Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 27

Circuits - all with Video Answers

Educators

Chapter Questions

Problem 1

In Fig. 27.8 , the ideal batteries have emfs $\mathscr{E}_1=12 \mathrm{~V}$ and $\mathscr{E}_2=$ $6.0 \mathrm{~V}$. What are (a) the current, the dissipation rate in (b) resistor $1(4.0 \Omega)$ and (c) resistor $2(8.0 \Omega)$, and the energy transfer rate in (d) battery 1 and (e) battery 2 ? Is energy being supplied or absorbed by (f) battery 1 and $(\mathrm{g})$ battery 2 ?
FIGURE CANT COPY
Fig. 27.8

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

In Fig. 27.9, the ideal batteries have emfs $\mathscr{E}_1=150 \mathrm{~V}$ and $\mathscr{E}_2=50 \mathrm{~V}$ and the resistances are $R_1=3.0 \Omega$ and $R_2=2.0 \Omega$. If the potential at $P$ is $100 \mathrm{~V}$, what is it at $Q$ ?
FIGURE CANT COPY
Fig. 27.9

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 3

A car battery with a $12 \mathrm{~V}$ emf and an internal resistance of $0.040 \Omega$ is being charged with a current of $50 \mathrm{~A}$. What are (a) the potential difference $V$ across the terminals, (b) the rate $P_r$ of energy dissipation inside the battery, and (c) the rate $P_{\mathrm{emf}}$ of energy conversion to chemical form? When the battery is used to supply $50 \mathrm{~A}$ to the starter motor, what are (d) $V$ and (e) $P_r$ ?

Salamat Ali

Numerade Educator

Problem 4

Figure 27.10 shows a circuit of four resistors that are connected to a larger circuit. The graph below the circuit shows the electric potential $V(x)$ as a function of position $x$ along the lower branch of the circuit, through resistor 4; the potential $V_A$ is $12.0 \mathrm{~V}$. The graph above the circuit shows the electric potential $V(x)$ versus position $x$ along the upper branch of the circuit, through resistors 1,2 , and 3 ; the potential differences are $\Delta V_B=2.00 \mathrm{~V}$ and $\Delta V_C=5.00 \mathrm{~V}$. Resistor 3 has a resistance of $200 \Omega$. What is the resistance of (a) resistor 1 and (b) resistor 2 ?
FIGURE CANT COPY
Fig. 27.10

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 5

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 6

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 US $$\$ 0.80$$, 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 the rate of US $$\$ 0.06$$ per kilowatt-hour?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 7

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

Salamat Ali

Numerade Educator

Problem 8

A certain car battery with a $12.0 \mathrm{~V}$ 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 many hours can it deliver energy at the rate of $100 \mathrm{~W}$ ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 9

(a) In electron-volts, how much work does an ideal battery with a $12.0 \mathrm{~V}$ emf do on an electron that passes through the battery from the positive to the negative terminal? (b) If $3.40 \times 10^{18}$ electrons pass through each second, what is the power of the battery in watts?

Salamat Ali

Numerade Educator

Problem 10

(a) In Fig. 27.11, what value must $R$ have if the current in the circuit is to be $1.0 \mathrm{~mA}$ ? Take $\mathscr{E}_1=2.0 \mathrm{~V}, \mathscr{E}_2=3.0 \mathrm{~V}$, and $r_1=r_2=3.0 \Omega$. (b) What is the rate at which thermal energy appears in $R$ ?
FIGURE CANT COPY
Fig. 27.11

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 11

In Fig. 27.12, circuit section $A B$ absorbs energy at a rate of $50 \mathrm{~W}$ when current $i=1.0 \mathrm{~A}$ through it is in the indicated direction. Resistance $R=2.0 \Omega$. (a) What is the potential difference between $A$ and $B$ ? Emf device $X$ lacks internal resistance. (b) What is its emf? (c) Is point $B$ connected to the positive terminal of $X$ or to the negative terminal?
FIGURE CANT COPY
Fig. 27.12

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 12

Figure 27.13 shows a resistor of resistance $R=6.00 \Omega$ connected to an ideal battery of emf $\mathscr{E}=12.0 \mathrm{~V}$ by means of two copper wires. Each wire has length $20.0 \mathrm{~cm}$ and radius $1.00 \mathrm{~mm}$. In dealing with such circuits in this chapter, we generally neglect the potential differences along the wires and the transfer of energy to thermal energy in them. Check the validity of this neglect for the circuit of Fig. 27.13: What is the potential difference 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 section of wire?
FIGURE CANT COPY
Fig. 27.13

Keshav Singh

Numerade Educator

Problem 13

A $10-\mathrm{km}$-long underground cable extends east to west and consists of two parallel wires, each of which has resistance $13 \Omega / \mathrm{km}$. An electrical short develops at distance $x$ from the west end when a conducting path of resistance $R$ connects the wires (Fig. 27.14). The resistance of the wires and the short is then $100 \Omega$ when measured from the east end and $200 \Omega$ when measured from the west end. What are (a) $x$ and (b) $R$ ?
FIGURE CANT COPY
Fig. 27.14

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 14

In Fig. 27.15a, both batteries have emf $\mathscr{E}=1.20 \mathrm{~V}$ and the external resistance $R$ is a variable resistor. Figure $27.15 b$ gives the electric potentials $V$ between the terminals of each battery as functions of $R$ : Curve 1 corresponds to battery 1 , and curve 2 corresponds to battery 2 . The horizontal scale is set by $R_s=0.20 \Omega$. What is the internal resistance of (a) battery 1 and (b) battery 2 ?
FIGURE CANT COPY
Fig. 27.15

