Electric Current in M etals Ohm’s Law 129 1.2 A D eterm ine the emf and th e in tern al resistance of th e

cu rren t source.

10.84. A d.c. generator w ith an emf of 150 V and an in te r­ nal resistance of 0.3 £2 supplies voltage to 20 incandescent lam ps h aving a resistance of 240 £2 each and connected in p arallel. The resistance of the leads is 2.7 Q. D eterm ine the voltage across the generator term in als and

across the lam ps.

10.85. The emergency lam ps of a tram car are fed by an accum ulator b a tte ry having an emf of 48 V and an in tern al resistance of 0.2 £2. Ten lam ps having a resistance of 39.5 £2 each are connected as shown in Fig. 47. D eterm ine the cu rren t in each lam p and in the leads.

10.86. A circu it contains 20 parallel- connected bulbs. The c u rre n t through a bulb is 1 A. The resistance of the wires

connecting the load w ith a generator is 0.2 £2. W h at m ust the emf of the generator be for the voltage across the bulbs to be 220 V? The in tern al resistance of the generator is 0.05 £2.

10.87. Three electric m otors and ten parallel-connected incandescent lam ps are connected to a generator w ith emf

F ig. 47

+ o

240 V and resistance 0.025 £2. A cu rren t of 50 A passes through each m otor w hile the cu rren t through each lam p is 1 A. The resistance of th e leads is 0.1 £2. D eterm ine the voltage across the generator term in als and across the loads.

10.88. A workshop receives electric power from a collective farm ’s electric power sta tio n . There are two electric m otors (M ) and four bulbs in the workshop connected as shown in Fig. 48. The cu rren t through each m otor is 10 A, and through

130 Ch. II. Fundam entals of E lectrodynam ics

each bulb 0.5 A. The distance I between th e s ta tio n and the workshop is 0.5 km. The in tern al resistance of the generator is 0.1 £2 and the voltage across its term in als is 220 V. D e­ term in e the emf of the generator and the cross-sectional area of th e copper leads if the adm issible voltage drop in them is 8% .

10.89. D eterm ine the counter emf of a tra c tio n m otor if th e resistance of its w indings is 0.1 £2 and th e voltage in th e c irc u it is 550 V a t a cu rren t of 150 A.

10.90. D eterm ine the emf of a generator w ith an in te rn al resistance of 0.05 £2 and the counter emf of a m otor if the cu rren t in the c irc u it is 100 A, the voltage across the gen­ erator term in als is 225 V, and the resistances of th e m otor w inding and the leads are 0.2 and 0.1 £2 respec­ tiv e ly .

10.91. The c irc u it diagram of a d.c. m otor is shown in Fig. 49. The voltage in th e circu it is U = 550 V, and the cu rren t is / = 102 A. The resistance of the arm atu re c irc u it is R a = 0.1 £2, th a t of the p arallel ex citatio n w inding R ex = 150 £2, and th a t of the rh eo static co ntro ller R r = 125 £2. D eterm ine the cu rren t in the parallel ex citatio n w inding when the rh eo static controller is com pletely on, and th e cou n ter emf if the electric m otor is sta rte d w ith o u t a s ta rte r rh eo stat.

10.92. Four loads h aving a resistance of 10 £2 each are con­ nected to an accum ulator b a tte ry h aving an emf of 48 V and an in te rn al resistance of 0.25 £2. D eterm ine the curren t through th e b a tte ry if the loads are connected (a) in series, (b) in p arallel, and (c) in two parallel branches containing two series-connected loads each.

10.93. One of two cells has an emf of 1.45 V and an in te rn al resistance of 0.5 £2 and supplies voltage to a c irc u it w ith an efficiency of 90% , w hile the other cell has an emf of 2 V and an in te rn al resistance of 0.5 £2 and operates w ith an effic­ iency of 80% in an id en tic a l circu it. D eterm ine the current in the two circu its.

10.94. An accu m u lator w ith an emf of 1.45 V produces a cu rren t of 0.5 A in a cond u ctor whose resistance is 2.5 £2. D eterm ine the sh o rt-circu it cu rren t.

10.95. D uring a sh ort c irc u it, the cu rren t from a source of 1.8 V is 6 A. W h at m u st th e extern al resistance be for th e current to be 2 A?

§ 10. E lectric Current in Metals. Ohm's Law 131 10.96. W hy should lead accum ulators n o t be short c ir ­ cu ited?

10.97. A galvanic cell is first connected to an ex tern al resistance of 1.9 £2 and the cu rren t in the c irc u it is 0.6 A. Then it is connected to an ex ternal resistance of 2.4 £2 and the cu rren t becomes 0.5 A. W h at is the sh o rt-circu it c u rre n t for th is cell?

10.98. Three galvanic cells, each having an emf of 1.5 V and an in te rn a l resistance of 0.6 £2, are connected in series to a conductor having a resistance of 1.8 £2. D eterm ine the cu rren t in the circu it. W h a t w ill the cu rren t in the same conductor be if the cells are connected in p arallel?

10.99. Using the d a ta in Problem 10.98, determ ine the connection for which the sh ort-circu it cu rren t is the largest.

10.100. How should six galvanic cells, each hav ing an emf of 1.5 V and an in tern al resistance of 0.5 £2, be connected in p arallel groups to obtain the m axim um c u rre n t when con­ nected to a 1.0-Q resistor?

10.101. How should 12 cells, each hav ing an emf of 1.2 V and an in te rn al resistance of 0.6 £2, be connected to produce a c u rre n t of 1.6 A in an ex tern al c irc u it whose resistance is 2.2 £2?

