Electricity in a uniform plane conducting Surface. 389 



are selected according to a definite and easily recognizable rule. 

 For this purpose the most convenient rule to adopt in relation 

 to the lines of flow, is to place them so that the total flow of 

 electricity between each pair of consecutive lines is the same — 

 and with regard to the equipotential lines, to place them so that 

 the change of potential in passing from any one to the next con- 

 secutive line is the same. In the sequel, whenever a system of 

 lines of flow, or of equipotential lines, is referred to, it is to be 

 understood that the lines are placed so as to fulfil the above 

 conditions. 



One Pole in an Infinite Sheet, 



4. In this case it is evident that the lines of flow are straight 

 lines radiating out from the pole, and that the spaces between 

 each pair of consecutive lines will convey equal currents if each 

 line makes the same angle with the next. If a circle of radius 

 r be drawn about the pole as centre, the quantity of electricity 

 which crosses the whole circumference in unit of time, will evi- 

 dently be the same whatever the value of r, and will be equal to 

 the quantity Q supplied in the same time by the source. Hence, 

 at distance r from the source, the strength of the current will be 



_Q_ 



or inversely proportional to the distance r. 



It is evident also, either from general considerations of sym- 

 metry, or from the condition that equipotential lines and lines 

 of flow must intersect each other at right angles, that the equi- 

 potential curves for the case we are considering are circles having 

 the source at their common centre. 



5. If R be the resistance to the radial flow of electricity across 

 the annular portion of the sheet bounded by circles at the po- 

 tentials V, and Y^ we have, by Ohm's law, 



whence, since Q is constant, equal differences of potential will 

 be found at the boundaries of annular belts of equal resistance. 

 The relation between the radii of successive equipotential circles 

 with a constant difference of potential may be deduced from this 

 condition as follows: — The resistance of any very narrow belt must 

 be proportional to its breadth, or the difference between the 

 radii of its external and internal circumferences ; and it must be 

 inversely proportional to the mean circumference, which, seeing 

 that the circumference varies directly as the radius, will be simply 

 the arithmetic mean of the external and internal circumferences. 

 That is, if r lt r 2 , r 3 are radii of successive circles, these circles 



