# Lecture 18

```Homework (lecture 14): 1, 7, 8, 12, 14, 15, 20, 27, 29,
32, 37, 40, 47, 55, 56, 63, 68, 73
1. In the figure below, a circular loop of wire 10 cm in diameter
(seen edge-on) is placed with its normal N at an angle  = 30° with
the direction of a uniform magnetic field B of magnitude 0.50 T.
The loop is then rotated such that N rotates in a cone about the
field direction at the rate 100 rev/min; angle  remains unchanged
during the process. What is the emf induced in the loop?
d B
 
dt
 B  B.A. cos   constant
Therefore:
 0
7. In the figure below, the magnetic flux through the loop
increases according to the relation B = 6.0t2 + 7.0t, where
B is in milliwebers and t is in seconds. (a) What is the
magnitude of the emf induced in the loop when t = 2.0 s?
(b) Is the direction of the current through R to the right or
left?
d B
(a) Using Faraday’s law:   
dt
|  | (12t  7)  31(mV )
(b) B increases, using Lenz’s law, Binduced
should be point into the page, so the
current through R is to the left
i
8. A uniform magnetic field B is perpendicular to the plane
of a circular loop of diameter 12 cm formed from wire of
diameter 2.5 mm and resistivity 1.69 x 10-8 .m. At what
rate must the magnitude of B change to induce a 10 A
current in the loop?
l
2 0.06

8
R    1.69 10 
 1.3 10 3 ()
A
0.0025 2

4

1 d B
r 2 dB
i 

R
R dt
R dt
dB
iR 10 1.3 10 3


 1.2(T / s)
2
2
dt
r
  (0.06)
12. In Fig. 30-45, a wire loop of lengths L = 50.0 cm and W =
20.0 cm lies in a magnetic field B. What are the (a) magnitude 
and (b) direction (clockwise or counterclockwise-or
"none“ if  = 0)

of the emf induced in the loop if
102 T / m) ykˆ ? What are
 B  (4.00 
2
ˆ .? What are (e) 
(c)  and (d) the direction
if
B

(
6
.
00

10
T
/
s
)
t
k

2
and (f) the direction if
B

(
8
.
00

10
T / m.s) ytkˆ ? What are (g)  and

(h) the direction if  B  (3.00 102 T / m.s) xtˆj ?
What are (i)  and
(j) the direction if B  (5.00 10 2 T / m.s) yt iˆ ?
(a) zero; (b) none
d B
 0.06  0.5  0.20  0.006(V )
(c)   
dt
(d) dB/dt increases, so the induced emf
direction is clockwise
14. In Fig. 30-41a, a uniform magnetic field B increases in
magnitude with time t as given by Fig. 30-41b. A circular
conducting loop of area 8.0 x 10-4 m2 lies in the field, in the plane
of the page. The amount of charge q passing point A on the loop is
given in Fig. 30-41c as a function of t. What is the loop's
resistance?
d B
dB
 
 A
 0.003 A (V )
dt
dt
dq
i
 0.002 ( A)
dt
|  | 0.003  8 10 4
R

 0.0012()
i
0.002
15. A square wire loop with 2.00 m sides is perpendicular to a
uniform magnetic field, with half the area of the loop in the field
as shown in the figure below. The loop contains an ideal battery
with emf  = 20.0 V. If the magnitude of the field varies with
time according to B = 0.0420 - 0.870t, with B in tesla and t in
seconds, what are (a) the net emf in the circuit and (b) the
direction of the (net) current around the loop?
(a) As time goes on, B decreases
2
d B
L
 

(0.87)  1.74(V )
dt
2
The induced B points out of the page, so the
induced current as well as the induced emf is
counterclockwise.
(b) The net current is counterclockwise:
 net     i
20. At a certain place, Earth's magnetic field has magnitude B =
0.590 gauss and is inclined downward at an angle of 70.0° to the
horizontal. A flat horizontal circular coil of wire with a radius of
10.0 cm has 2500 turns and a total resistance of 85.0 . It is
connected in series to a meter with 140  resistance. The coil is
flipped through a half-revolution about a diameter, so that it is
again horizontal. How much charge flows through the meter during
the flip?
d B
dq
 
;i 
dt
dt
d B

B
| |
dq
dt
i


R
dt
R
| d B | N
| dq |
 BA cos   ( BA cos  )
R
R
2 NBA cos 
0

4
| dq |
with   20 ; 1 gauss  10 T
R
27. As seen in Fig. 30-49, a square loop of wire has sides of
length 2.0 cm. A magnetic field is directed out of the page; its
magnitude is given by B = 4.0t2y, where B is in teslas, t is in
seconds, and y is in meters. At t = 2.5 s, what are the (a)
magnitude and (b) direction of the emf induced in the loop?
d B
 
dt
(a)
l
 
 B  BdA  4t 2 y (ldy)  4lt 2 ydy  2l 3t 2


d B
 
 4l 3t
dt

0
dA
At t = 2.5 s:
|  | 4l 3t  8 10 5 (V )
(b) B increases, the current direction is clockwise, so the induced
emf direction is clockwise
29. In Fig. 30-52, a metal rod is forced to move with constant
velocity v along two parallel metal rails, connected with a strip of
metal at one end. A magnetic field of magnitude B = 0.350 T
points out of the page. (a) If the rails are separated by L = 25.0
cm and the speed of the rod is 55.0 cm/s, what emf is generated?
