In the free space, we have (or the atomsphericair) we have perm activity of the free species suitability of the free is, volume charge density and density of the conduction current is there for, the differential forms of Max well's equations, are Which are the different tail forms of Max well's equations in the free space By taking for both sides of eq. 3, we obtain By substituting from eqns 1 and 4 in eqn (5, 4 ) which is the wave equation for the electric field in the free space. The electro magnetic waves in the free space

By taking for both sides of eqn 4 we get By substituting g from eqns 2 and 3 in eqn 6 we obtain : x which is the wave equation for the magnetic field in the free space. For simplicity, the electromagnetic y Field sin one dimension in the carte Sian coordinal atiesare : B ( x, y, z, t ) = E ( z, t ) z ( x, y, z, t ) = B ( z, t ) There fore, the wave equation stake the forms : and Which are the wave equations for the electric and the magnetic fields in one dimension

The general form for the wave equation is By comparing the general equation with the wave equations of the electric and the magnetic fields, we find that : ɛ ̥ µ̥ C The result indicates that there are electromagnetic waves move in the free space by the speed light this in addition to the light consists of electromagnetic waves that is could explained these facts experiment enthalpy the solution of the wave equations is the wave functions: and By Where and are the maximum values for the electric and the magnetic fields, K is the wave number and W is angular frequency for the electric field we find that 2,, )

By substituting in the wave equation where these equation in dictate that the wave function of satisfies the wave equation of, the previous method applied also to the magnetic filed Equations the wave functions show propagation of the electrom a gneti cwaves in the free space as in the figure.

The relation between the electric, and the magnetic fields in the free space Faradays lawis - In cartesian coordinaties, we have : Then

In general, we get : therelation between B, h space

The electromagnetic waves in isotro pic in sulating medium : Suppose that we have an isotropic insulating mediums has,, and there fore, the differential forms of Max well’s equation are: By taking for both sides of eq. 3, weget. By substituting from eqns 1 and 4 in eqn (5, 4 )

It is know that where Which is the wave equation for the electric field in is tropic insulating medium. Similarly, be taking for both, sides of eqn. 4, we get : By substituting from eqns 2 and 3 in eqn 6 We get : It is known that Which is the wave equation for the magnetic field in an isotropic insulating medium

It is known that, the general wave equation is By comparing the general wave equation with the wave equation of the electric and the magnetic fields, we get :, where, and is called C= The refractive in dex of the is otropic in sulfating medium. Which is the law of refraction of light

Suppose that : and There for : and which are the wave equations in one dimension only. The solution of these equations are :

The relation between the electric, and the magnetic fields in isotropic in sulating medium faraday’s law is In one dimension, we have : and Then ampere’s law takes the form : Where But In general,

Energy of the electrom gnetic waves in the free space The energy density perunit volume of the electric fields The energy density per unit volume of the magnetic, fieldis The total energy density inside the boxes shown in the figure is Where v is the volume of the box we know that and there fore The average energy density per unit time with in the box Where T is the periodic time.

but There fore the average energy passing the unit area per the unit time in the direction of propagation of the wave is called pointing vector, as follows. pointing vector Isotropic insulating medium.

Isotropic insulating medium. In isotropic insulfating we have. There fore, the pouting vector for an electromagnetic wave propagates in an istropic in sulating medium

Absorption of the electromagnetic waves in the conductors : The conductors (the metals) μ Have, and Subsequently, the max well’s equations for the conductors are By taking for both sides of eq. 3, we get: From Eqn 1 and 4 in Eq. 5, we obtain :

Which is the wave equation for the electric field in the conductor materials similarly, by taking for both sides of eq. 4 we get From Eq. ns 2 and 3 in Eq. 6, we obtain: (Which the wave equation for the magnatic field in the conductor materials (metals For simplicity, suppose that.

And There for, the wave equations are : Which are the wave functions in one dimension. Supposethat the solution of the wave function of the wave furiction of the electric field is : where There for : By substituting in the wave equation, we get, Suppose that

Real : Imaginary : There for, the wave equation of the electric field takes the form : For most of the conductor materials, we have : then : the equation tends to : and Subsequently, the equation ginesus :, and the wave function in equation becomes : Similarly, the wave function of the magnetic field is :

The wave function for the electric and the magnetic field in dicate that the field decreases exponentially With the increase of z, as show in the figure The attend auction depth sis the distant trance at which The electric (or the magnetic) Field component approaches zero, where

Speed of the electromagnetic waves in the conductor when But And Imaginary…. reel

There for And the speed

Spherical capacitor has capacitance and is connected to an ac source of correct of is passing through the. Capacitor. The electric field vector (component) of an electric tromagnatic wave propagates in the positive direction of is, where the physical quantities, are expressed in (cm, gm, s ) system. Find : a)The wave length of the wave b) Frequency of the wave c) Component of the magnetic field Sols a)

A TV station Tran smelted its programs with frequency of Find the wave lung the of the trans milted electromagnetic waves.