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 15

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 16

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 the (a) internal resistance and (b) 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 17

In Fig. 27.16, battery 1 has emf $\mathscr{E}_1=12.0 \mathrm{~V}$ and internal resistance $r_1=0.016 \Omega$ and battery 2 has emf $\mathscr{E}_2=12.0 \mathrm{~V}$ and internal resistance $r_2=0.012 \Omega$. The batteries are connected in series with an external resistance $R$. (a) What $R$ value makes the terminal-to-terminal potential difference of one of the batteries zero? (b) Which battery is that?
FIGURE CANT COPY
Fig. 27.16

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 18

In Fig. 27.2.1, what is the potential difference $V_d-V_c$ between points $d$ and $c$ if $\mathscr{E}_1=4.0 \mathrm{~V}, \mathscr{E}_2=1.0 \mathrm{~V}, R_1=R_2=10 \Omega$, and $R_3=5.0 \Omega$, and the batteries are ideal?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 19

A total resistance of $3.00 \Omega$ is to be produced by connecting an unknown resistance to a $12.0 \Omega$ resistance. (a) What must be the value of the unknown resistance, and (b) should it be connected in series or in parallel?

Salamat Ali

Numerade Educator

Problem 20

When resistors 1 and 2 are connected in series, the equivalent resistance is $16.0 \Omega$. When they are connected in parallel, the equivalent resistance is $3.0 \Omega$. What are (a) the smaller resistance and (b) the larger resistance of these two resistors?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 21

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 22

Figure 27.17 shows five $5.00 \Omega$ resistors. Find the equivalent resistance between points (a) $F$ and $H$ and (b) $F$ and $G$.
FIGURE CANT COPY
Fig. 27.17

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 23

In Fig. 27.18, $R_1=100 \Omega$, $R_2=50 \Omega$, and the ideal batteries have emfs $\mathscr{E}_1=6.0 \mathrm{~V}, \mathscr{E}_2=5.0 \mathrm{~V}$, and $\mathscr{E}_3=4.0 \mathrm{~V}$. Find (a) the current in resistor 1, (b) the current in resistor 2, and (c) the potential difference between points $a$ and $b$.
FIGURE CANT COPY
Fig. 27.18

Neelesh Sharma

Numerade Educator

Problem 24

In Fig. $27.19, R_1=R_2=4.00 \Omega$ and $R_3=2.50 \Omega$. Find the equivalent resistance between points $D$ and $E$.
FIGURE CANT COPY
Fig. 27.19

Alex Garger

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

Figure 27.20 shows a battery connected across a uniform resistor $R_0$. A sliding contact can move across the resistor from $x=0$ at the left to $x=10 \mathrm{~cm}$ at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in resistor $R$ as a function of $x$. Plot the function for $\mathscr{E}=50 \mathrm{~V}, R=2000 \Omega$, and $R_0=100 \Omega$.
FIGURE CANT COPY
Fig. 27.20

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 27

Figure 27.21 indicates one reason no one should stand under a tree during a lightning storm. If lightning comes down the side of the tree, a portion can jump over to the person, especially if the current on the tree reaches a dry region on the bark and thereafter must travel through air to reach the ground. In the figure, part of the lightning jumps through distance $d$ in air and then travels through the person (who has negligible resistance relative to that of air because of the highly conducting salty fluids within the body). The rest of the current travels through air alongside the tree, for a distance $h$. If $d / h=$ 0.400 and the total current is $I=$ $5000 \mathrm{~A}$, what is the current through the person?
FIGURE CANT COPY
Fig. 27.21

Keshav Singh

Numerade Educator

Problem 28

The ideal battery in Fig. $27.22 a$ has emf $\mathscr{E}=6.0 \mathrm{~V}$. Plot 1 in Fig. $27.22 b$ gives the electric potential difference $V$ that can appear across resistor 1 versus the current $i$ in that resistor when the resistor is individually tested by putting a variable potential across it. The scale of the $V$ axis is set by $V_s=18.0 \mathrm{~V}$, and the scale of the $i$ axis is set by $i_s=3.00 \mathrm{~mA}$. Plots 2 and 3 are similar plots for resistors 2 and 3 , respectively, when they are individually tested by putting a variable potential across them. What is the current in resistor 2 in the circuit of Fig. 27.22a?
FIGURE CANT COPY
Fig. 27.22

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 29

In Fig. $27.23, R_1=6.00 \Omega$, $R_2=18.0 \Omega$, and the ideal battery has emf $\mathscr{E}=12.0 \mathrm{~V}$. What are the (a) size and (b) direction (left or right) of current $i_1$ ? (c) How much energy is dissipated by all four resistors in $1.00 \mathrm{~min}$ ?
FIGURE CANT COPY
Fig. 27.23

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 30

In Fig. 27.24, the ideal batteries have emfs $\mathscr{E}_1=10.0 \mathrm{~V}$ and $\mathscr{E}_2=0.500 \mathscr{E}_1$, and the resistances are each $4.00 \Omega$. What is the current in (a) resistance 2 and (b) resistance 3 ?
FIGURE CANT COPY
Fig. 27.24

Vishal Gupta

Numerade Educator

Problem 31

In Fig. 27.25, the ideal batteries have emfs $\mathscr{E}_1=$ $5.0 \mathrm{~V}$ and $\mathscr{E}_2=12 \mathrm{~V}$, the resistances are each $2.0 \Omega$, and the potential is defined to be zero at the grounded point of the circuit. What are potentials (a) $V_1$ and (b) $V_2$ at the indicated points?
FIGURE CANT COPY
Fig. 27.25

