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Chap 27
8. (a) We solve i = (2 – 1)/(r1 + r2 + R) for R:
R
 2  1
i
 r1  r2 
3.0 V  2.0 V
 3.0   3.0   9.9  102 .
1.0  103 A
(b) P = i2R = (1.0  10–3 A)2(9.9  102 ) = 9.9  10–4 W.
11. Let the emf be V. Then V = iR = i'(R + R'), where i = 5.0 A, i' = 4.0 A and R' =
2.0 . We solve for R:
R
iR (4.0) (2.0)

 8.0 .
i  i 5.0  4.0
14. (a) Let the emf of the solar cell be  and the output voltage be V. Thus,
V    ir   
VI
F
G
HR J
Kr
for both cases. Numerically, we get
0.10 V =  – (0.10 V/500 )r
0.15 V =  – (0.15 V/1000 )r.
We solve for  and r.
(a) r = 1.003 .
(b)  = 0.30 V.
(c) The efficiency is
V2 /R
0.15V

 2.3103  0.23%.
2
3
2
Preceived 1000   5.0cm   2.0 10 W/cm 
29. Let the resistors be divided into groups of n resistors each, with all the resistors in
the same group connected in series. Suppose there are m such groups that are
connected in parallel with each other. Let R be the resistance of any one of the
resistors. Then the equivalent resistance of any group is nR, and Req, the equivalent
resistance of the whole array, satisfies
m
1
1
m


.
Req
nR
1 nR
Since the problem requires Req = 10  = R, we must select n = m. Next we make use
of Eq. 27-16. We note that the current is the same in every resistor and there are n · m
= n2 resistors, so the maximum total power that can be dissipated is Ptotal = n2P, where
P 1.0 W is the maximum power that can be dissipated by any one of the resistors.
The problem demands Ptotal  5.0P, so n2 must be at least as large as 5.0. Since n must
be an integer, the smallest it can be is 3. The least number of resistors is n2 = 9.
32. (a) By symmetry, when the two batteries are connected in parallel the current i
going through either one is the same. So from  = ir + (2i)R with r = 0.200  and R =
2.00r, we get
iR  2i 
2
2(12.0V)

 24.0 A.
r  2 R 0.200  2(0.400)
(b) When connected in series 2 – iRr – iRr – iRR = 0, or iR = 2/(2r + R).
iR  2i 
2
2(12.0V)

 30.0 A.
2r  R 2(0.200)  0.400
(c) In series, since R > r.
(d) If R = r/2.00, then for parallel connection,
iR  2i 
2
2(12.0V)

 60.0 A.
r  2 R 0.200  2(0.100)
(e) For series connection, we have
iR  2i 
2
2(12.0V)

 48.0 A.
2r  R 2(0.200)  0.100
(f) In parallel, since R < r.
33. (a) We first find the currents. Let i1 be the current in R1 and take it to be positive if
it is to the right. Let i2 be the current in R2 and take it to be positive if it is to the left.
Let i3 be the current in R3 and take it to be positive if it is upward. The junction rule
produces
i1  i2  i3  0 .
The loop rule applied to the left-hand loop produces
1  i1R1  i3 R3  0
and applied to the right-hand loop produces
 2  i2 R2  i3 R3  0.
We substitute i3 = –i2 – i1, from the first equation, into the other two to obtain
1  i1R1  i2 R3  i1R3  0
and
 2  i2 R2  i2 R3  i1R3  0.
Solving the above equations yield
i1 
i2 
1 ( R2  R3 )   2 R3
R1 R2  R1 R3  R2 R3
 2 ( R1  R3 )  1 R3
R1 R2  R1 R3  R2 R3

(3.00 V)(2.00   5.00 )  (1.00 V)(5.00 )
 0.421 A.
(4.00 )(2.00 )  (4.00 )(5.00)  (2.00 )(5.00 )

(1.00 V)(4.00   5.00 )  (3.00 V)(5.00 )
 0.158 A.
(4.00 )(2.00 )  (4.00 )(5.00)  (2.00 )(5.00 )
i3  
 2 R1  1 R2
R1 R2  R1 R3  R2 R3

