420 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



Accordingly for points not on the axis, at a distance r from the 

 center of the circle, 



T r / ~D\ a. ~t O * i ~D\ R ^ 



r>R, 



- 



274 3 2.4.6 



In order to find the direction of the force we must differentiate 

 this in the directions parallel and perpendicular to the axis, and 

 take the resultant. A figure of the lines of force is given by 

 Maxwell, Plate 18. 



216. Infinite Straight Current. Law of Biot and 

 Savart. If we have a current flowing through a straight linear 

 conductor of infinite length, we may consider the circuit com- 

 pleted by conductors lying at an infinite distance all in the same 

 plane. The solid angle subtended by the circuit at a point P 

 will be that sector of the unit sphere with center at P included 

 between the plane through the straight conductor and P, and a 

 plane through P parallel to a given plane, which is assumed to 

 be the plane of the circuit. This angle being <, we have the 

 ratio of the solid angle &> to the surface of the unit sphere equal 

 to the ratio of the plane angle to the circumference of the unit 

 circle, 



But $ is equal to the angle made by a plane through P and 

 the conductor with a fixed plane through the conductor. Conse- 

 quently the equipotential surfaces are planes through the con- 

 ductor, and the lines of force are circles whose planes are per- 

 pendicular to the conductor. 



The line integral of force about a circle of radius r is the value 

 of the force H, which is tangential to the circle, times the length 

 of the circumference, and this must be equal to 4t7rl, 



47T/ = 2-irrH. 



Accordingly the value of the force is 



H&- 



r 

 This is the law of Biot and Savart*. 



* Biot et Savart, Ann. Chim. Phys. 15, p. 222, 1820. 



