31 
Given a long straight cylindrical conductor of radius a m, in free space, carrying a constant current I distributed uniformly over its cross section. Plot the magnetic field intensity H vs distance from the center of the conductor in terms of the current I for values of distance varying between 0 and 4a. Assume there are no magnetic fields other than the one produced by the conductor in this problem.

32 
Figure 344 shows a crosssectional view of a straight cylindrical conductor of radius a m carrying a constant current of I amp distributed uniformly over its cross section. Assume the conductor to be in free space.
(a) Determine the value of the line integral of the magnetic field intensity H taken (1) around the triangular path 1 and (2) around the circular path 2.
(b) How would the values of part (a) above be affected if the current I in the conductor remained unchanged but the conductor were surrounded by, and electrically insulated from, a medium having a relative magnetic permeability of μ_{r}?
(c) Another identical conductor runs parallel to the conductor in Fig. 344, the spacing being 5a between centers. The second conductor carries a current of 10a. Repeat part (a) if the current in the conductors is flowing (1) in the same direction (2) in opposite directions.

Figure 344. Cross section of a cylindrical conductor for Problem 32. 

33 
A magnetic circuit linking a winding of two coils having N_{1} and N_{2} turns respectively is shown in Fig. 345. The two coils are connected in series aiding, i.e., the direction of the current in each coil is such as to produce flux in the same direction around the core. The current in both coils is I amp.
(a) Determine the value of the line integral of the magnetic field intensity H taken around paths 1, 2, 3, 4, and 5 in Fig. 345.
(b) Repeat part (a) above but with the polarity of the lower coil (the one with N_{2} turns) reversed.

Figure 345. Magnetic core with a winding of 2 coils for Problem 33.


34 
A toroid as shown in Fig. 39 has a ratio of R_{2}/R_{1} = 1.5 and has a uniform magnetic permeability throughout. Determine the percent error in using Eq. 342(a) instead of 342 to determine the flux in the core.

35 
(a) Determine the current in the exciting winding of the magnetic circuit shown in Fig. 316 when the flux in the core is 19,000 maxwells. The core is comprised of U.S.S. Transformer 72, 29 Gage Steel. Stacking factor = 0.95.
(b) What is the ratio of this value of current to the one in Example 31 for the same magnetic structure ?
(c) How does this ratio compare with the corresponding ratio of fluxes ?

36 
The threelegged core in Fig. 317 is stacked to a depth of 1.50 in. The laminations are of 0.0140in. U.S.S. Electrical Sheet Steel. Use Fig. 311 to determine the flux in the center leg and in the outer legs for a current of
(a) 10.0 amp.
(b) 3.0 amp. Assume a stacking factor of 0.90.

37 
Plot in rectangular coordinates on the same graph curves of flux vs ampere turns for the magnetic circuit of Fig, 319, using the curve in Fig. 311 if the iron is U.S.S. Electrical Sheet Steel, for
(a) The iron alone.
(b) The air gap alone for g =0.100 in.
(c) The combination of iron and air for g =0.100 in. for values of flux up to 25,000 lines using the circuit of Fig. 319 and assuming a stacking factor of 0.90. Correct for fringing but neglect leakage.

38 
(a) In Problem 37 what is the mmf when the flux is 12,500 lines, required by
(i) The iron?
(ii) The air gap ?
(iii) The entire magnetic circuit ?
(b) If the permeability of the iron were doubled for the conditions above at 12,500 lines, what would be the mmf required by
(i) The iron?
(ii) The air gap ?
(iii) The entire magnetic circuit ?

39 
Determine the mmf for a flux of 10,000 maxwells in the core of Fig. 319 for an air gap 0.100 in. long. Neglect leakage flux, but correct for fringing.

310 
Using the graphical construction shown in Fig. 322 determine the values of flux in the magnetic structure of Example 34 and Fig. 319 for the following values of mmf: 200, 400, 600, 800, and 1,200 amp turns. Plot, in rectangular coordinates, these values of flux for the various mmfs above.

311 
Determine the flux in the magnetic structure of Problem 310 if the air gap has a length of 0.100 in. and the current in the 600turn exciting winding is 2.5 amp.

312 
The core of a large 60cycle power transformer weighs 100 tons. The core material is comprised of 4.25 percent silicon steel laminations 0.014 in. thick. The specific weight of the core material is 465 lb per cu ft. The magnetic characteristics of this material are shown graphically in Fig. 311. Determine:
(a) The energy stored magnetically in a cubic inch of the core when the flux density is 80 kilolines per sq in. (Assume a linear relationship between B and H.)
(b) The energy stored in the entire core when the flux density is 80 kilolines per sq in.
(c) The energy in joules per kilogram for a flux density of 80 kilolines per sq in.

313 
If the permeance, i.e., the ratio of flux to mmf, of the core in Problem 312 above were reduced 20 percent, what would be the energy stored in the entire magnetic circuit when the flux density is 80 kilolines per sq in. ? (Assume a linear relationship between B and H.)

314 
The electromagnet in Fig. 346 is made up of laminations stacked to a height of 1 in. The winding consists of 8,000 turns of No. 28 enamel wire. Neglect the reluctance of the iron, the effect of leakage, and fringing and determine the current required to develop a force of 37.5 lb when
(a) The air gap length is 0.05 in.
(b) The air gap length is 0.10 in.

Figure 346. Electromagnet for Problem 314. 