10.102. A “K rona” b a tte ry is the power source for tr a n ­ sisto r radios. I t contains seven series-connected g alvanic cells. W h a t is the emf of a cell if the emf of th e b a tte ry is 9 V? W h a t w ill the change in the emf of the b a tte ry be if the c o n stitu e n t cells are connected in parallel?

10.103. Two alk alin e accum ulators, each h aving an emf of 1.5 V and an in te rn al resistance of 0.3 £2, are connected first in series and then in p a ra lle l and used to su p p ly voltage to a c irc u it w ith a resistance of 6 £2. W hich connection produces th e higher cu rren t in the ex tern al c irc u it, i.e. w hich m ethod is more advantageous?

10.104. Solve Problem 10.103 for the case when th e e x te r­ nal resistance of the c irc u it is 0.9 £2. Using the resu lts of the two problem s, find the ra tio betw een th e ex te rn al and in te r ­ nal resistances for w hich connecting the c u rre n t sources in series is more advantageous.

10.105. A passenger carriage is lighted using a b a tte ry co n tain in g 26 series-connected lead accum ulators. The emf of an accum ulator is 2 V and th e in te rn al resistance is

132___________ Ch. II. F undam entals of E lectrodynam ics

0.004 £2. D eterm ine the emf and the voltage across the te r­ m inals of the b a tte ry if the current in th e c irc u it is 20 A.

10.106. W h at is the reading of a v o ltm eter connected to th e term in als of a b a tte ry consisting of three series-connected a lk a lin e accum ulators, each w ith an emf of 1.2 V and an in te rn a l resistance of 0.3 £2? The ex tern al c irc u it consists

F ig. 50 F ig. 51

of a bulb w ith a resistance of 16 £2 and 2-m long alum in ium leads w ith a cross-sectional area of 0.2 m m 2. D eterm ine the voltage on th e bulb.

10.107. An accu m u lator b a tte ry , having an emf of 22 V

and an in te rn al resistance of 0.1 £2, is charged from a re c ti­ fier th e voltage across whose term in als is 24 V. D eterm ine th e resistance introduced in to the circ u it by a poten tiom eter connected in series w ith th e b a tte ry if the charging cu rren t is 4 A.

10.108. Two accum ulators, having em f’s of 1.3 and 1.8 V

and in te rn al resistances of 0.1 and 0.15 £2 respectively, are connected in p arallel. D eterm ine the cu rren t in the c irc u it and the voltage across the term in als (Fig. 50).

10.109. D eterm ine th e cu rre n t through a resistor R = 2 £2

connected in a circu it as shown in Fig. 51. Given: = 2 V, rj = 0.50 £2, g 2 = 4.0 V, and r 2 = 0.70 £2.

§ 11. WORK, POWER, AND THE THERMAL EFFECT OF CURRENT

Basic Concepts and Form ulas

E lectric energy is transform ed into o th er kinds of energy so th a t energy is conserved. The m easure of the transform ation is th e work done by electric current:

§ 11. Work, Power, and the Therm al Effect of Current 133 If the current or voltage are su b stitu te d using O hm ’s law for a conductor we obtain

A = IU t ---- I 2R t = ^ - t .

Power is the ratio of the work done by an electric c u rre n t d uring tim e t to the tim e t:

p = ^ - f _t p *=i u = i-r = !£-._n

The u n it of work is the joule (J). In electrical engineering and everyday life, electrical work is m easured in kilo w att- hours (1 kW h = 3.6 M J). The u n it of power is the w a tt (1 W - 1 J/s).

J o u le ’s law: the am ount of h eat liberated in a conductor carrying a current is proportional to the square of the c u rre n t, the tim e of its passage, and the resistance of the conductor:

Q = P R t .

T his form ula is for series-connected loads. For a p a ra lle l connection, the form ula is

W orked Problem s

Problem 63. An electric m otor operating for 5 h is driven a t the m ains voltage of 380 V and a curren t of 35 A. The resistance of the m otor w inding is 0.5 £2. D eterm ine the am ount of energy consum ed, the am ount of h eat lib erated in the w inding during the o p eration, and the m echanical work done by the m otor.

Given: U = 380 V is the voltage a t the m otor term in a ls, / = 35 A is the cu rren t, R = 0.5 £2 is the resistance of the m otor w inding, and t = 5 h = 5 X 3600 s is the operation

tim e.

F in d : the energy A consumed by the m otor, the am ount of h eat Q liberated in the w inding, and the m echanical work Amech-

Solution. The energy consumed or the to ta l work done by the c u rre n t can be determ ined from the form ula

A = I U t , A = 35 A x 380 V X 5 x 3600 s ~ 2.4 x 108 J .

134 Ch. II. Fundam entals of E lectrodynam ics

The am ount of heat liberated in the w inding of the m otor can be determ ined from Jo u le ’s law:

Q = P R t ,

Q = (35 A)2 x 0.5 Q X 5 X 3600 s = 1.1 X 107 J. The m echanical work done by the m otor can be d e te r­ mined by su b tractin g the energy spent in h eatin g the w ind­ ing from the whole energy spent:

^1 mech — A Q,

^ mech = 2.4 x 108 J - 1.1 x 107 J = 2.29 X 108 J . Answer. The energy spent by the m otor is ap proxim ately 2.4 X 108 J , the am ount of h eat lib erated in the w inding is 11 M J, and the m echanical work is 2.3 X 108 J .