32 STATIC ELECTRICITY 



dealing with an electric system in which the charges are separated 

 by air. Some of the propositions will, however, apply at once to 

 magnetic and gravitative systems if we use corresponding units of 

 measurement. 



The field. The space in which the action of a system is 

 manifested is termed its field. 



Force between quantities m l and w >2 . Experiments which 

 are described in Chapter V show that a small body charged with /// t 

 acts upon a small body charged with m 2 and distant d from it with 

 a force proportional to m Wg/rf 2 , whatever be the unit in terms of 

 which we measure the charges. 



Unit quantity. The inverse square law enables us to fix on 

 the following convenient unit: If two small bodies charged with 

 equal quantities would act on each other with a force of 1 dyne 

 if placed 1 cm. apart in air, each body has the unit charge on it. 



This unit is termed the Electrostatic or E.S. unit. If two 

 bodies have charges m t and m 2 in E.S. units and are d cm. apart, 

 the inverse square law shows that the force between them is 

 m i m 2/d' 2 degrees. 



Intensity of field. The intensity at a point is the force 

 which would act on a small body placed at the point if the body 

 carried a unit charge. 



The intensity due to a charge m at a distance d is m/d?. 



Lines of force and tubes of force. If a line is drawn in t In- 

 field so that the tangent to it at any point is in the direction of the 

 intensity at that point, the line is termed a line of force. A bundle 

 of lines of force is termed a tube of fom . < )r we may think of a 

 tube of force as enclosed by the surface obtained by drawing tin- 

 lines of force through every point of a small closed curve. 



We shall now prove a theorem due to Gauss which enables u> to 

 obtain the intensities in certain cases very simply, and which also 

 'shows us that lines and tubes of force indie ate for us the magnitude 

 as well as the direction of the intensity at 

 every point in their length. 



Gauss's theorem. If we take any 



closed surface S, and if N is the resolved part 

 of the intensity normal to the surface at the 

 element ^/S. positive when outwards, negative 



when inwards, then f^d* = 4--^, where (J is 



the quantity of charge within the surface S. 

 Any charge without the surface makes on 



the whole no contribution to JNdS. 



To prove this let us consider an element 

 FIG. 28. of charge q, situated at a point O within a 



closed surface, such as is indicated by S in 

 Fig. 23. Let an elementary cone of solid angle dw be drawn from 

 O, intercepting an element of area dS of the surface S. Let this cone 



