Waves at the Surface of a Moving Mirror. 515 



surface o£ the reflector, while the opposite side is just inside 

 the reflector. The work done in carrying a unit N pole 

 round this circuit is equal to 4-7T times the electric current 

 that flows through the circuit. Notice that the circuit 

 moves with the mirror. Then, if C denotes the instantaneous 

 value of the superficial current that flows vertically upwards 

 across a horizontal line of unit length drawn on the surface 

 of the mirror, 



c 



■'■ C =^{(* + v)^-(c-»)V2}- ■ ■ (9) 



(9) gives the value of the superficial screening current in- 

 duced in the mirror. The function of this current is to 

 annul the magnetic force that would otherwise be exerted by 

 the waves, on a unit N pole placed inside the substance of 

 the reflector. 



Now at points on opposite sides of the current sheet, the 

 magnetic forces due to the current are numerically equal but 

 oppositely directed. Inside the conductor the magnetic 

 force due to the waves is annulled by the magnetic force 

 due to the current sheet ; therefore it follows that just outside 

 the conductor the magnetic force due to the current sheet is 

 added to the equal magnetic force not derived from the 

 current, the resultant magnetic force being equal to 



(l/c 2 ){(c + v)E 1 ^{c-v)E 2 }. 



Thus the magnetic force not derived from the current is 

 equal to 



^{(c+fl)Ei-(c-iOE 2 } (10) 



When an electric current flows in a direction perpendicular 

 to an independent magnetic field, the force exerted on unit 

 length of the conductor is equal to the product of the current 

 and the strength of the independent magnetic field. In the 

 present case it is not quite clear, at first sight, whether the value 

 of the current given by (9) should be multiplied by (10), or 

 only by that part of the magnetic force which is independent 

 of v. This point can be settled by reverting to the energy 

 equations. 



In a short time dt, during which the values of the electric 

 field-strengths of the incident and reflected waves have the 

 respective values E] and E 2 at the surface of the mirror, 



2L2 



