(This is, incidentally, the simplest possible spherical wave. For notational convenience, let in your calculations.)
(a) Show that obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field.
(b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector . (Does it point in the expected direction? Does it fall off like , as it should?)
(c) Integrate over a spherical surface to determine the total power radiated. [Answer:]
The value of divergence of electric field of Maxwell’s equation is .
The value of curl of electric field is
The value of magnetic field is.
The value of Gauss law of magnetism is .
The value of Ampere’s law.is .
The value of Intensity vector is and the pointing vector S over the full
(c) The value of total power radiated is.
Consider this, incidentally, the simplest possible spherical wave. For notational convenience, let in your calculations.
Write the formula of electric field is,
Here,role="math" localid="1658485458510" is the electric field component of a spherical wave, role="math" localid="1658485452985" is electric field component of a spherical wave, is radius, electric field component of a spherical wave andis electric field component of a spherical wave.
Write the formula of curl of electric field.
Here, is the magnetic field strength andis radius.
Write the formula of magnetic field.
Here, is radius,represent the wave number and is constant.
Write the formula of Gauss law of magnetism.
Here, is the magnetic field strength.
Write the formula of Ampere’s law.
Here, is the electric field component of a spherical wave, is permeability, denotes the speed of light.
Write the formula of intensity vector.
Here, S is Poynting vector.
Write the formula of total power radiated.
Here,I is intensity vector.
From Gauss’s law,
Here, is the free charge density.
Determine the divergence of electric field is,
As there is no free charge density here, therefore .
Hence, Gauss’s law is obeyed.
According to Faraday’s Law.
Determine the curl of electric field is,
Therefore, the value of curl of electric field is
Substitute for .
Integrate equation (8),
Substitute role="math" localid="1658488205000" for and for into equation (9).
Therefore, the value of magnetic field is
Determine the Gauss’s law of magnetism,
Substitute for B into equation (4).
Solve further as
Hence, the Gauss law of magnetism is obeyed.
Determine the Ampere’s law,
Substitute for B .
Solve the term ,
Hence, Ampere’s law is obeyed.
Determine the Poynting vector is given by the following equation.
Substitute for B and for E into above equation (10).
Average over a full cycle is,
Determine the Intensity vector.
Substitute for into equation (6).
The intensity fluctuates as and faces in the direction of . A spherical wave is predicted to behave in this way.
Therefore, the intensity vector is.
Determine the total power radiated is,
Substitute for I.
Therefore, the value of total power radiated is .
Show that the mode cannot occur in a rectangular wave guide. [Hint: In this case role="math" localid="1657512848808" , so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that role="math" localid="1657512928835" is a constant, and hence—applying Faraday’s law in integral form to a cross section—that role="math" localid="1657513040288" , so this would be a TEM mode.]
If you take the model in Ex. 4.1 at face value, what natural frequency do you get? Put in the actual numbers. Where, in the electromagnetic spectrum, does this lie, assuming the radius of the atom is 0.5 Å? Find the coefficients of refraction and dispersion, and compare them with the measured values for hydrogen at and atmospheric pressure: , .
In writing Eqs. 9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave—along the x direction. Prove that this must be so. [Hint: Let the polarization vectors of the transmitted and reflected waves be
prove from the boundary conditions that .]
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