(1.00 V)(4.00 )  (3.00 V)(2.00 )
 0.263 A.
(4.00 )(2.00 )  (4.00 )(5.00)  (2.00 )(5.00 )
Note that the current i3 in R3 is actually downward and the current i2 in R2 is to the
right. The current i1 in R1 is to the right.
(a) The power dissipated in R1 is P1  i12 R1   0.421A   4.00    0.709 W.
2
(b)
The
power
dissipated
in
R2
is
P2  i22 R2  (0.158 A) 2 (2.00 )  0.0499 W  0.050 W.
(c) The power dissipated in R3 is P3  i32 R3   0.263A   5.00    0.346 W.
2
(d) The power supplied by 1 is i31 = (0.421 A)(3.00 V) = 1.26 W.
(e) The power “supplied” by 2 is i22 = (–0.158 A)(1.00 V) = –0.158 W. The negative
sign indicates that 2 is actually absorbing energy from the circuit.
36. (a) We use P = 2/Req, where
Req  7.00  
12.0   4.00   R
.
12.0   4.0    12.0   R   4.00   R
Put P = 60.0 W and  = 24.0 V and solve for R: R = 19.5 .
(b) Since P  Req, we must minimize Req, which means R = 0.
(c) Now we must maximize Req, or set R = .
(d) Since Req, min = 7.00 , Pmax = 2/Req, min = (24.0 V)2/7.00  = 82.3 W.
(e) Since
Req, max = 7.00  + (12.0 )(4.00 )/(12.0  + 4.00 ) = 10.0 ,
Pmin = 2/Req, max = (24.0 V)2/10.0  = 57.6 W.
41. Since the current in the ammeter is i, the voltmeter reading is V’ =V+ i RA= i (R +
RA), or R = V’/i – RA = R' – RA, where R' = V’/i is the apparent reading of the
resistance.
Now, from the lower loop of the circuit diagram, the current through the voltmeter is
iV   /( Req  R0 ) , where
R  R  RA   300   85.0   3.00  
1
1
1


 Req  V

 68.0  .
Req RV RA  R
RV  R  RA
300   85.0   3.00 
The voltmeter reading is then
V '  iV Req 
 Req
Req  R0

(12.0 V)(68.0 )
 4.86 V.
68.0   100
(a) The ammeter reading is
i
V'
4.86 V

 0.0552 A.
R  RA 85.0   3.00
(b) As shown above, the voltmeter reading is V '  4.86 V.
(c) R' = V’/i = 4.86 V/(5.52  10–2 A) = 88.0 .
(d) Since R  R ' RA , if RA is decreased, the difference between R’ and R decreases.
In fact, when RA=0, R’=R.
42. The currents in R and RV are i and i' – i, respectively. Since V = iR = (i' – i)RV we
have, by dividing both sides by V, 1 = (i' /V – i/V)RV = (1/R' – 1/R)RV. Thus,
1 1 1
 
R R RV
 R'
RRV
.
R  RV
The equivalent resistance of the circuit is Req  RA  R0  R '  RA  R0 
RRV
.
R  RV
(a) The ammeter reading is
i 

Req


RA  R0  RV R  R  RV 

12.0V
3.00  100    300   85.0    300   85.0  
 7.09 102 A.
(b) The voltmeter reading is
V = – i' (RA + R0) = 12.0 V – (0.0709 A) (103.00 ) = 4.70 V.
(c) The apparent resistance is R' = V/i' = 4.70 V/(7.09  10–2 A) = 66.3 .
(d) If RV is increased, the difference between R and R’ decreases. In fact, R '  R as
RV   .
53. At t = 0 the capacitor is completely uncharged and the current in the capacitor
branch is as it would be if the capacitor were replaced by a wire. Let i1 be the current
in R1 and take it to be positive if it is to the right. Let i2 be the current in R2 and take it
to be positive if it is downward. Let i3 be the current in R3 and take it to be positive if
it is downward. The junction rule produces i1  i2  i3 , the loop rule applied to the
left-hand loop produces
  i1 R1  i2 R2  0 ,
and the loop rule applied to the right-hand loop produces
i2 R2  i3 R3  0 .
Since the resistances are all the same we can simplify the mathematics by replacing R1,
R2, and R3 with R.
(a) Solving the three simultaneous equations, we find
c
c
h
h
2 12
.  103 V
2
i1 

 11
.  103 A
6
3R 3 0.73  10 
(b) i2 

3R

1.2 103 V
 5.5 104 A.
6
3  0.7310  
(c) i3  i2  5.5 104 A.
At t =  the capacitor is fully charged and the current in the capacitor branch is 0.
Thus, i1 = i2, and the loop rule yields
  i1 R1  i1 R2  0 .
(d) The solution is
i1 

