26 Electrostatics Field of Force [OH. n 



to the curve at that point. Such a curve is called a Line of Force. We 

 may therefore define a line of force as follows : 



A line of force is a curve in the electric field, such that the tangent at every 

 point is in the direction of the electric intensity at that point. 



If we suppose the motion of a charged particle to be so much retarded by frictional 

 resistance that it cannot acquire any appreciable momentum, then a charged particle set 

 free in the electric field would trace out a line of force. In the same way, we should have 

 lines of current on the surface of a lake, such that the tangent to a line of current at any 

 point coincided with the direction of the current, and a small float set free on the lake 

 would describe a current-line. 



32. The resultant of a number of known forces has a definite direction, 

 so that there is a single direction for the electric intensity at every point of 

 the field. It follows that two lines of force can never intersect ; for if they 

 did there would be two directions for the electric intensity at the point of 

 intersection (namely, the two tangents to the lines of force at this point) so 

 that the resultant of a number of known forces would be acting in two 

 directions at once. An exception occurs, as we shall see, when the resultant 

 intensity vanishes at any point. 



The intensity R may be regarded as compounded of three components 

 X, Y, Z, parallel to three rectangular axes Ox, Oy, Oz. 



The magnitude of the electric intensity is then given by 



E 2 = X 2 +F 2 4-^ 2 , 

 and the direction cosines of its direction are 



X Y Z_ 



R' R' R' 



These, therefore, are also the direction cosines of the tangent at a?, y, z 

 to the line of force through the point. The differential equation of the 

 system of lines of force is accordingly 



dx _ dy _ dz J 

 ~~~~ 



III. The Potential. 



: 



33. In moving the small test-charge e about in the field, we may either 

 have to do work against electric forces, or we may find that these forces 

 will do work for us. A small charged particle which has been placed at a 

 point in the electric field may be regarded as a store of energy, this 

 energy being equal to the work (positive or negative) which has been done 

 in taking the charge to in opposition to the repulsions and attractions of 

 the field. The energy can be reclaimed by allowing the particle to retrace 

 its path. Assume the charge on the moving particle to be so small that 