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 32

Both batteries in Fig. $27.26 a$ are ideal. Emf $\mathscr{E}_1$ of battery 1 has a fixed value, but emf $\mathscr{E}_2$ of battery 2 can be varied between $1.0 \mathrm{~V}$ and $10 \mathrm{~V}$. The plots in Fig. $27.26 b$ give the currents through the two batteries as a function of $\mathscr{E}_2$. The vertical scale is set by $i_s=0.20 \mathrm{~A}$. 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 emf. What are (a) emf $\mathscr{E}_1$, (b) resistance $R_1$, and (c) resistance $R_2$ ?
FIGURE CANT COPY
Fig. 27.26

Keshav Singh

Numerade Educator

Problem 33

In Fig. 27.27, the current in resistance 6 is $i_6=1.40 \mathrm{~A}$ and the resistances are $R_1=R_2=R_3=2.00 \Omega, R_4=16.0 \Omega$, $R_5=8.00 \Omega$, and $R_6=4.00 \Omega$. What is the emf of the ideal battery?
FIGURE CANT COPY
Fig. 27.27

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

The resistances in Figs. $27.28 a$ and $b$ are all $6.0 \Omega$, and the batteries are ideal $12 \mathrm{~V}$ batteries. (a) When switch $\mathrm{S}$ in Fig. $27.28 a$ is closed, what is the change in the electric potential $V_1$ across resistor 1 , or does $V_1$ remain the same? (b) When switch $\mathrm{S}$ in Fig. $27.28 b$ is closed, what is the change in $V_1$ across resistor 1 , or does $V_1$ remain the same?
FIGURE CANT COPY
Fig. 27.28

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 35

In Fig. $27.29, \mathscr{E}=12.0 \mathrm{~V}$, $R_1=2000 \Omega, \quad R_2=3000 \Omega, \quad$ and $R_3=4000 \Omega$. What are the potential differences (a) $V_A-V_B$, (b) $V_B-V_C$, (c) $V_C-V_D$, and (d) $V_A-V_C$ ?
FIGURE CANT COPY
Fig. 27.29

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 36

In Fig. 27.30, $\mathscr{E}_1=6.00 \mathrm{~V}$, $\mathscr{E}_2=12.0 \mathrm{~V}, R_1=100 \Omega, R_2=200 \Omega$, and $R_3=300 \Omega$. One point of the circuit is grounded $(V=0)$. What are the (a) size and (b) direction (up or down) of the current through resistance 1, the (c) size and (d) direction (left or right) of the current through resistance 2 , and the (e) size and (f) direction of the current through resistance 3 ? (g) What is the electric potential at point $A$ ?
FIGURE CANT COPY
Fig. 27.30

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 37

In Fig. 27.31, the resistances are $R_1=2.00 \Omega, \quad R_2=$ $5.00 \Omega$, and the battery is ideal. What value of $R_3$ maximizes the dissipation rate in resistance 3 ?
FIGURE CANT COPY
Fig. 27.31

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 38

Figure 27.32 shows a section of a circuit. The resistances are $R_1=2.0 \Omega, R_2=4.0 \Omega$, and $R_3=6.0 \Omega$, and the indicated current is $i=6.0 \mathrm{~A}$. The electric potential difference between points $A$ and $B$ that connect the section to the rest of the circuit is $V_A-V_B=78 \mathrm{~V}$. (a) Is the device represented by "Box" absorbing or providing energy to the circuit, and (b) at what rate?
FIGURE CANT COPY
Fig. 27.32

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 39

In Fig. 27.33, two batteries with an $\operatorname{emf} \mathscr{E}=12.0 \mathrm{~V}$ and an internal resistance $r=0.300 \Omega$ are connected in parallel across a resistance $R$. (a) For what value of $R$ is the dissipation rate in the resistor a maximum? (b) What is that maximum?
FIGURE CANT COPY
Fig. 27.33

Konstantin Pavlovskii

Konstantin Pavlovskii

Numerade Educator

Problem 40

Two identical batteries of emf $\mathscr{E}=12.0 \mathrm{~V}$ and internal resistance $r=0.200 \Omega$ are to be connected to an external resistance $R$, either in parallel (Fig. 27.33) or in series (Fig. 27.34). If $R=2.00 r$, what is the current $i$ in the external resistance in the (a) parallel and (b) series arrangements? (c) For which arrangement is $i$ greater? If $R=r / 2.00$, what is $i$ in the external resistance in the (d) parallel arrangement and (e) series arrangement? (f) For which arrangement is $i$ greater now?
FIGURE CANT COPY
Fig. 27.34

Keshav Singh

Numerade Educator

Problem 41

In Fig. 27.24, $\varepsilon_1=3.00 \mathrm{~V}$, $\mathscr{E}_2=1.00 \mathrm{~V}, R_1=4.00 \Omega, R_2=$ $2.00 \Omega, R_3=5.00 \Omega$, 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 ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 42

In Fig. 27.35, an array of $n$ parallel resistors is connected in series to a resistor and an ideal battery. All the resistors have the same resistance. If an identical resistor were added in parallel to the parallel array, the current through the battery would change by $1.25 \%$. What is the value of $n$ ?
FIGURE CANT COPY
Fig. 27.35

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 43

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 44

In Fig. 27.36, $R_1=100 \Omega, R_2=R_3=50.0 \Omega, R_4=$ $75.0 \Omega$, and the ideal battery has emf $\mathscr{E}=6.00 \mathrm{~V}$. (a) What is the equivalent resistance? What is $i$ in (b) resistance 1 , (c) resistance 2 , (d) resistance 3 , and (e) resistance 4 ?
FIGURE CANT COPY
Fig. 27.36