2R

1.2 103 V
 8.2 104 A.
6
2  0.7310  
(e) i2  i1  8.2 104 A.
(f) As stated before, the current in the capacitor branch is i3 = 0.
We take the upper plate of the capacitor to be positive. This is consistent with current
flowing into that plate. The junction equation is i1 = i2 + i3, and the loop equations are
  i1 R  i2 R  0 and 
q
 i3 R  i2 R  0 .
C
We use the first equation to substitute for i1 in the second and obtain  – 2i2R – i3R = 0.
Thus i2 = ( – i3R)/2R. We substitute this expression into the third equation above to
obtain –(q/C) – (i3R) + (/2) – (i3R/2) = 0. Now we replace i3 with dq/dt to obtain
3R dq q 
  .
2 dt C 2
This is just like the equation for an RC series circuit, except that the time constant is 
= 3RC/2 and the impressed potential difference is /2. The solution is
q
C
1  e 2 t 3 RC .
2
c
h
The current in the capacitor branch is
i3 (t ) 
dq  2 t 3 RC

e
.
dt 3R
The current in the center branch is
i2 (t ) 

i3 
 2 t 3 RC 


e

 3  e2 t 3 RC 
2R 2 2R 6R
6R

and the potential difference across R2 is
V2 (t )  i2 R 
(g) For t  0, e2 t 3 RC is 1 and

3  e
6
2 t 3 RC
.
V2   3  1.2 103 V  3  4.0 102 V .
(h) For t  , e2 t 3 RC is 0 and V2   2  1.2  203 V  2  6.0 102 V .
(i) A plot of V2 as a function of time is shown in the following graph.
56. We apply Eq. 27-39 to each capacitor, demand their initial charges are in a ratio of
3:2 as described in the problem, and solve for the time: we obtain
t=
3
ln2
 
1  = 162 s .
 1
R C - R C 
 2 2 1 1
Chap 28
5. Using Eq. 28-2 and Eq. 3-30, we obtain

F  q v x By  v y Bx k  q v x 3Bx  v y Bx k
d
i
db g
i
where we use the fact that By = 3Bx. Since the force (at the instant considered) is Fz k
where Fz = 6.4  10–19 N, then we are led to the condition
d
i
q 3v x  v y Bx  Fz  Bx 
Fz
.
q 3v x  v y
d
i
Substituting Vx = 2.0 m/s, vy = 4.0 m/s and q = –1.6  10–19 C, we obtain Bx = –2.0 T.
10. (a) The force due to the electric field


(F
= qE
) is distinguished from that


associated with the magnetic field ( F
= q
v B
) in that the latter vanishes at the
speed is zero and the former is independent of speed. The graph (Fig.28-34) shows
that the force (y-component) is negative at v = 0 (specifically, its value is –2.0  10–19
N there) which (because q = –e) implies that the electric field points in the +y
direction. Its magnitude is
E = (2.0  10–19)/(1.60  10–19) = 1.25 V/m.
(b) We are told that the x and z components of the force remain zero throughout the
motion, implying that the electron continues to move along the x axis, even though
magnetic forces generally cause the paths of charged particles to curve (Fig. 28-11).
The exception to this is discussed in section 28-3, where the forces due to the electric
and magnetic fields cancel.
This implies (Eq. 28-7) B = E/v = 2.50  102 T.




For F
= q
vB
to be in the opposite direction of F
= qE
we must have



v  B in the opposite direction from E which points in the +y direction, as
discussed in part (a).
Since the velocity is in the +x direction, then (using the
right-hand rule) we conclude that the magnetic field must point in the +z direction
( i  k
^
^
^
= j ). In unit-vector notation, we have B  (2.50 102 T)kˆ .

14. We note that B
must be along the x axis because when the velocity is along that
axis there is no induced voltage. Combining Eq. 28-7 and Eq. 28-9 leads to
 

d V / vB where one must interpret the symbols carefully to ensure that d , v and B
are mutually perpendicular. Thus, when the velocity if parallel to the y axis the
absolute value of the voltage (which is considered in the same “direction” as d ) is
0.012 V, and
d = dz = (0.012)/(3)(0.02) = 0.20 m.
And when the velocity if parallel to the z axis the absolute value of the appropriate
voltage is 0.018 V, and d = dy = (0.018)/(3)(0.02) = 0.30 m. Thus, our answers are
(a) dx = 25 cm (which we arrive at “by elimination” – since we already have figured
out dy and dz ),
(b) dy = 30 cm, and
(c) dz = 20 cm
20. Referring to the solution of problem 19 part (b), we see that r  2mK qB
implies K = (rqB)2/2m  q2m–1. Thus,
(a) K   q q p   mp m  K p   2  1 4  K p  K p  1.0MeV;
2
2
(b) K d   qd q p   mp md  K p  1 1 2  K p  1.0 MeV 2  0.50MeV.
2
2
23. (a) If v is the speed of the positron then v sin  is the component of its velocity in
the plane that is perpendicular to the magnetic field. Here  is the angle between the
velocity and the field (89°). Newton’s second law yields eBv sin  = me(v sin )2/r,
where r is the radius of the orbit. Thus r = (mev/eB) sin . The period is given by
2  9.111031 kg 
2me
2r
T


