interpret the brightness of bulbs $A$ and $B$ to decide which resistor is larger.

Sanu Kumar

Numerade Educator

Problem 72

Which of the three circuits shown in Fig. $27-59$, if any, are electrically identical? Which are different? Explain your answers.

Shelby Mohamed

Numerade Educator

Problem 73

Bulb Examine the circuit shown in Fig. 27 60 . (a) Rank the bulbs according to brightness and explain your reasoning. (b) How will the brightness of bulbs 1 and 3 change if bulb 4 is unscrewed? Explain. (c) How will the brightness of bulbs $1,3,5$, and 6 change if a conducting wire is connected between points $A$ and $F$ ? Explain.

Hunza Gilgit

Numerade Educator

Problem 74

Examine the circuit shown in Fig. 27-61. (a) Assume that the switch is open. State which bulbs or combination of bulbs are in series, and in parallel. (b) Assume that the switch is closed. State whether the bulbs in the circuit are arranged in series or parallel.

Colin Gold

Numerade Educator

Problem 75

Two Examine the circuit shown in Fig. 27-62. (a) Assume that the switch is open. Rank the bulbs according to brightness and explain your reasoning.
(b) Assume that the switch is closed. Rank the bulbs according to brightness and explain your reasoning.

Luis Rios

Numerade Educator

Problem 76

If the batteries in Fig. $27-63$ are identical, which circuit draws more current? Circuit $A ?$ Circuit $B ?$ Neither? Show your calculations and reasoning.

Manish Kumar

Numerade Educator

Problem 77

In the circuits shown in Fig. $27-64$, state which resistors are connected in series with which other resistors, which are connected in parallel with which other resistors, and which are neither in series nor parallel.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

Lots of Batteries and a Bulb Figure $27-65$ shows identical batteries connected in different arrangements to the same lightbulb. Assume the batteries have negligible internal resistances. The positive terminal of each battery is marked with a plus. Rank these arrangements on the basis of bulb brightness from the highest to the lowest. Please explain your reasoning.

Sheh Lit Chang

University of Washington

Problem 79

We have studied batteries that provide a fixed voltage across their terminals. In that case, we had to examine our circuit and use our physical principles in order to calculate the current through the battery. In neuroscience, it is sometimes useful to use a constant current source (CCS), which instead provides a fixed amount of current through itself. In this case, we have to use our physical principles in order to calculate the voltage drop across the source.
Suppose we have a constant current source (denoted CSS) that always provides a current of $i_{c}=10^{-6}$ amps. For the three circuits shown in Fig. $27-66$, find the voltage drop across the current source. Each resistor has a resistance $R=2000 \Omega$. (If you prefer, you may leave your answer in terms of the symbols $i_{c}$ and $R$.)

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 80

Circuit The circuit shown in Fig. $27-67$ contains an ideal battery and three resistors. The battery has an emf of $1.5 \mathrm{~V}, R_{1}=2 \Omega, R_{2}=3 \Omega$, and $R_{3}=5 \Omega .$ Also shown in Fig. 27-67 is a graph tracking some quantity around the circuit. Make threecopies of this graph. On the first, plot the voltage a test charge would experience as it moved through the circuit. On the second, plot the electric field a test charge would experience as it moved through the circuit. On the third, plot the current one would measure crossing a plane perpendicular to the wire of the circuit as one goes through the circuit.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 81

(From a homework set in a graduate course in synaptic physiology) As a result of a complex set of biochemical reactions, the cell membrane of a nerve cell pumps ions $\left(\mathrm{Na}^{+}\right.$ and $\left.\mathrm{K}^{+}\right)$ back and forth across itself, thereby maintaining an electrostatic potential difference from the inside to the outside of the membrane. Modifications on the conditions can result in changes in those potentials. Part of the process can be modeled by treating the membrane as if it were a simple electric circuit consisting of batteries, resistors, and a switch. A simple model of the membrane of $\underline{a}$ nerve cell is shown in Fig. $27-68$. It consists of two batteries (ion pumps) with voltages $\Delta V_{1}=$ $100 \mathrm{mV}$ and $V_{2}=50 \mathrm{mV}$. The resistance to flow across the membrane is represented by two resistors with resistances $R_{1}=10 \mathrm{~K} \Omega$ and $R_{2}=90 \mathrm{~K} \Omega$. The variability is represented by a switch, $S_{1}$.

Four points on the circuit are labeled by the letters $a-d$. The point $b$ represents the outside of the membrane and the point $d$ the inside of the membrane.
(a) What is the voltage difference across the membrane (i.e., between $d$ and $b$ ) when the switch is open?
(b) What is the current flowing around the loop when the switch is closed?
(c) What is the voltage drop across the resistor $R_{1}$ when the switch is open? Closed?
(d) What is the voltage drop across the resistor $R_{2}$ when the switch is open? Closed?
(e) What is the potential difference across the membrane (i.e., between $d$ and $b$ ) when the switch is closed?
(f) If the locations of resistances $R_{1}$ and $R_{2}$ were reversed, would the voltages across the cell membrane be different?

Shoukat Ali

Other Schools

Problem 82

Consider the circuit in Fig. 27-69. (a) Apply the junction rule to junctions $d$ and $a$ and the loop rule to the three loops to produce five simultaneous, linearly independent equations. (b) Represent the five linear equations by the matrix equation $[A][B]=[C]$
where
$$
[B]=\left[\begin{array}{c}
i_{1} \\
i_{2} \\
i_{3} \\
i_{4} \\
i_{5}
\end{array}\right]
$$
What are the matrices $[A]$ and $[C] ?$ (c) Have the calculator perform $[A]^{-1}[C]$ to find the values of $i_{1}, i_{2}, i_{3}, i_{4}$, and $i_{5}$

PK

Pramod Kumar

Numerade Educator

Problem 83

Currents For the same situation as in Problem 82 and having already solved for the five unknown currents, do the following. (a) Find the electric potential difference across the $9 \Omega$ resistor. (b) Find the rate at which work is being done on the $7 \Omega$ resistor. (c) Find the rate at which the $12 \mathrm{~V}$ battery is doing work on the circuit. (d) Find the rate at which the $4 \mathrm{~V}$ battery is doing work on the circuit. (e) Of the points in the circuit labeled $a$ and $c$, which is at the higher electric potential?

Mayukh Banik

Numerade Educator