433-436] Force inside a Magnetised Body 375 



Then by Gauss' Theorem ( 409), 



Q .............................. (360), 



where N is the component of force in the direction of the outward normal to 

 S y acting on a unit pole placed at any point of the surface S. This force, 

 however, is exactly identical with that considered in 433, and its normal 

 component has been seen to be identical with the normal component of the 

 induction. Thus N, in equation (360), will be the normal component of 

 induction, so that this equation proves the theorem. 



Analytically, the theorem may be stated in the form 



Q ..................... (361), 



and this by Green's Theorem ( 179), is identical with 



^ + ^ + 8 = (362). 



435. DEFINITION. By a line of induction is meant a curve in the 

 magnetic field such that the tangent at every point is in the direction of 

 the magnetic induction at that point. 



DEFINITION. A tube of induction is a tubular surface of small cross- 

 section, which is bounded entirely by lines of induction. 



By a proof exactly similar to that of 409, it can be shewn that the 

 product of the induction and cross-section of a tube retains a constant value 

 along the tube. This constant value is called the strength of the tube. 



In free space the lines and tubes of induction become identical with the 

 lines and tubes of force, and the foregoing definition of the strength of a tube of 

 induction is such as to make the strengths of the tubes also become identical. 



436. At any point of a surface let B be the induction, and let e be the 

 angle between the direction of the induction and the normal to the surface. 

 The aggregate cross-section of all the tubes which pass through an element 

 dS of this surface is dS cos 6, so that the aggregate strength of all these tubes 

 is BcosedS. Since B cose = N, where N is the normal induction, this may 

 be written in the form NdS. Thus the aggregate strength of the tubes of 

 induction which cross any area is equal to 



NdS. 



This, we may say, is the number of unit-tubes of induction which cross 

 this area. 