315 
In Problem 314 how much energy is stored in the air gap when its length is
0.10 in. and the force is 37.5 lb?

316 
Figure 347 shows an Alnico V permanent magnet with soft iron pole shoes and an air gap g. The demagnetization curve for this material is given below.
H oersteds

0

100

200

300

400

500

550

600

B kilogauss

12.5

12.3

12.0

11.5

10.7

8.8

6.6

0


Figure 347. Alnico magnet with soft iron pole shoes and air gap for Problem 316. 
Plot a demagnetization curve similar to that shown in Fig. 331 (a) and determine the flux for an air gap length.
(a) g =0.125 in.
(b) g = 0.250 in.
(c) g = 0.060 in.
Neglect leakage but correct for fringing at the air gap.

317 
If the reluctance of the iron can be neglected and the alnico in the magnet of Problem 316 were to be replaced by silicon steel with negligible residual flux, how many ampere turns would need to be furnished to produce the same values of flux in the respective air gaps of Problem 316?

318 
A permanent magnetmoving coil instrument has a magnetic circuit similar in configuration to that shown in Fig. 327. The volume of the double air gap (the two air gaps in series) is 0.24 cu in. The magnet flux in the structure is furnished by an alnico magnet that has the demagnetization characteristic shown in Fig. 331 (a). Neglect leakage, fringing, and the reluctance of the soft iron portions and determine the smallest volume of alnico that will furnish the air gap with a flux density of 70 kilolines per sq in.

319 
The rotor of a 2pole permanentmagnet generator consists of Alnico V material and has an effective length of 1.00 in. (see Fig. 335(b) for rotor shape). The effective crosssectional area is 1/8 sq in. (1/4 x 1/2). The minimum double air gap has an equivalent length of 0.020 in. and a maximum equivalent gap length of 0.065 in. The characteristic of the Alnico V material is listed in the table of Problem 316. Determine
(a) The flux when the rotor position is such that the air gap is a maximum.
(b) The flux when the rotor position is such that the air gap is minimum after having occupied a position of maximum air gap.

320 
The following data define the upper half of the hysteresis loop for a sample of U.S.S. Transformer 72, 29 Gage Steel.
B Kilolines per square inch

0 
10 
20 
30 
40 
45 
50 
55 
60 
64.3 
H ampere turns per inch

0.85 
1.00 
1.17 
1.45 
1.87 
2.14 
2.50 
2.97 
3.65 
4.45 
B Kilolines per square inch

60 
55 
50 
45 
40 
30 
20 
10 
0 
H ampere turns per inch

2.65 
1.47 
0.80 
0.37 
0.05 
0.35 
0.60 
0.73 
0.85 
Plot the complete hysteresis loop and determine
(a) The residual flux density in (i) Kilolines per square inch, (ii) Gausses.
(b) The coercive force in
(i) Ampere turns per inch, (ii) Oersteds.
(c) The energy product in joules for a flux density of B = +24 kilolines per square inch on the decreasing portion of the loop.

321 
On the basis of the hysteresis loop of Problem 320 determine the hysteresis loss in joules per cycle per cubic inch of iron at a maximum flux density of 64.3 kilolines per sq in.

322 
A transformer core has a mean length of 50 in. and a uniform gross crosssectional area of 20 sq in. The core material is the same as that of Problem 320.
(a) Determine the hysteresis loss in watts for a frequency of 60 cps and a maximum flux density of 64.3 kilolines per sq in. The stacking factor is 0.90.
(b) What is the hysteresis loss in watts per pound if the density is 0.272 lb per cu in. ?

323 
The thickness of 29 gage sheets is 0.0140 in. and the resistivity of the core material in Problem 322 is 22 microhms per cu in. Determine the eddycurrent loss in the core of Problem 322 for a frequency of 60 cps and a maximum flux density of 64.3 kilolines per sq in.

324 
Determine the total core loss for a transformer core of Problem 322 if the flux density is 64.3 kilolines per sq in. and the frequency is 120 cps.

325 
Assume the hysteresis loss in the core of Problem 322 to vary in accordance with the equation
and determine the total core loss for a frequency of 60 cps and a flux density of 90,000 maxwells per sq in.

326 
The core loss in an ironcore reactor is 548 w of which 402 w is hysteresis loss when the applied voltage is 240 v and the frequency is 30 cps. What is the core loss when the voltage and the frequency are both doubled? Neglect the resistance of the winding.

327 
The core loss in a transformer is 500 w of which 100 w is eddycurrent loss. If the number of turns in the exciting winding were to remain unchanged but the volume of the core were doubled by increasing the crosssection of the iron, what would be the core loss for the same applied voltage and frequency ? Neglect winding resistance and assume
for a given volume.

328 
The dimensions of the core in a coretype transformer are as follows: the outside dimensions of the core parallel to the plane of the window are 8 3/4 in. x 10 in.; cross section of iron = 2 5/8 in. x 4 1/2 in. The laminations are 26 (t = 0.0185 in.) gage and are stacked to a depth of 4 1/2 in. with a stacking factor of 0.93. The primary winding has two coils of 87 turns each. The rated primary voltage when the two coils are connected in series is 440 v at 60 cps. The density of the iron is 0.272 lb per cu in. The following noload tests were made by energizing the primary winding.
Volts

Frequency

Watts

440

60

103.5

220

30

45.2

Determine
(a) The 60cycle hysteresis loss at rated voltage and rated frequency.
(b) The 60cycle eddycurrent loss at rated voltage and rated frequency.
(c) The core loss per pound at rated voltage and rated frequency.
(d) The core loss per pound if the same iron were used but if the laminations were 24 gage (T = 0.0250 in.).

329 
Determine the magnetic flux in each of the three legs of the electromagnet shown in Fig. 343 if the air gaps in the outer legs are as indicated and an air gap of 0.05 in. is inserted in the middle leg and when the current in the 900turn winding is 3.5 amp. Neglect the reluctance of the iron, and magnetic leakage but correct for fringing.

330 
Determine the forces on each of the faces of the air gaps in the electromagnet of Problem 329.