(b) If the rod has a resistance of 18.0  and the rails and
connector have negligible resistance, what is the current in the
rod? (c) At what rate is energy being transferred to thermal
energy?
(a)
(b)
d B
d ( BLvt )


 BvL
dt
dt
  0.35  0.55  0.25  0.048(V )
 0.048
3
i
R

18
 2.7 10
( A)
using Lenz’s law: the current direction is clockwise
(c)
P  i 2 R  0.13(mW )
32. A loop antenna of area 2.00 cm2 and resistance 5.21  is
perpendicular to a uniform magnetic field of magnitude 21.0 T.
The field magnitude drops to zero in 2.96 ms. How much thermal
energy is produced in the loop by the change in field?
d B
B  A 2110 6  2 10 4

6
|  |


 1.4 10 (V )
dt
t
2.96 10 3
Ethermal  i 2 Rt 
2
R
t 
1.4 2 10 12
5.2110 6
2.96 10 3  1110 10 ( J )
37. A long solenoid has a diameter of 12.0 cm. When a current i
exists in its windings, a uniform magnetic field of magnitude B =
30.0 mT is produced in its interior. By decreasing i, the field is
caused to decrease at the rate of 6.50 mT/s. Calculate the
magnitude of the induced electric field (a) 2.20 cm and (b) 8.20 cm
from the axis of the solenoid (see also Sample Problem 30-4, page
804)


(a)
(b)

d B

E.ds  
; E.ds  E 2r; B  BA  Br 2
dt
r dB 2.2 10 2
E

6.5 10 3  7.15 10 5 (V / m)
2 dt
2
 
E.ds  E 2r; B  BA  BR 2



R 2 dB
6.0 2 10 4
E

6.5 10 3
2r dt 2  8.2 10 2
 1.43 10 4 (V / m)
Figure 30-15 of Sample Problem 30-4
40. The inductance of a closely packed coil of 400 turns is 8.0 mH.
Calculate the magnetic flux through the coil when the current is
12.0 mA.
• B: magnetic flux
Li 8.0 10 3 12.0 10 3
B 

 2.4 10 7 (Wb)
N
400
47. Inductors in series. Two inductors L1 and L2 are connected in
series and are separated by a large distance so that the magnetic
field of one cannot affect the other. (a) Show that the equivalent
inductance is given by Leq = L1 + L2. (Hint: Review the derivations
for resistors in series and capacitors in series. Which is similar
here?) (b) What is the generalization of (a) for N inductors in
series?
di
 L  L
dt
Consider two inductors in series:
di
di
 L1   L1 ;  L2   L2
dt
dt
di 
 Leq   Leq ; Leq   L1   L2
dt
 Leq  L1  L2
55. A battery is connected to a series RL circuit at time t = 0. At
what multiple of L will the current be 0.100% less than its
equilibrium value?

i  1  e t /  L 
The equilibrium value:
R
iequil. 
Time at which:


R
i  0.999iequil.  0.999

R
t  6.91 L

t /  L 
 1  e

R

Homework (lecture 15): 1, 2, 7, 9, 10, 17, 23, 25, 29, 32,
38, 41, 48, 53, 57, 62
1. An oscillating LC circuit consists of a 75.0 mH inductor and a
3.60 F capacitor. If the maximum charge on the capacitor is 2.90
C, what are (a) the total energy in the circuit and (b) the
maximum current?
q 2 Li 2
U  U E U B 

2C
2
(a) When q is maximum:
2
qmax
U  U E ,max 
2C
(b) i is maximum when q = 0:
2
Limax
U B,max 
 U E ,max
2
2. The frequency of oscillation of a certain LC circuit is 220 kHz.
At time t = 0, plate A of the capacitor has maximum positive
charge. At what earliest time t > 0 will (a) plate A again have
maximum positive charge, (b) the other plate of the capacitor have
maximum positive charge, and (c) the inductor have maximum
magnetic field?
q  Q cos(t   )
Determine  from the conditions given in the problem, at t = 0:
q  Q cos  i s maxi mum, so   0
(a) So, q is max again as T = (2/) x n
T  2 LC  1 / f  4.6 10
6
(b) plate B has maximum positive charge at:
(c)
(s)  4.6(s)
1
t  T  (n  1)T  t  2.3( s)
2
2
Q
T (n  1)
2
UB 
sin (t   )  UB max as t  
T
2C
4
2
 t  1.15(s)
7. The energy in an oscillating LC circuit containing a 1.25 H
inductor is 5.70 J. The maximum charge on the capacitor is 175
C. For a mechanical system with the same period, find the (a)
mass, (b) spring constant, (c) maximum displacement, and (d)
maximum speed.