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 45

In Fig. 27.37, the resistances are $R_1=1.0 \Omega$ and $R_2=2.0 \Omega$, and the ideal batteries have emfs $\mathscr{E}_1=2.0 \mathrm{~V}$ and $\mathscr{E}_2=\mathscr{E}_3=4.0 \mathrm{~V}$. What are the (a) size and (b) direction (up or down) of the current in battery 1 , the (c) size and (d) direction of the current in battery 2 , and the (e) size and (f) direction of the current in battery 3 ? (g) What is the potential difference $V_a-V_b$ ?
FIGURE CANT COPY
Fig. 27.37

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 46

In Fig. 27.38a, resistor 3 is a variable resistor and the ideal battery has emf $\mathscr{E}=12 \mathrm{~V}$. Figure $27.38 b$ gives the current $i$ through the battery as a function of $R_3$. The horizontal scale is set by $R_{3 s}=20 \Omega$. The curve has an asymptote of $2.0 \mathrm{~mA}$ as $R_3 \rightarrow \infty$. What are (a) resistance $R_1$ and (b) resistance $R_2$ ?
FIGURE CANT COPY
Fig. 27.38

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 47

A copper wire of radius $a=0.250 \mathrm{~mm}$ has an aluminum jacket of outer radius $b=0.380 \mathrm{~mm}$. There is a current $i=2.00 \mathrm{~A}$ in the composite wire. Using Table 26.3.1, calculate the current in (a) the copper and (b) the aluminum. (c) If a potential difference $V=12.0 \mathrm{~V}$ between the ends maintains the current, what is the length of the composite wire?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 48

In Fig. 27.36, the resistors have the values $R_1=7.00 \Omega$, $R_2=12.0 \Omega$, and $R_3=4.00 \Omega$, and the ideal battery's emf is $\mathscr{E}=24.0 \mathrm{~V}$. For what value of $R_4$ will the rate at which the battery transfers energy to the resistors equal (a) $60.0 \mathrm{~W}$, (b) the maximum possible rate $P_{\max }$, and (c) the minimum possible rate $P_{\min }$ ? What are (d) $P_{\max }$ and (e) $P_{\min }$ ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 49

(a) In Fig. 27.39, what current does the ammeter read if $\mathscr{E}=5.0 \mathrm{~V}$ (ideal battery), $R_1=$ $2.0 \Omega, R_2=4.0 \Omega$, and $R_3=6.0 \Omega$ ? (b) The ammeter and battery are now interchanged. Show that the ammeter reading is unchanged.
FIGURE CANT COPY
Fig. 27.39

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 50

In Fig. 27.40, $R_1=2.00 R$, the ammeter resistance is zero, and the battery is ideal. What multiple of $\mathscr{E} / R$ gives the current in the ammeter?
FIGURE CANT COPY
Fig. 27.40

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 51

In Fig. 27.41, a voltmeter of resistance $R_{\mathrm{V}}=300 \Omega$ and an ammeter of resistance $R_{\mathrm{A}}=$ $3.00 \Omega$ are being used to measure a resistance $R$ in a circuit that also contains a resistance $R_0=100 \Omega$ and an ideal battery with an emf of $\mathscr{E}=12.0 \mathrm{~V}$. Resistance $R$ is given by $R=V / i$, where $V$ is the potential across $R$ and $i$ is the ammeter reading. The voltmeter reading is $V^{\prime}$, which is $V$ plus the potential difference across the ammeter. Thus, the ratio of the two meter readings is not $R$ but only an apparent resistance $R^{\prime}=V^{\prime} / i$. If $R=85.0 \Omega$, what are (a) the ammeter reading, (b) the voltmeter reading, and (c) $R^{\prime \prime}$ ? (d) If $R_{\mathrm{A}}$ is decreased, does the difference between $R^{\prime}$ and $R$ increase, decrease, or remain the same?
FIGURE CANT COPY
Fig. 27.41

Keshav Singh

Numerade Educator

Problem 52

A simple ohmmeter is made by connecting a $1.50 \mathrm{~V}$ flashlight battery in series with a resistance $R$ and an ammeter that reads from 0 to $1.00 \mathrm{~mA}$, as shown in Fig. 27.42. 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) $10.0 \%$, (b) $50.0 \%$, and (c) $90.0 \%$ 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 $R$ ?
FIGURE CANT COPY
Fig. 27.42

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 53

In Fig. 27.3.1, assume that $\mathscr{E}=3.0 \mathrm{~V}, \quad r=100 \Omega, R_1=250 \Omega$, and $R_2=300 \Omega$. If the voltmeter resistance $R_{\mathrm{V}}$ is $5.0 \mathrm{k} \Omega$, what percent error does it introduce into the measurement of the potential difference across $R_1$ ? Ignore the presence of the ammeter.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 54

When the lights of a car are switched on, an ammeter in series with them reads $10.0 \mathrm{~A}$ and a voltmeter connected across them reads $12.0 \mathrm{~V}$ (Fig. 27.43). When the electric starting motor is turned on, the ammeter reading drops to $8.00 \mathrm{~A}$ and the lights dim somewhat. If the internal resistance of the battery is $0.0500 \Omega$ 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?
FIGURE CANT COPY
Fig. 27.43

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 55

In Fig. 27.44, $R_s$ 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: $R_x=R_s R_2 / R_1$. 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.
FIGURE CANT COPY
Fig. 27.44