 3.58 1010 s.
19
v sin 
eB
1.60 10 C  0.100T 
The equation for r is substituted to obtain the second expression for T.
(b) The pitch is the distance traveled along the line of the magnetic field in a time
interval of one period. Thus p = vT cos . We use the kinetic energy to find the speed:
K  21 me v 2 means
2  2.00 103 eV 1.60 1019 J eV 
2K
v

 2.65 107 m s .
31
me
9.1110 kg
Thus



p  2.65 107 m s 3.58 1010 s cos 89  1.66 104 m .
(c) The orbit radius is



9.111031 kg 2.65 107 m s sin 89
me v sin 
R

 1.51103 m .
19
eB
1.60 10 C  0.100 T 


25. (a) We solve for B from m = B2qx2/8V (see Sample Problem 28-3):
B
8Vm
.
qx 2
We evaluate this expression using x = 2.00 m:
c
hc
h 0.495T .
c3.20  10 Chb2.00 mg
8 100  103 V 3.92  1025 kg
B
19
2
(b) Let N be the number of ions that are separated by the machine per unit time. The
current is i = qN and the mass that is separated per unit time is M = mN, where m is
the mass of a single ion. M has the value
M
100  106 kg
 2.78  108 kg s .
3600 s
Since N = M/m we have
c
hc
h
3.20  1019 C 2.78  108 kg s
qM
i

 2.27  102 A .
m
3.92  1025 kg
(c) Each ion deposits energy qV in the cup, so the energy deposited in time t is given
by
E  NqV t 
iqV
t  iV t .
q
For t = 1.0 h,
c
hb g
hc
E  2.27  102 A 100  103 V 3600 s  817
.  106 J .
To obtain the second expression, i/q is substituted for N.
29. We approximate the total distance by the number of revolutions times the
circumference of the orbit corresponding to the average energy. This should be a good
approximation since the deuteron receives the same energy each revolution and its
period does not depend on its energy. The deuteron accelerates twice in each cycle,
and each time it receives an energy of qV = 80  103 eV. Since its final energy is 16.6
MeV, the number of revolutions it makes is
n
16.6  106 eV
 104 .
2 80  103 eV
c
h
Its average energy during the accelerating process is 8.3 MeV. The radius of the orbit
is given by r = mv/qB, where v is the deuteron’s speed. Since this is given by
v  2 K m , the radius is
r
m 2K
1

2 Km .
qB m
qB
For the average energy
r
c
hc
hc
.  10 Ch
. Tg
b157
c160
2 8.3  106 eV 160
.  1019 J eV 3.34  1027 kg
19
h 0.375 m .
The total distance traveled is about n2r = (104)(2)(0.375) = 2.4  102 m.
36. The magnetic force on the wire is



FB  iL  B  iLˆi  By ˆj  Bz kˆ  iL  Bz ˆj  By kˆ

  0.500A   0.500m     0.0100T  ˆj   0.00300T  kˆ 
 2.50 103 ˆj  0.750 103 kˆ N.





 
38. We use dFB  idL  B , where dL  dxi and B  Bx i  By j . Thus,


FB   idL  B   idxˆi  Bx ˆi  By ˆj  i  By dxkˆ
xf
  5.0A 

xi
3.0
1.0
xf
xi
8.0 x dx   m  mT   kˆ  (0.35N)k.ˆ
2

45. We use Eq. 28-37 where  is the magnetic dipole moment of the wire loop and

B is the magnetic field, as well as Newton’s second law. Since the plane of the loop
is parallel to the incline the dipole moment is normal to the incline. The forces acting
on the cylinder are the force of gravity mg, acting downward from the center of mass,
the normal force of the incline FN, acting perpendicularly to the incline through the
center of mass, and the force of friction f, acting up the incline at the point of contact.
We take the x axis to be positive down the incline. Then the x component of Newton’s
second law for the center of mass yields
mg sin   f  ma.
For purposes of calculating the torque, we take the axis of the cylinder to be the axis
of rotation. The magnetic field produces a torque with magnitude B sin, and the
force of friction produces a torque with magnitude fr, where r is the radius of the
cylinder. The first tends to produce an angular acceleration in the counterclockwise
direction, and the second tends to produce an angular acceleration in the clockwise
direction. Newton’s second law for rotation about the center of the cylinder,  = I,
gives
fr  B sin   I .
Since we want the current that holds the cylinder in place, we set a = 0 and  = 0, and
use one equation to eliminate f from the other. The result is mgr = B. The loop is
rectangular with two sides of length L and two of length 2r, so its area is A = 2rL and
the dipole moment is  = NiA = 2NirL. Thus, mgr = 2NirLB and
b gc h
b gb gb g
0.250 kg 9.8 m s2
mg
i

 2.45 A .
2 NLB 2 10.0 0100
.
m 0.500 T
52. Let a = 30.0 cm, b = 20.0 cm, and c = 10.0 cm. From the given hint, we write
   
  0.150jˆ  0.300kˆ  A  m .