(a) mass m = 1.25 kg
(b) spring constant k = 1/C
Q2
Q 2 (175 10 6 ) 2
U
C 

 2.69 10 3 ( F )
2C
2U 2  5.7 10 6
1
k
 372( N / m)

3
2.69 10
(c) xmax = 175 m = 1.75 x 10-4 (m)
(d)
Li 2 Q 2
Q

i 
 3.02 10 3 ( A)
2
2C
LC
vmax  3.02 10 3 (m / s)
9. In an oscillating LC circuit with L = 50 mH and C = 4.0 F, the
current is initially a maximum. How long will it take before the
capacitor is fully charged for the first time?
i   I sin(t   )
At t = 0, i is max:
   / 2
i   I sin(t   / 2)
when the capacitor is fully charged, i = 0:
t 
T
t   / 2  2   t 
T 2
4
T 2 LC 2 50 10 3  4 10 6
t 

 7 10 4 ( s)
4
4
4
10. LC oscillators have been used in circuits connected to
loudspeakers to create some of the sounds of electronic music.
What inductance must be used with a 3.4 F capacitor to produce
a frequency of 10 kHz, which is near the middle of the audible
range of frequencies?
T  2 LC  1 / f  L 
1
4 2 f 2C
17. In Fig. 31-25, R = 14.0 , C = 6.20 F, and L = 54.0 mH,
and the ideal battery has emf  = 34.0 V. The switch is kept at a
for a long time and then thrown to position b. What are the (a)
frequency and (b) current amplitude of the resulting oscillations?
(a)
T  2 LC  1 / f  f 
1
2 LC
(b) When the capacitor is fully charge
(the switch is on a):
Q  C
The maximum current when the switch is on b:
I  Q  2fQ
23. In an oscillating LC circuit, L = 25.0 mH and C = 7.80 F.
At time t = 0 the current is 9.20 mA, the charge on the capacitor
is 3.80 C, and the capacitor is charging. What are (a) the total
energy in the circuit, (b) the maximum charge on the capacitor, and
(c) the maximum current? (d) If the charge on the capacitor is
given by q = Qcos(t + ), what is the phase angle ? (e) Suppose
the data are the same, except that the capacitor is discharging at
t = 0. What then is ?
(a) t = 0:
q 2 Li 2
U  U E U B 
2C

2
(b) the maximum charge:
Q2
U  U E ,max 
 Q  2CU
2C
(c) the maximum current:
(d) the charge
LI 2
U  U B,max 
I 
2
is given:
At t = 0: q = 3.8 C
q  Q cos(t   )
2U
L
q
cos      47 0
Q
The capacitor is charging at t = 0:
dq
 Q sin(t   )  Q sin   0
dt
  47 0
(d) if the capacitor is discharging:
dq
 Q sin(t   )  Q sin   0
dt
  47 0
25. What resistance R should be connected in series with an
inductance L = 220 mH and capacitance C = 12.0 F for the
maximum charge on the capacitor to decay to 99.0% of its initial
value in 50.0 cycles? (Assume ‘  ).
q  Qe  Rt / 2L cos( ' t   )
 '   2  ( R / 2 L) 2
We have:
  1 / LC
qmax
 e  Rt / 2 L  0.99
Q
t  50T  50
2

 100 LC
qmax
2 L qmax
 Rt / 2 L  ln(
)R
ln(
)
Q
t
Q
29. A 50.0 mH inductor is connected as in Fig. 31-10a to an ac
generator with m = 30.0 V. What is the amplitude of the resulting
alternating current if the frequency of the emf is (a) 1.00 kHz and
(b) 8.00 kHz?
iL  I L sin(d t   )
VL  I L X L   m
X L  d L  2fL
m
m
IL 

X L 2f d L
32. An ac generator has emf  = msindt, with m = 25.0 V and d
= 377 rad/s. It is connected to a 12.7 H inductor. (a) What is the
maximum value of the current? (b) When the current is a maximum,
what is the emf of the generator? (c) When the emf of the
generator is -12.5 V and increasing in magnitude, what is the
current?
m
m
25
3
I




5
.
22

10
( A)
L
(a)
X L d L 377 12.7

iL  I L sin(d t  )
2

(b) when iL is maximum, sin(d t  )  1  d t  (2n  1)
2
   m sin(d t )  0
(c)
1
7
11
sin(d t )    d t  2n   or d t  2n  
2
6
6
d
7
  md cos(d t )  0  d t  2n  
dt
6
2
iL  I L sin(2n   )  5.22 10 3  0.866  4.52 10 3 ( A)
3
38. The current amplitude I versus driving angular frequency d for
a driven RLC circuit is given in Fig. 31-26. The inductance is 200
H, and the emf amplitude is 8.0 V. What are (a) C and (b) R?
(a) The current I is maximum when:
d  25 103 (rad / s)
d   
1
C 
LC
(b) At resonance:
1
d2 L
ZR
m
8
RZ 
  2()
I
4
41. In Fig. 31-7, set R = 200 , C = 70.0 F, L = 230 mH, fd =
60.0 Hz, and m = 36.0 V. What are (a) Z, (b) , and (c) I? (d)
Draw a phasor diagram.
Z  R2  ( X L  X C )2
X L  XC
tan  
R
If  > 0: m leads the current
If  < 0: the current leads m
I
m
Z
```