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 56

In Fig. 27.45, a voltmeter of resistance $R_{\mathrm{V}}=300 \Omega$ and an ammeter of resistance $R_{\mathrm{A}}=3.00 \Omega$ are being used to measure a resistance $R$ in a circuit that also contains a resistance $R_0=100 \Omega$ and an ideal battery of emf $\mathscr{E}=12.0 \mathrm{~V}$. Resistance $R$ is given by $R=V / i$, where $V$ is the voltmeter reading and $i$ is the current in resistance $R$. However, the ammeter reading is not $i$ but rather $i^{\prime}$, which is $i$ plus the current through the voltmeter. Thus, the ratio of the two meter readings is not $R$ but only an apparent resistance $R^{\prime}=V / i^{\prime}$. If $R=85.0 \Omega$, what are (a) the ammeter reading, (b) the voltmeter reading, and (c) $R^{\prime}$ ? (d) If $R_{\mathrm{V}}$ is increased, does the difference between $R^{\prime}$ and $R$ increase, decrease, or remain the same?
FIGURE CANT COPY
Fig. 27.45

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 57

Switch in Fig. 27.46 is closed at time $t=0$, to begin charging an initially uncharged capacitor of capacitance $C=15.0 \mu \mathrm{F}$ through a resistor of resistance $R=20.0 \Omega$. At what time is the potential across the capacitor equal to that across the resistor?
FIGURE CANT COPY
Fig. 27.46

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 58

In an $R C$ series circuit, emf $\mathscr{E}=12.0 \mathrm{~V}$, resistance $R=1.40 \mathrm{M} \Omega$, and capacitance $C=1.80 \mu \mathrm{F}$. (a) Calculate the time constant. (b) Find the maximum charge that will appear on the capacitor during charging. (c) How long does it take for the charge to build up to $16.0 \mu \mathrm{C}$ ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 59

What multiple of the time constant $\tau$ gives the time taken by an initially uncharged capacitor in an $R C$ series circuit to be charged to $99.0 \%$ of its final charge?

Salamat Ali

Numerade Educator

Problem 60

A capacitor with initial charge $q_0$ is discharged through a resistor. What multiple of the time constant $\tau$ gives the time the capacitor takes to lose (a) the first one-third of its charge and (b) two-thirds of its charge?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 61

A $15.0 \mathrm{k} \Omega$ resistor and a capacitor are connected in series, and then a $12.0 \mathrm{~V}$ potential difference is suddenly applied across them. The potential difference across the capacitor rises to $5.00 \mathrm{~V}$ in $1.30 \mu \mathrm{s}$. (a) Calculate the time constant of the circuit. (b) Find the capacitance of the capacitor.

Salamat Ali

Numerade Educator

Problem 62

Figure 27.47 shows the circuit of a flashing lamp, like those attached to barrels at highway construction sites. The fluorescent lamp L (of negligible capacitance) is connected in parallel across the capacitor $C$ of an $R C$ circuit. There is a current through the lamp only when the potential difference across it reaches the breakdown voltage $V_{\mathrm{L}}$; then the capacitor discharges completely through the lamp and the lamp flashes briefly. For a lamp with breakdown voltage $V_{\mathrm{L}}=72.0 \mathrm{~V}$, wired to a $95.0 \mathrm{~V}$ ideal battery and a $0.150 \mu \mathrm{F}$ capacitor, what resistance $R$ is needed for two flashes per second?
FIGURE CANT COPY
Fig. 27.47

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 63

In the circuit of Fig. $27.48, \mathscr{E}=1.2 \mathrm{kV}, \quad C=6.5 \mu \mathrm{F}$, $R_1=R_2=R_3=0.73 \mathrm{M} \Omega$. With $C$ completely uncharged, switch $\mathrm{S}$ is suddenly closed (at $t=0)$. At $t=0$, what are (a) current $i_1$ in resistor 1 , (b) current $i_2$ in resistor 2, and (c) current $i_3$ in resistor 3? At $t=\infty$ (that is, after many time constants), what are (d) $i_1$, (e) $i_2$, and (f) $i_3$ ? What is the potential difference $V_2$ across resistor 2 at (g) $t=0$ and (h) $t=\infty$ ? (i) Sketch $V_2$ versus $t$ between these two extreme times.
FIGURE CANT COPY
Fig. 27.48

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 64

A capacitor with an initial potential difference of $100 \mathrm{~V}$ is discharged through a resistor when a switch between them is closed at $t=0$. At $t=10.0 \mathrm{~s}$, the potential difference across the capacitor is $1.00 \mathrm{~V}$. (a) What is the time constant of the circuit? (b) What is the potential difference across the capacitor at $t=17.0 \mathrm{~s}$ ?
$65 \mathrm{M}$ @ In Fig. 27.49, $R_1=$ $10.0 \mathrm{k} \Omega, R_2=15.0 \mathrm{k} \Omega, C=0.400$ $\mu \mathrm{F}$, and the ideal battery has emf $\mathscr{E}=20.0$ V. First, the switch is closed a long time so that the steady state is reached. Then the switch is opened at time $t=0$. What is the current in resistor 2 at $t=4.00 \mathrm{~ms}$ ?
FIGURE CANT COPY
Fig. 27.49

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 66

Figure 27.50 displays two circuits with a charged capacitor that is to be discharged through a resistor when a switch is closed. In Fig. $27.50 a, R_1=20.0 \Omega$ and $C_1=5.00 \mu \mathrm{F}$. In Fig. $27.50 b, R_2=$ $10.0 \Omega$ and $C_2=8.00 \mu \mathrm{F}$. The ratio of the initial charges on the two capacitors is $q_{02} / q_{01}=1.50$. At time $t=0$, both switches are closed. At what time $t$ do the two capacitors have the same charge?
FIGURE CANT COPY
Fig. 27.50

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 67

The potential difference between the plates of a leaky (meaning that charge leaks from one plate to the other) $2.0 \mu \mathrm{F}$ capacitor drops to one-fourth its initial value in $2.0 \mathrm{~s}$. What is the equivalent resistance between the capacitor plates?

