General Wave Equation

General Wave Equation

AU : May-16, Dec.-17, 18

Now according to the vector identity,

• In general the wave equations can be obtained by relating the space and time variations of the electric and magnetic fields, using the Maxwell's equations.

• To obtain general wave equations, let us assume that the electric and magnetic fields exist in a linear, homogeneous and isotropic medium with the parameters µ₁ ε  and σ. Also assume that the medium is source free which clearly gives the idea about the charge free medium. Assume that the medium obeys the Ohm's law i.e.  Then the Maxwell's equations are given by,

To eliminate  from equation (10.2.1), taking curl on both the sides of equation (10.2.1), we get,

•  operator indicates differentiation with respect to space while ∂ /∂t indicates differentiation with respect to time. Both are independent of each other, the operators can be interchanged.

So we get,

Now according to the vector identity,

• This is the wave equation for the electric field  . Now multiplying both the sides of equation (10.2.10) by ε,

• This is the wave equation for  in uniform medium.

• Exactly on the similar lines, the wave equation for  can be obtained by taking curl on both the sides of equation (10.2.2), we get,

Substituting  from equation (10.2.4) in equation (10.2.15), we get

• This is the wave equation for the magnetic field . Now multiplying both the sides by µ, we get,

• This is the wave equation for  in the uniform medium. Hence in general we can write,

• Above equation is three dimensional equation for all the field vectors.

Review Question

1. Derive general wave equations for  fields using Maxwell's equations. AU : May-16, Dec.-17, 18, Marks 8