  1  2  iab kˆ  iac ˆj  ia cˆj  bkˆ   5.00A  0.300m   0.100m  ˆj   0.200m  kˆ 
2
55. If N closed loops are formed from the wire of length L, the circumference of each
loop is L/N, the radius of each loop is R = L/2N, and the area of each loop is
b
A  R 2   L 2N
g L
2
2
4N 2 .
(a) For maximum torque, we orient the plane of the loops parallel to the magnetic
field, so the dipole moment is perpendicular (i.e., at a 90 angle) to the field.
(b) The magnitude of the torque is then
L I
iL B
B
b gF
G
J
H4N K 4N .
  NiAB  Ni
2
2
2
To maximize the torque, we take the number of turns N to have the smallest possible
value, 1. Then  = iL2B/4.
(c) The magnitude of the maximum torque is

iL2 B (4.51103 A)(0.250 m) 2 (5.71103 T)

 1.28 107 N  m
4
4
Chap 29
13. Each of the semi-infinite straight wires contributes  0i 4 R (Eq. 29-7) to the
field at the center of the circle (both contributions pointing “out of the page”). The
current in the arc contributes a term given by Eq. 29-9 pointing into the page, and this
is able to produce zero total field at that location if Barc  2.00 Bsemiinfinite , or
 0i
 i 
 2.00  0 
4pR
 4pR 
which yields  = 2.00 rad.
17. Our x axis is along the wire with the origin at the midpoint. The current flows in
the positive x direction. All segments of the wire produce magnetic fields at P1 that
are out of the page. According to the Biot-Savart law, the magnitude of the field any
(infinitesimal) segment produces at P1 is given by
dB 
 0i sin 
4 r 2
dx
where  (the angle between the segment and a line drawn from the segment to P1) and
x 2  R 2 and sin 
r (the length of that line) are functions of x. Replacing r with
with R r  R
B
0iR
4p
x 2  R 2 , we integrate from x = –L/2 to x = L/2. The total field is

dx
L2
L 2
 4p 10

-7
x

0iR 1
4p R  x

T  m A   0.0582 A 
2
R
2 32
2p  0.131 m 
2
x
2
L2
R

2 1 2 L 2

 0i
L
2pR
L2  4 R 2
0.180m
(0.180m)  4(0.131m)
2
2
 5.03 108 T.
19. Our x axis is along the wire with the origin at the right endpoint, and the current is
in the positive x direction. All segments of the wire produce magnetic fields at P2 that
are out of the page. According to the Biot-Savart law, the magnitude of the field any
(infinitesimal) segment produces at P2 is given by
dB 
 0i sin 
4 r 2
dx
where  (the angle between the segment and a line drawn from the segment to P2) and
r (the length of that line) are functions of x. Replacing r with
with R r  R
B
0iR
4p
x 2  R 2 and sin 
x 2  R 2 , we integrate from x = –L to x = 0. The total field is

dx
0
x
L
 4p 10

-7
2
R

2 32

0iR 1
4p R
2
x
x
2
T  m A   0.693 A 
4p  0.251 m 
0
R

2 1 2 L

 0i
L
4pR
L  R2
2
0.136m
(0.136m)  (0.251m)
2
2
 1.32 107 T.
25. (a) Recalling the straight sections discussion in Sample Problem 29-1, we see that
the current in the straight segments collinear with P do not contribute to the field at
that point. We use the result of problem 16 to evaluate the contributions to the field at
P, noting that the nearest wire-segments (each of length a) produce magnetism into
the page at P and the further wire-segments (each of length 2a) produce magnetism
pointing out of the page at P. Thus, we find (into the page)
2  4p  10- 7 T  m A  13 A 
 2  0i   2  0i 
2  0i
BP  2 

  2 
 
8
p
a
8
p
2
a
8
p
a
8  0.047 m 



 

 1.96  10- 5 T  2.0  10- 5 T.
(b) The direction of the field is into the page.
29. We label these wires 1 through 5, left to right, and use Eq. 29-13. Then,
(a) The magnetic force on wire 1 is
-7
1
1
1  ˆ 25 0i 2l ˆ 25  4p 10 T  m A   3.00A  (10.0m) ˆ
F1 


j
j
 
 j=
2p  d 2d 3d 4d 
24pd
24p  8.00 102 m 
2
 0i 2l  1
 (4.69 104 N)jˆ
(b) Similarly, for wire 2, we have
F2 
 0i 2l  1

