512 Mr. E. Eclser on the Reflexion of Electromagnetic 



The auxiliary force exerted on the charge in virtue o£ its 

 motion is easily obtained. 



Let H denote the value of the resultant magnetic field in 

 the plane at x, the positive direction of H being from South 

 to North. Then if a vertical conducting filament is moving 

 from West to East through the plane at x with velocity n, 

 the electromotive force per unit length of the conductor, 

 induced by its motion, is equal to Hu, its direction, when 

 positive, being vertically upwards. A conducting filament 

 owes its characteristic properties to the presence of charged 

 particles which can move within the substance of the con- 

 ductor; therefore the force per unit charge induced by the 

 motion of the charge must be equal to Hw. 



In the plane at x, the magnetic field is equal to (E x — E 2 )/c, 

 its positive direction being from South to North. Therefore 

 a unit positive charge, moving through the plane at x from 

 West to East with velocity u, will be subjected to a vertical 

 force equal to (E 2 + E 2 ) due to the electric field at x, together 

 with an auxiliary force (E x — E 2 )(it/c) due to the motion of 

 the charge. At the surface of the mirror, the total force 

 that would be exerted on a unit positive charge which moves 

 with the mirror is therefore equal to 



Bx+E 2 + t -(E 1 -E 2 ) = E 1 (^) +%,(—). 



It is this force that must vanish at the surface of the mirror, 

 and not the force (E! + E 2 ) that would be exerted on a unit 

 positive charge in a stationary position. To see this, let a 

 rectangular circuit be described, in a plane perpendicular to 

 the surface of tiie mirror, one side of the circuit being- 

 parallel to and just outside the reflecting surface, and the 

 opposite side just inside the reflector. By making the sides 

 close enough together, the area of the circuit can be made as 

 small as we please, and therefore the magnetic induction 

 interlinked with the circuit can be diminished indefinitely. 

 Therefore, on carrying a unit positive charge round the 

 circuit, which moves with the mirror, no work can be done ; 

 for if work were done, we could accumulate mechanical 

 energy at pleasure, without any compensating loss of mag- 

 netic energy, by carrying a charge round the circuit in a 

 suitable direction. Then, since the force on unit charge 

 must vanish inside the conductor, it follows that the force 

 on a unit charge must vanish just outside the conductor, 

 and 



