100 ELEMENTS OF ELECTRICITY AND MAGNETISM. 



53. Contribution to the magnetic field at a given point by one element of an 

 electric wire. The region surrounding an electric circuit is a magnetic field and 

 each element of the wire which constitutes the circuit may be considered as contribu- 

 ting its share to the field intensity at each point. Imagine a magnet pole of strength 

 m to be placed at the point at which it is desired to find the field intensity &H which 

 is produced by a given element A/ of the wire. 



Let r be the distance from m to A/ and 

 let be the angle between r and A/, as 

 shown in Fig. 64. The field intensity at the 

 element due to the pole is m /r 1 according to 

 equation (17). The component of this field 

 which is at right angles to the element is 

 w/r*X s ' n an d this component of the 

 field pushes sidewise on the wire with a force 

 which is given by 



Fig. 64. 



according to equation (30). This is the 

 force with which the pole m acts on the 

 element, and therefore it is also the force 

 (disregarding sign) with which the element 



acts upon the pole. But the force with which the element acts upon the pole must be 



equal to the product of the strength of the pole and the field intensity at the pole due 



to the element, that is, 



&F-= m &H 



whence 



^ (30 



in which A// is the field intensity at the point m in Fig. 64 due to the element 

 A/. This field A/7 is perpendicular to r and to A/. 



Note. It is evident from the above discussion that the magnetic field at a given 

 point in the neighborhood of a given coil of wire, or a circuit of any form, in which 

 an electric current is flowing is proportional to the strength of the current, and that 

 its direction is fixed. That is to say, if the strength of the current is doubled the field 

 intensity is doubled everywhere, but the direction of the field is everywhere unaltered. 

 The trend of the lines of force of the magnetic field due to a given coil or circuit de- 

 pends only upon the shape and size of the coil. 



54. The intensity of the magnetic field at the center of a circu- 

 lar loop of wire. If we can calculate the force with which a cir- 

 cular loop of wire with given current acts on a magnet pole of 

 given strength placed at the center of the circular loop, we can 

 derive an expression for the intensity of the field at the center 

 of the loop due to the current, because the force exerted on the 