Salamat Ali

Numerade Educator

Problem 68

A $1.0 \mu \mathrm{F}$ capacitor with an initial stored energy of $0.50 \mathrm{~J}$ is discharged through a $1.0 \mathrm{M} \Omega$ resistor. (a) What is the initial charge on the capacitor? (b) What is the current through the resistor when the discharge starts? Find an expression that gives, as a function of time $t$, (c) the potential difference $V_C$ across the capacitor, (d) the potential difference $V_R$ across the resistor, and (e) the rate at which thermal energy is produced in the resistor.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 69

A $3.00 \mathrm{M} \Omega$ resistor and a $1.00 \mu \mathrm{F}$ capacitor are connected in series with an ideal battery of emf $\mathscr{E}=4.00 \mathrm{~V}$. At $1.00 \mathrm{~s}$ after the connection is made, what is the rate at which (a) the charge of the capacitor is increasing, (b) energy is being stored in the capacitor, (c) thermal energy is appearing in the resistor, and (d) energy is being delivered by the battery?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 70

Each of the six real batteries in Fig. 27.51 has an emf of $20 \mathrm{~V}$ and a resistance of $4.0 \Omega$. (a) What is the current through the (external) resistance $R=4.0 \Omega$ ? (b) What is the potential difference across each battery? (c) What is the power of each battery? (d) At what rate does each battery transfer energy to internal thermal energy?
FIGURE CANT COPY
Fig. 27.51

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 71

In Fig. 27.52, $R_1=20.0 \Omega$, $R_2=10.0 \Omega$, and the ideal battery has emf $\mathscr{E}=120 \mathrm{~V}$. What is the current at point $a$ if we close (a) only switch $\mathrm{S}_1$, (b) only switches $S_1$ and $S_2$, and (c) all three switches?
FIGURE CANT COPY
Fig. 27.52

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 72

In Fig. 27.53, the ideal battery has emf $\mathscr{E}=30.0 \mathrm{~V}$, and the resistances are $R_1=R_2=14 \Omega$, $R_3=R_4=R_5=6.0 \Omega, R_6=2.0 \Omega$, and $R_7=1.5 \Omega$. What are currents (a) $i_2$, (b) $i_4$, (c) $i_1$, (d) $i_3$, and (e) $i_5$ ?
FIGURE CANT COPY
Fig. 27.53

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 73

Wires $A$ and $B$, having equal lengths of $40.0 \mathrm{~m}$ and equal diameters of $2.60 \mathrm{~mm}$, are connected in series. A potential difference of $60.0 \mathrm{~V}$ is applied between the ends of the composite wire. The resistances are $R_A=0.127 \Omega$ and $R_B=0.729 \Omega$. For wire $A$, what are (a) magnitude $J$ of the current density and (b) potential difference $V$ ? (c) Of what type material is wire $A$ made (see Table 26.3.1)? For wire $B$, what are (d) $J$ and (e) $V$ ? (f) Of what type material is $B$ made?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 74

What are the (a) size and (b) direction (up or down) of current $i$ in Fig. 27.54, where all resistances are $4.0 \Omega$ and all batteries are ideal and have an emf of 10 V?
FIGURE CANT COPY
Fig. 27.54

Keshav Singh

Numerade Educator

Problem 75

Suppose that, while you are sitting in a chair, charge separation between your clothing and the chair puts you at a potential of $200 \mathrm{~V}$, with the capacitance between you and the chair at $150 \mathrm{pF}$. When you stand up, the increased separation between your body and the chair decreases the capacitance to $10 \mathrm{pF}$. (a) What then is the potential of your body? That potential is reduced over time, as the charge on you drains through your body and shoes (you are a capacitor discharging through a resistance). Assume that the resistance along that route is $300 \mathrm{G} \Omega$. If you touch an electrical component while your potential is greater than $100 \mathrm{~V}$, you could ruin the component. (b) How long must you wait until your potential reaches the safe level of $100 \mathrm{~V}$ ?
If you wear a conducting wrist strap (Fig. 27.55) that is connected to ground, your potential does not increase as much when you stand up; you also discharge more rapidly because the resistance through the grounding connection is much less than through your body and shoes. (c) Suppose that when you stand up, your potential is $1400 \mathrm{~V}$ and the chair-to-you capacitance is $10 \mathrm{pF}$. What resistance in that wrist-strap grounding connection will allow you to discharge to $100 \mathrm{~V}$ in $0.30 \mathrm{~s}$, which is less time than you would need to reach for, say, your computer?
FIGURE CANT COPY
Fig. 27.55

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 76

In Fig. 27.56, the ideal batteries have emfs $\mathscr{E}_1=20.0 \mathrm{~V}$, $\mathscr{E}_2=10.0 \mathrm{~V}$, and $\mathscr{E}_3=5.00 \mathrm{~V}$, and the resistances are each $2.00 \Omega$. What are the (a) size and (b) direction (left or right) of current $i_1$ ? (c) Does battery 1 supply or absorb energy, and (d) what is its power? (e) Does battery 2 supply or absorb energy, and (f) what is its power? (g) Does battery 3 supply or absorb energy, and (h) what is its power?
FIGURE CANT COPY
Fig. 27.56

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 77

A temperature-stable resistor is made by connecting a resistor made of silicon in series with one made of iron. If the required total resistance is $1000 \Omega$ in a wide temperature range around $20^{\circ} \mathrm{C}$, what should be the resistance of the (a) silicon resistor and (b) iron resistor? (See Table 26.3.1.)