2p  2d

1  ˆ 50i 2l ˆ
ˆ
j=(1.88104 N)j.
 j
3d  12pd
(c) F3 = 0 (because of symmetry).
(d) F4  F2 =( 1.88104 N)jˆ , and
(e) F5   F1  (4.69 104 N)jˆ .
33. The magnitudes of the forces on the sides of the rectangle which are parallel to the
long straight wire (with i1 = 30.0 A) are computed using Eq. 29-13, but the force on
each of the sides lying perpendicular to it (along our y axis, with the origin at the top
wire and +y downward) would be figured by integrating as follows:
F sides 
z
a b
a
i2  0i1
dy.
2y
Fortunately, these forces on the two perpendicular sides of length b cancel out. For the
remaining two (parallel) sides of length L, we obtain
F

0i1i2 L  1
0i1i2b
1 
 

2  a a  d  2a  a  b 
 410
7
T  m/A   30.0A  20.0A 8.00cm   300 10 2 m 
2 1.00cm  8.00cm 
 3.20 103 N,

and F
points toward the wire, or  ˆj . In unit-vector notation, we have
F  (3.20 103 N)ˆj
38. (a) The field at the center of the pipe (point C) is due to the wire alone, with a
magnitude of
BC 
0iwire
2  3R 

0iwire
6pR
.
For the wire we have BP, wire > BC, wire. Thus, for BP = BC = BC, wire, iwire must be into
the page:
BP  BP ,wire  BP ,pipe 
 0iwire
2R

 0i
b g.
22 R
Setting BC = –BP we obtain iwire = 3i/8 = 3(8.00 103 A) / 8  3.00 103 A .
(b) The direction is into the page.
39. For r  a ,
B r  
0ienc
2pr

0
2pr

r
0
J  r  2prdr 
0
r
2p 0
 J r2
r
J 0   2prdr  0 0 .
3a
a
(a) At r  0, B  0 .
(b) At r  a / 2 , we have
B r  
0 J 0 r 2
3a

(4107 T  m/A)(310A/m2 )(3.1103 m / 2) 2
 1.0 107 T.
3
3(3.110 m)
(c) At r  a,
B r  a 
0 J 0 a (4107 T  m/A)(310A/m2 )(3.1103 m)
3

3
 4.0 107 T.
43. (a) We use Eq. 29-24. The inner radius is r = 15.0 cm, so the field there is
B
 0iN
2 r
c4  10

7
hb gb g 5.33  10
2 b
0150
.
mg
T  m / A 0.800 A 500
4
T.
(b) The outer radius is r = 20.0 cm. The field there is
B
 0iN
2 r
c4  10

7
hb gb g 4.00  10
2 b
0.200mg
T  m / A 0.800 A 500
4
T.
53. (a) We denote the large loop and small coil with subscripts 1 and 2, respectively.
B1 
 0i1
2 R1
hb g 7.9  10
2b
012
. mg
c4  10

7
T  m A 15 A
5
T.
(b) The torque has magnitude equal to
 | 2  B1 | 2 B1 sin 90  N 2i2 A2 B1  N 2i2 r22 B1
    1.3A   0.82 102 m   7.9 105 T   1.1106 N  m.
2
Chap 30
9. (a) Let L be the length of a side of the square circuit. Then the magnetic flux
through the circuit is  B  L2 B / 2 , and the induced emf is
i  
dB
L2 dB

.
dt
2 dt
Now B = 0.042 – 0.870t and dB/dt = –0.870 T/s. Thus,
(2.00 m) 2
i 
(0.870 T / s) = 1.74 V.
2
The magnetic field is out of the page and decreasing so the induced emf is
counterclockwise around the circuit, in the same direction as the emf of the battery.
The total emf is
 + i = 20.0 V + 1.74 V = 21.7 V.
(b) The current is in the sense of the total emf (counterclockwise).
11. (a) It should be emphasized that the result, given in terms of sin(2 ft), could as
easily be given in terms of cos(2 ft) or even cos(2 ft + ) where  is a phase
constant as discussed in Chapter 15. The angular position  of the rotating coil is
measured from some reference line (or plane), and which line one chooses will affect
whether the magnetic flux should be written as BA cos, BA sin or BA cos( + ).
Here our choice is such that  B  BA cos . Since the coil is rotating steadily, 
increases linearly with time. Thus,  = t (equivalent to  = 2 ft) if  is understood
to be in radians (and would be the angular velocity ). Since the area of the
rectangular coil is A=ab , Faraday’s law leads to
  N
d  BA cos 
d cos  2 ft 
  NBA
 N Bab2 f sin  2 ft 
dt
dt
which is the desired result, shown in the problem statement. The second way this is
written (0 sin(2ft)) is meant to emphasize that the voltage output is sinusoidal (in its
time dependence) and has an amplitude of 0 = 2f N abB.
(b) We solve 0 = 150 V = 2f N abB when f = 60.0 rev/s and B = 0.500 T. The three
unknowns are N, a, and b which occur in a product; thus, we obtain N ab = 0.796 m2.
15. (a) The frequency is
f
 (40 rev/s)(2 rad/rev)

