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电报方程[Telegraphy Equations] [2005-8-14]
iamet 发表在 Ω〖物理〗
| The telegraphy equations describe the propagation of electric signals, and predict traveling damped electromagnetic waves. To derive them, begin with the Maxwell equations in a dielectric medium. In cgs,
   (1)
  (2)
  (3)
  (4)
where E is the electric field, is the charge density, is the electric permittivity, c is the speed of light, B is the magnetic field, is the magnetic permeability, and J is the current density. In MKS, the corresponding equations are
 (5)
 (6)
(7)
 (8)
Assume we are in a regime where Ohm's law holds, so that
 (9)
where  is the electrical conductivity. Then (4) becomes in cgs,
 (10)
and in MKS,
 (11)
Consider the electric field in cgs,
 
]
] (12)
and in MKS,
]
] (13)
Using equation (4) to eliminate  in cgs
 (14)
 (15)
and in MKS
 (16)
Now, if is constant, then  and the equations reduce to in cgs
 (17)
and in MKS
 (18)
Consider the magnetic field in cgs
 
]
] (19)
 (20)
and in MKS
]
] (21)
 (22)
Using equation (2) to eliminate  . In cgs
 (23)
and in MKS
 (24)
Equations (17) to (18) and (23) to (24) are known as the equations of telegraphy.
A harmonic traveling wave in the dielectric will be of the form
 (25)
Therefore, in cgs
 (26)
so the propagation vector is
  [/right](27)[/right]
  [/right](28)[/right]
  [/right](29)[/right]
If  , then equations (5) and (6) simplify further to in cgs
 (30)
 (31)
and in MKS
 (32)
 (33)
which are wave equations for the coupled E and B fields, predicting the existence of electromagnetic radiation. |
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