Keshav Singh

Numerade Educator

Problem 78

In Fig. 27.3.1, assume that $\mathscr{E}=5.0 \mathrm{~V}, r=2.0 \Omega, R_1=5.0 \Omega$, and $R_2=4.0 \Omega$. If the ammeter resistance $R_{\mathrm{A}}$ is $0.10 \Omega$, what percent error does it introduce into the measurement of the current? Assume that the voltmeter is not present.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 79

An initially uncharged capacitor $C$ is fully charged by a device of constant emf $\mathscr{E}$ connected in series with a resistor $R$. (a) Show that the final energy stored in the capacitor is half the energy supplied by the emf device. (b) By direct integration of $i^2 R$ over the charging time, show that the thermal energy dissipated by the resistor is also half the energy supplied by the emf device.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 80

In Fig. 27.57, $R_1=5.00 \Omega, R_2$ $=10.0 \Omega, R_3=15.0 \Omega, C_1=5.00 \mu \mathrm{F}$, $C_2=10.0 \mu \mathrm{F}$, and the ideal battery has $\operatorname{emf} \mathscr{E}=20.0 \mathrm{~V}$. Assuming that the circuit is in the steady state, what is the total energy stored in the two capacitors?
FIGURE CANT COPY
Fig. 27.57

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 81

In Fig. 27.1.5a, find the potential difference across $R_2$ if $\mathscr{E}=12 \mathrm{~V}, R_1=3.0 \Omega, R_2=4.0 \Omega$, and $R_3=5.0 \Omega$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 82

In Fig. 27.1.8a, calculate the potential difference between $a$ and $c$ by considering a path that contains $R, r_1$, and $\mathscr{E}_1$.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 83

A controller on an electronic arcade game consists of a variable resistor connected across the plates of a $0.220 \mu \mathrm{F}$ capacitor. The capacitor is charged to $5.00 \mathrm{~V}$, then discharged through the resistor. The time for the potential difference across the plates to decrease to $0.800 \mathrm{~V}$ is measured by a clock inside the game. If the range of discharge times that can be handled effectively is from $10.0 \mu \mathrm{s}$ to $6.00 \mathrm{~ms}$, what should be the (a) lower value and (b) higher value of the resistance range of the resistor?

Salamat Ali

Numerade Educator

Problem 84

An automobile gasoline gauge is shown schematically in Fig. 27.58. The indicator (on the dashboard) has a resistance of $10 \Omega$. The tank unit is a float connected to a variable resistor whose resistance varies linearly with the volume of gasoline. The resistance is $140 \Omega$ when the tank is empty and $20 \Omega$ when the tank is full. Find the current in the circuit when the tank is (a) empty, (b) half-full, and (c) full. Treat the battery as ideal.
FIGURE CANT COPY
Fig. 27.58

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 85

The starting motor of a car is turning too slowly, and the mechanic has to decide whether to replace the motor, the cable, or the battery. The car'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?

Salamat Ali

Numerade Educator

Problem 86

Two resistors $R_1$ and $R_2$ may be connected either in series or in parallel across an ideal battery with emf $\mathscr{E}$. We desire the rate of energy dissipation of the parallel combination to be five times that of the series combination. If $R_1=100 \Omega$, what are the (a) smaller and (b) larger of the two values of $R_2$ that result in that dissipation rate?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 87

The circuit of Fig. 27.59 shows a capacitor, two ideal batteries, two resistors, and a switch $S$. Initially $\mathrm{S}$ has been open for a long time. If it is then closed for a long time, what is the change in the charge on the capacitor? Assume $C=10 \mu \mathrm{F}, \mathscr{E}_1=1.0 \mathrm{~V}, \mathscr{E}_2=3.0 \mathrm{~V}$, $R_1=0.20 \Omega$, and $R_2=0.40 \Omega$. $R_1=0.20 \Omega$, and $R_2=0.40 \Omega$.
FIGURE CANT COPY
Fig. 27.59

Keshav Singh

Numerade Educator

Problem 88

In Fig. 27.24, $R_1=10.0 \Omega, R_2=20.0 \Omega$, and the ideal batteries have emfs $\mathscr{E}_1=20.0 \mathrm{~V}$ and $\mathscr{E}_2=50.0 \mathrm{~V}$. What value of $R_3$ results in no current through battery 1 ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 89

In Fig. 27.60, $R=10 \Omega$. What is the equivalent resistance between points $A$ and $B$ ?
FIGURE CANT COPY
Fig. 27.60

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 90

(a) In Fig. 27.1.4a, 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 91

In Fig. 27.61, the ideal batteries have emfs $\mathscr{E}_1=12.0 \mathrm{~V}$ and $\mathscr{E}_2=4.00 \mathrm{~V}$, and the resistances are each $4.00 \Omega$. What are the (a) size and (b) direction (up or down) of $i_1$ and the (c) size and (d) direction of $i_2$ ? (e) Does battery 1 supply or absorb energy, and (f) what is its energy transfer rate? (g) Does battery 2 supply or absorb energy, and (h) what is its energy transfer rate?
FIGURE CANT COPY
Fig. 27.61

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 92

Figure 27.62 shows a portion of a circuit through which there is a current $I=6.00 \mathrm{~A}$. The resistances are $R_1=R_2=2.00 R_3=$ $2.00 R_4=4.00 \Omega$. What is the current $i_1$ through resistor 1 ?
FIGURE CANT COPY
Fig. 27.62

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 93

Thermal energy is to be generated in a $0.10 \Omega$ resistor at the rate of $10 \mathrm{~W}$ by connecting the resistor to a battery whose emf is $1.5 \mathrm{~V}$. (a) What potential difference must exist across the resistor? (b) What must be the internal resistance of the battery?

