CHAP X] ENERGY AND INDUCTANCE 187 



Prob. 10. It is desired to make a standard of inductance of one 

 millihenry by winding uniformly one layer of thin flat conductor upon 

 a toroidal wooden ring of circular cross-section. How many turns are 

 needed if the diameter of the cross-section of the ring is 10 cm. and 

 the mean diameter of the ring itself is 50 cm.? Ans. 400. 



Prob. 11. An iron ring of circular cross-section is uniformly wound 

 with n turns of wire, the total thickness of the winding being t\ the 

 mean diameter of the ring is D and the radius of its cross-section r. 

 What is the inductance of the apparatus, assuming the permeability 

 of the iron to be constant and equal to 1500 times that of the air? 

 Hint: dff p ~ti2n(r+x)dx/itD; np = n(x/t). 



Ans. 1.25(nVD) [1500r J -M(fr + #)] X 10-' henry. 



Prob. 12. A ring is made of non-magnetic material, and has a 



rectangular cross-section of dimensions b and h', the mean diameter 



of the ring is D. It is uniformly wound with n turns of wire, the total 



thickness of the winding being t. What is the inductance of the winding? 



Ans. 0.4(n'/) [bh + $t(b + h +3<)]1(>- 4 millihenry. 



Prob. 13. The ring in the preceding problem has the following 

 dimensions: Z) = 50 cm.; /i = 6 = 10 cm.; it is to be wound with a con- 

 ductor 3 mm. thick (insulated). How many turns are required in order 

 to get an inductance of 0.43 of a henry? Hint : Solve by trials, assum- 

 ing reasonable values for t ; the number of turns per layer decreases as 

 the thickness of the winding increases. Ans. About 5300. 



Prob. 14. It is desired to design a choke coil which will cause a 

 reactive drop of 250 volts, at 10 amp. and 50 cycles. The cross-section 

 of the core (Fig. 12) is 120 sq.cm., and the mean length of the path 

 130 cm.; the maximum flux density in the iron must be not over 7 

 kl. per sq. cm. What is the required number of turns and the length 

 of the air-gap in the core? 1 Ans. 150; 3.7 mm. 



Prob. 15. An electrical circuit, which consists of a Leyden jar 

 battery of 0.01 mf. capacity and of a coil having an inductance of 10 

 millihenrys, undergoes free electrical oscillations in such a way that the 

 maximum instantaneous voltage across the condenser is 10,000 volts. 

 What is the current through the inductance one-quarter of a cycle 

 later, neglecting any loss of energy during the interval ? 



Ans. 10 amp. 



Prob. 16. Suggest a practical experiment which would prove directly 

 that the stored electromagnetic energy is proportional to the square 

 of the current. 



Prob. 17. Prove that the inductance of a coil of given external 

 dimensions is proportional to the square of the number of turns, taking 

 into account the complete :m<l tin* partial linkages. Show thqt the 

 ohmic resistance of tin- >il is also proportional to the square of the 

 number of tun is, provided that the >paee I'artor is constant. 



NOTE 1. The theoretical calculation of the inductance of short 



1 For a complete design of a reactive coil see G. Kapp, Transformers (1908), 



