The Boundary Conditions. 479. When light is reflected and refracted at the surface of separation of two isotropic or crystalline media, the boundary conditions are, (i) that the components of the electromotive and magnetic forces parallel to the surface of separation must be continuous; (ii) that the components of electric displacement and magnetic induction perpendicular to the surface of separation must likewise be continuous. We have, therefore, six equations to determine four unknown quantities; but inasmuch as two pairs of these equations are identical, the total number reduces to four, which is just sufficient to determine the four unknown quantities. If, however, we were to assume these six conditions in the case of a magnetized medium, we should find that we should be led to inconsistent results, and we shall, therefore, proceed to prove the boundary conditions. Since the electric displacement and the magnetic induction both satisfy the equation df dg dh which is an equation of the same form as the equation of continuity of an incompressible fluid in Hydrodynamics; it follows that the components of the electric displacement and magnetic induction perpendicular to the surface of separation must be continuous. To obtain the other conditions, let us suppose, as before, that the plane x=0 is the surface of separation, and that the plane z = 0 contains the direction of propagation. Then, since the coefficients of y and t in the exponential factor must be the same in all four waves, d/dy and d/dt of any continuous function will also be continuous, and conversely. Since none of the quantities are functions of z, dy dy which shows that y is continuous. Since the continuity of y follows from that of f, the conditions of continuity of both these quantities will be expressed by the same equation. THE BOUNDARY CONDITIONS. 401 it follows that H is continuous, whence if the accents refer to the second medium, we obtain from (2) This equation shows that the electromotive force parallel to z is discontinuous. This circumstance may, at first sight, appear somewhat strange, and may perhaps be regarded as an objection to the theory; but since the p's are exceedingly small quantities, the discontinuity is also very small. We have, moreover, assumed that the transition from one medium to the other is abrupt, whereas, if we were better acquainted with the conditions at the confines of two different media, we should probably find that this was not the case; but that there would be a rapid but continuous change in the component of the electromotive force parallel to the boundary, in passing from one medium to the other. We have, therefore, as yet, only obtained two independent boundary equations. Now, we shall presently see that when plane polarized light is reflected and refracted at the surface of a magnetized medium, the reflected light is elliptically polarized; whilst, as we have already shown, the two refracted waves are circularly polarized in opposite directions. We have, therefore, four unknown quantities to determine, viz., the amplitudes of the two components of the reflected vibration, and the amplitudes of the two refracted waves. We, therefore, require two more equations. To find a third equation, we shall assume, that the component of magnetic force parallel to the axis of y is continuous. A fourth equation will be obtained from the condition of continuity of energy; for since there is no conversion of energy into heat, or any form of energy other than the electrical kind, it follows that the rate of increase of the electrostatic and electrokinetic energies within any closed surface must be equal to the rate at which energy flows in across the boundary. 480. We must now obtain an expression for the energy. It is a general principle of Dynamics, that if equations are given which are sufficient to completely determine the motion of a system, the Principle of Energy can be deduced from these equations. The proper form of the Principle of Energy in the case of a dielectric medium is this:-Describe any closed surface in the medium, then the rate at which energy increases within the surface, is equal to the rate at which energy flows in across the boundary. B. O. 26 If E be the electric energy per unit of volume, the rate at which energy increases within the surface is fff Edxdydz, and, consequently, this quantity must be capable of being expressed as a surface integral taken over the boundary; and any form of E which is not capable of being so expressed must certainly be wrong. If the medium were a conductor, in which there is a conversion of energy into heat, fff Edxdydz would not be expressible in the form of a surface integral', but this case need not be considered, since we are dealing with a transparent dielectric. Since P=4fK, = 4πkAf, equations (6) may be written in the form Multiply this equation and the two corresponding ones by a, B, y; then add and integrate throughout any closed surface, and we shall obtain W = 2πk Let = 4πk dh dxdydz...(15). +C*h2) dædydz ...(16), · [[[ (A2oƒƒ + B3gg + Chh) dædydz = [[[ (Pƒ + Qÿ + Rh) dædydz. Substituting the values of f, g, h in terms of a, ß, y, and integrating by parts, we obtain we substitute the values of f, g, h from the equations of 4πƒ = dy/dy – dẞ/dz, &c., in the coefficients of the terms in brackets, and integrate by parts, we shall find that the last volume integral in (15) is equal to (18) − [[[ {(P.ƒ — μ‚¿) B − (ph− p.ƒ) v} {(pģ−ph) y − (p.ƒ—p‚j) a} + n {(p,h − p.ƒ) a − (pj-ph) B}]dS ...(18). Accordingly (15) becomes on substitution from (16), (17) and 1 (a ↓ [ [ [ { (R + p: ƒ − pò) B − ( Q + ph− p ̧Ï) v} 4π +m {(P + pġ − ph) y − (R+p2ƒ— p.ÿ) a} + n {(Q+p‚h − p ̧ƒ) a − (P+p.g− på) 8}]dS ...(19). The physical interpretation of this equation is, that the rate at which something increases within the closed surface must be equal to the rate at which something flows into the surface. This cannot be anything else but energy; we are therefore led to identify the expression k 8π - (a2 + B2 + y2) + 2πk (A2ƒ2 + B3g2 + C3h2), as representing the energy of the electric field per unit of volume. The first term represents the electrokinetic energy, and the second term the electrostatic energy. The above expressions are the same as those obtained by Maxwell by a different method, and it thus appears that the expressions for each species of energy are not altered by the additional terms, which have been introduced into the general equations of electromotive force. The right-hand side of (19) represents the rate at which work is done by the electric and magnetic forces, which act upon the surface of S. x 481. In the optical problem which we are considering, the bounding surface is the plane = 0; whence if the quantities in the magnetized medium be denoted by accented letters, the condition of continuity of energy becomes RB − Qy = (R' + p2ƒ' — μ‚Ï') B′ − (Q' + p‚Ì' − p‹ƒ' ́) √'· Since BB' and y = y', it follows from (14) that this equation reduces to which shows that the component of the electromotive force in the plane of incidence is also discontinuous. The boundary conditions are therefore the following; (i) continuity of electric displacement perpendicular to the reflecting surface, which is equivalent to continuity of magnetic force parallel to z; (ii) continuity of magnetic induction perpendicular to the reflecting surface, which is also equivalent to equation (14); (iii) continuity of magnetic force parallel to y; (iv) equation (20), which follows partly from (i), (ii) and (iii), and partly from the condition that the flow of energy must be continuous. We have therefore four equations, and no more, to determine the four unknown quantities. 482. Reflection and Refraction. We shall now calculate the amplitudes of the reflected and refracted waves, when light is reflected and refracted at the surface of a transparent medium which is magnetized normally, so that p2p3=0. Let A, B be the amplitudes of the two components of the incident light perpendicular to, and in the plane of incidence; then the displacements in the four waves may be written. |