Salamat Ali

Numerade Educator

Problem 94

Figure 27.63 shows three $20.0 \Omega$ resistors. Find the equivalent resistance between points (a) $A$ and $B$, (b) $A$ and $C$, and (c) $B$ and $C$.
FIGURE CANT COPY
Fig. 27.63

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 95

A $120 \mathrm{~V}$ power line is protected by a $15 \mathrm{~A}$ fuse. What is the maximum number of $500 \mathrm{~W}$ lamps that can be simultaneously operated in parallel on this line without "blowing" the fuse because of an excess of current?

Salamat Ali

Numerade Educator

Problem 96

Figure 27.46 shows an ideal battery of emf $\mathscr{E}=12 \mathrm{~V}$, a resistor of resistance $R=4.0 \Omega$, and an uncharged capacitor of capacitance $C=4.0 \mu \mathrm{F}$. After switch $\mathrm{S}$ is closed, what is the current through the resistor when the charge on the capacitor is $8.0 \mu \mathrm{C}$ ?

Keshav Singh

Numerade Educator

Problem 97

Figure 27.64 shows a cube made of 12 resistors, each of resistance $R$. What is the equivalent resistance $R_{12}$ that the combination would present to a battery attached across points 1 and 2? (Although this problem can be attacked by "brute force" methods using the loop and junction rules and solving multiple simultaneous equations, the symmetry of the connections suggests that there must be a slicker method.
FIGURE CANT COPY
Fig. 27.64

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 98

An active area of research involves measuring the resistivities of brain tissue and brain tumors, to aid in certain types of surgeries and in the placement of deepbrain electrodes for treatment of epilepsy and Parkinson's disease. One concern is the determination of an electric current through adjacent tumor and healthy tissue. Figure 27.65 shows one research group's simple model of the electric pathway for a current $i$ through the resistance $R_1=160 \Omega$ of gliomas (which accounts for about $30 \%$ of all brain tumors) and resistance $R_2=372 \Omega$ of healthy white matter. In a living brain, what percent of the current is through (a) the gliomas and (b) the white matter? If the same arrangement is in a cadaver in which formaldehyde increases each resistance by $2700 \Omega$, what percent of the current is through (c) the gliomas and (d) the white matter? The results reveal that studies on cadavers must be adjusted to account for the electrical properties of a living brain.
FIGURE CANT COPY
Fig. 27.65

Vishal Gupta

Numerade Educator

Problem 99

Wire for electric heater. You are to construct a heating coil but will first test two wires, both of length $L=10 \mathrm{~cm}$ and diameter $d=2.5$ mils (a mil is a common unit that is $1 / 1000$ of an inch): Wire 1 consists of copper with resistivity $\rho_1=1.7 \times$ $10^{-8} \Omega \cdot \mathrm{m}$ and wire 2 consists of Nichrome (an alloy of nickel and chromium) with resistivity $\rho_2=1.1 \times 10^{-6} \Omega \cdot \mathrm{m}$. You will put a potential difference of $V=110 \mathrm{~V}$ across four arrangements of the wires. What power will be dissipated as heat for (a) wire 1 alone, (b) wire 2 alone, (c) wires 1 and 2 in series, and (d) wires 1 and 2 in parallel?

Vishal Gupta

Numerade Educator

Problem 100

Electric eels are known to leap at people and animals to shock them. In a recent research experiment, a juvenile electric eel was allowed to leap to a volunteer's arm in order to measure the current set up along the arm. Figure $27.66 a$ is a photo of the eel striking the arm with the clenched hand immersed in the water. (The strikes always caused the volunteer to withdraw the arm.) Figure $27.66 b$ gives a circuit diagram of eel, arm, and water. The emf generated by the eel is $200 \mathrm{~V}$. The resistance of the eel's body is $R_1=1000 \Omega$, the resistance from the front of the eel down along its body's surface to the water is $R_3=2.3 \mathrm{k} \Omega$, the resistance along the arm from the strike point to the water is $R_4=2.2 \mathrm{k} \Omega$, and the resistance of the current's return path through the water is $R_2=400 \Omega$. Find (a) the current $i$ generated by the eel and (b) the current $i_1$ along the arm. (c) How much power was delivered to the arm?
FIGURE CANT COPY
Fig. 27.66

Sanjeev Kumar

Numerade Educator

Problem 101

When gasoline is loaded onto a tanker truck or dispensed from it into an underground tank at a gasoline station, great care must be taken so that a spark from electrostatic charges does not ignite the vapor. That charge is produced by the sloshing of the gasoline as it moves through hoses or when the truck travels along a road. For gasoline vapor, the critical value for the spark energy is $U_{\text {critical }}=24 \mathrm{~mJ}$. To avoid fire and explosion, the truck must be grounded before gasoline is poured into it or poured out of it. The grounding is by means of a conducting cable with a resistance of $10 \Omega$, with one end buried in the ground and the other clipped to the truck (Fig. 27.67). If a truck has a $600 \mathrm{pF}$ capacitance and an initial potential energy of $2.25 \mathrm{~J}$, how much time is required to discharge the truck to the critical value of potential energy?
FIGURE CANT COPY
Fig. 27.67

Shoukat Ali

Other Schools