 40 Hz .
2
2
(b) First, we define angle relative to the plane of Fig. 30-44, such that the semicircular
wire is in the  = 0 position and a quarter of a period (of revolution) later it will be in
the  = /2 position (where its midpoint will reach a distance of a above the plane of
the figure). At the moment it is in the  = /2 position, the area enclosed by the
“circuit” will appear to us (as we look down at the figure) to that of a simple rectangle
(call this area A0 which is the area it will again appear to enclose when the wire is in
the  = 3/2 position). Since the area of the semicircle is a2/2 then the area (as it
appears to us) enclosed by the circuit, as a function of our angle , is
A  A0 
a 2
cos
2
where (since  is increasing at a steady rate) the angle depends linearly on time,
which we can write either as  = t or  = 2ft if we take t = 0 to be a moment when

the arc is in the  = 0 position. Since B is uniform (in space) and constant (in time),
Faraday’s law leads to
d
i
b g
d A0  a2 cos
d B
dA
a 2 d cos 2 ft

 B
 B
 B
dt
dt
dt
2
dt
2
which yields  = B2 a2 f sin(2ft). This (due to the sinusoidal dependence) reinforces
the conclusion in part (a) and also (due to the factors in front of the sine) provides the
voltage amplitude:
 m  B 2 a 2 f  (0.020 T) 2 (0.020 m)2 (40 / s)  3.2 103 V.
23. (a) Consider a (thin) strip of area of height dy and width   0.020 m . The strip is
located at some 0  y   . The element of flux through the strip is
c hb g
d B  BdA  4t 2 y dy
where SI units (and 2 significant figures) are understood. To find the total flux
through the square loop, we integrate:
B 
Thus, Faraday’s law yields
z zc
d B 

0
h
4t 2 y dy  2t 2  3 .
 
d B
 4 t 3 .
dt
At t = 2.5 s, we find the magnitude of the induced emf is 8.0  10–5 V.
(b) Its “direction” (or “sense’’) is clockwise, by Lenz’s law.
30. Noting that Fnet = BiL – mg = 0, we solve for the current:
i
mg |  | 1 d B
B dA Bvt L
 


,
BL R R dt
R dt
R
which yields vt = mgR/B2L2.
32. (a) The “height” of the triangular area enclosed by the rails and bar is the same
as the distance traveled in time v: d = vt, where v = 5.20 m/s. We also note that the
“base” of that triangle (the distance between the intersection points of the bar with the
rails) is 2d. Thus, the area of the triangle is
A
1
1
( base)(height)  (2vt )(vt )  v 2 t 2 .
2
2
Since the field is a uniform B = 0.350 T, then the magnitude of the flux (in SI units) is
B = BA = (0.350)(5.20)2t2 = 9.46t2.
At t = 3.00 s, we obtain B = 85.2 Wb.
(b) The magnitude of the emf is the (absolute value of) Faraday’s law:
d B
dt 2

 9.46
 18.9t
dt
dt
in SI units. At t = 3.00 s, this yields  = 56.8 V.
(c) Our calculation in part (b) shows that n = 1.
37. The magnetic field B can be expressed as
bg
b
g
B t  B0  B1 sin t   0 ,
where B0 = (30.0 T + 29.6 T)/2 = 29.8 T and B1 = (30.0 T – 29.6 T)/2 = 0.200 T. Then
from Eq. 30-25
E
F
I
G
HJ
K
b
b
g
g
1 dB
r d
1
r
B0  B1 sin t   0  B1r cos t   0 .
2 dt
2 dt
2
We note that  = 2 f and that the factor in front of the cosine is the maximum value
of the field. Consequently,
E max 
b g b
gb gb gc
h
1
1
B1 2 f r  0.200 T 2  15 Hz 16
.  102 m  015
. V / m.
2
2
41. We refer to the (very large) wire length as  and seek to compute the flux per
meter:  B / . Using the right-hand rule discussed in Chapter 29, we see that the
net field in the region between the axes of antiparallel currents is the addition of the
magnitudes of their individual fields, as given by Eq. 29-17 and Eq. 29-20. There is
an evident reflection symmetry in the problem, where the plane of symmetry is
midway between the two wires (at x = d/2); the net field at any point 0 < x < d/2 is the
same at its “mirror image” point
d – x. The central axis of one of the wires passes through the origin, and that of the
other passes through x = d. We make use of the symmetry by integrating over 0 < x <
d/2 and then multiplying by 2:
 B  2
d /2
0
B dA  2  B  dx   2 
a
d /2
0
a
B  dx 
where d = 0.0025 m is the diameter of each wire. We will use r instead of x in the
following steps. Thus, using the equations from Ch. 29 referred to above, we find
B
a   i
d /2
 0i 
 2   0 2 r 
 dr  2 a
0
2  d  r  
 2 a
i
 d  a    0i  d  a 
 0 1  2 ln 
ln 
 

