360 MAGNETIC METHOD [Chap. 8 



angle of the coil with the normal to the frame {z" direction), the flux due 

 to z" is and that due to y" is at a maximum when ^ = 0. Hence, the 

 current due to z" is max.+ and that due to y" is 0. If (p = 90° (coil 

 plane in plane of frame), the flux due to z" is a max. that due to y" is 0. 

 The current due to z" is and that due to y" is max. + , and so on. In 

 Fig. 8-37, the e.m.f.'s induced in the coil are indicated by a dotted hne. 

 For the y" component the maxima occur at 90° and 270°, while for z" 

 the maxima occur at 0° and 180°. The plane of commutation being at 

 right angles to the coil, reversals in sign occur at the 90° positions for the 

 next 180°; therefore, the y" component is neutralized (provided the gal- 

 vanometer period is large compared with the commutation period) and 

 only the e.m.f. due to the z" component remains. Its value is obtained 

 from the last equation in (8-14). Changing the astronomic to magnetic 

 components so that X' = H, Y' = 0, and Z' = Z, 



z" = — H sin a sin I + Z cos i, (8-43) 



where a is the magnetic azimuth of the axis aa' in Fig. 8-36, and i the 

 inclination of the axis 66'. If the cross-sectional area of the coil is S and 

 the number of turns A'', the total magnetic flux, when the coil is normal 

 to the field, is * = SiVz". Hence, for any coil position given by the angle 

 with the y" direction or "phase" angle, 5 = 90 — ^ = w^, the flux $ = SATz" 



cos (Jit and the induced e.m.f. E = —^~ — SiVz" w sin w^. Substituting 



S = irr (in which r is the mean turn radius of the coil) and co = 2Tn (in 

 which n is the number of revolutions per second), 



E = 27rV^iVz"n sin <^t 



■ I = 27rViVz'^nsinco / 



(8-44) 



where R is the resistance of the coil plus the resistance of the galvanometer. 

 Instantaneous current values, speed of revolution, and phase angle are 

 difficult to measure. However, by using a ballistic galvanometer the 

 quantity of electricity (which is independent of the angular velocity) may 

 be determined. Since E = —d^/dt, I = {l/"Si)—d^/dt. Idt = the quan- 

 tity of electricity, equals dQ = —d^/R. At two instances when the 

 fluxes are $2 (at the end) and $1 (at the beginning of the motion), Q = 



/ dQ = ~ — ^ — - . Since <l> = SA'z" cos 5 in any position, 

 •'♦1 A 



Q = ''^-^ — (cos 5i - cos 52). (8-45) 



