### Home Work Solutions 10 F HG I KJFHGIKJ F HG I KJ F HGIKJ

```Home Work Solutions 10
10-1 Figure A shows two parallel loops of wire having a common axis. The smaller
loop (radius r) is above the larger loop (radius R) by a distance x>>R. Consequently,
the magnetic field due to the counterclockwise current i in the larger loop is nearly
uniform throughout the smaller loop. Suppose that x is increasing at the constant rate
dx/dt = v. (a) Find an expression for the magnetic flux through the area of the smaller
loop as a function of x. (Hint: See Eq. 29-27.) In the smaller loop, find (b) an
expression for the induced emf and (c) the direction of the induced current.
Sol: (a) In the region of the smaller loop the magnetic field produced by the larger
loop may be taken to be uniform and equal to its value at the center of the
smaller loop, on the axis. Eq. 29-27, with z = x (taken to be much greater than
R), gives
  0iR 2
i
B
2x3
where the +x direction is upward in Fig. 30-50. The magnetic flux through the
smaller loop is, to a good approximation, the product of this field and the area
(r2) of the smaller loop:
 0ir 2 R 2
.
2x3
(b) The emf is given by Faraday’s law:
B 

F
G
H
d B
 0ir 2 R 2

dt
2
IJd F
F ir R IJF
1I
3 dx I 3 ir R v
 G

G
J
G
K 2 x .
Kdt Hx K H 2 KHx dt J
2
2
2
0
3
0
4
(c) As the smaller loop moves upward, the flux through it decreases, and we have
a situation like that shown in Fig. 30-5(b). The induced current will be directed
so as to produce a magnetic field that is upward through the smaller loop, in
the same direction as the field of the larger loop. It will be counterclockwise as
viewed from above, in the same direction as the current in the larger loop.
10-2 Figure B shows a copper strip of width W = 16.0 cm that has been bent to form a
shape that consists of a tube of radius R = 1.8 cm plus two parallel flat extensions.
Current i = 35 mA is distributed uniformly across the width so that the tube is
effectively a one-turn solenoid. Assume that the magnetic field outside the tube is
negligible and the field inside the tube is uniform. What are (a) the magnetic field
magnitude inside the tube and (b) the inductance of the tube (excluding the flat
extensions)?
4
2
Sol: (a) We imagine dividing the one-turn solenoid into N small circular loops placed
along the width W of the copper strip. Each loop carries a current i = i/N.
Then the magnetic field inside the solenoid is
7
 N  i   i (410 T  m/A)(0.035A)
B  0 ni  0     0 
 2.7 107 T.
0.16m
 W  N  W
(b) Eq. 30-33 leads to
L
2
 B R 2 B R  0i / W  0 R 2  (4107 T  m/A)(0.018m)2




 8.0 109 H.
i
i
i
W
0.16m
10-3 A coil C of N turns is placed around a long solenoid S of radius R and n turns per
unit length, as in Figure C. (a) Show that the mutual inductance for the coil–solenoid
combination is given by M = μ0πR2nN. (b) Explain why M does not depend on the
shape, size, or possible lack of close packing of the coil.
Sol: (a) The coil-solenoid mutual inductance is
M  M cs 
c
h
2
N cs N  0is nR

  0 R 2 nN .
is
is
(b) As long as the magnetic field of the solenoid is entirely contained within the
cross-section of the coil we have sc = BsAs = BsR2, regardless of the shape,
size, or possible lack of close-packing of the coil.
Fig. A
Fig. B
Fig. C
10-4 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
(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?
Sol: (a) Voltage is proportional to inductance (by Eq. 30-35) just as, for resistors, it is
proportional to resistance. Since the (independent) voltages for series elements
add (V1 + V2), then inductances in series must add, Leq  L1  L2 , just as was the
case for resistances.
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 Section 30-12). The requirement is that magnetic field lines from one
inductor should not have significant presence in any other.
(b) Just as with resistors, Leq  n 1 Ln .
N
10-5 Inductors in parallel. Two inductors L1 and L2 are connected in parallel and
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
(Hint: Review the derivations for resistors in parallel and capacitors in parallel.
Which is similar here?) (b) What is the generalization of (a) for N inductors in
parallel?
Sol: (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
  di t   di t .
di1 t
dt
2
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 apply 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 Section 30-12). The requirement is that the field of one inductor not to
have significant influence (or “coupling’’) in the next.
N
1
1
 .
(b) Just as with resistors,
Leq n 1 Ln
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