2 
 d    a 
  0i
 0i 


 dr
 2r 2  d  r  
where the first term is the flux within the wires and will be neglected (as the problem
cb g h
suggests). Thus, the flux is approximately   B   0i /  ln d  a / a . Now, we use
Eq. 30-33 (with N = 1) to obtain the inductance per unit length:
L

7
 B 0
 d  a  (410 T  m/A)  142 1.53 
6

ln 

ln 

  1.8110 H/m.
i


 a 
 1.53 
46. (a) Voltage is proportional to inductance (by Eq. 30-35) just as, for resistors, it is
proportional to resistance. Now, the (independent) voltages for parallel elements are
equal (V1 = V2), and the currents (which are generally functions of time) add (i1 (t) + i2
(t) = i(t)). This leads to the Eq. 27-21 for resistors. We note that this condition on the
currents implies
bg bg bg
di1 t di2 t
di t


.
dt
dt
dt
Thus, although the inductance equation Eq. 30-35 involves the rate of change of
current, as opposed to current itself, the conditions that led to the parallel resistor
formula also applies to inductors. Therefore,
1
1
1
  .
Leq L1 L2
Note that to ensure the independence of the voltage values, it is important that the
inductors not be too close together (the related topic of mutual inductance is treated in
§30-12). The requirement is that the field of one inductor not have significant
influence (or “coupling’’) in the next.
(b) Just as with resistors,
1
Leq
 n 1 L1n .
N
55. Applying the loop theorem
L
di I
F
G
Hdt J
K iR ,
we solve for the (time-dependent) emf, with SI units understood:
di
d
 iR  L  3.0  5.0t    3.0  5.0t  R   6.0  5.0    3.0  5.0t  4.0 
dt
dt
  42  20t  .
 L
62. (a) The energy delivered by the battery is the integral of Eq. 28-14 (where we use
Eq. 30-41 for the current):

t
0
Pbattery dt  
t
2
0
R
1  e Rt L  dt 
2 
L  Rt L

t

e
 1 


R R


 

 5.50 H  e6.70  2.00 s 5.50 H  1
 2.00 s 
6.70  
6.70 

 18.7 J.
10.0 V 

2


(b) The energy stored in the magnetic field is given by Eq. 30-49:
2
1 2
1  
Li  t   L   1  e  Rt L 
2
2 R
2
UB 
2
2
 10.0 V 
1
 6.70   2.00 s  5.50 H

  5.50 H  
 1  e

2
 6.70  
 5.10 J .
(c) The difference of the previous two results gives the amount “lost” in the resistor:
18.7 J – 5.10 J = 13.6 J.
67. (a) The energy per unit volume associated with the magnetic field is
7
2
0i 2  410 H m  10 A 
B2
1   0i 
3
uB 


 1.0 J m .

 
2
2

3
20 20  2 R 
8R
8  2.5 10 m 2 
2
(b) The electric energy density is
2

  iR  1
1
2
uE   0 E 2  0   J   0     8.85 1012 F m  10A   3.3  103 m  
2
2
2  2
2
 4.8 1015 J m .
3
Here we used J = i/A and R   A to obtain J  iR  .
71. (a) We assume the current is changing at (nonzero) rate di/dt and calculate the
total emf across both coils. First consider the coil 1. The magnetic field due to the
current in that coil points to the right. The magnetic field due to the current in coil 2
also points to the right. When the current increases, both fields increase and both
changes in flux contribute emf’s in the same direction. Thus, the induced emf’s are
b
 1   L1  M
gdtdi
b
and  2   L2  M
gdtdi .
Therefore, the total emf across both coils is
b
   1   2   L1  L2  2 M
gdtdi
which is exactly the emf that would be produced if the coils were replaced by a single
coil with inductance Leq = L1 + L2 + 2M.
(b) We imagine reversing the leads of coil 2 so the current enters at the back of coil
rather than the front (as pictured in the diagram). Then the field produced by coil 2 at
the site of coil 1 is opposite to the field produced by coil 1 itself. The fluxes have
opposite signs. An increasing current in coil 1 tends to increase the flux in that coil,
but an increasing current in coil 2 tends to decrease it. The emf across coil 1 is
b
 1   L1  M
Similarly, the emf across coil 2 is
gdtdi .
b
 2   L2  M
gdtdi .
The total emf across both coils is
b
   L1  L2  2 M
gdtdi .
This the same as the emf that would be produced by a single coil with inductance
Leq = L1 + L2 – 2M.