Write out Maxwell\'s Equations component wise in cylindrical coordinates in thei

Question

Write out Maxwell's Equations component wise in cylindrical coordinates in their most general form. Remember that each field component can be a function of all three spatial dimensions as well as time. You should end up with eight equations: the two divergence equations, and three equations for the curl equations when they are separated component wise. At this point, assume that all source terms are present-that there exists both a charge density and a current density throughout the region of interest. This is the form you would need if you had to solve a fully general program for a wave form on a coaxial cable.

Explanation / Answer

First, the two divergence equations:

( abla .ec{B}=0)

thus, (rac{1}{s}rac{partial s B_{s}}{partial s}+rac{1}{s}rac{partial B_{phi}}{partial phi}+rac{partial B_z}{partial z}=0)

Then,

( abla .ec{E}= ho/epsilon_0)

thus, (rac{1}{s}rac{partial s E_{s}}{partial s}+rac{1}{s}rac{partial E_{phi}}{partial phi}+rac{partial E_z}{partial z}= ho/epsilon_0)

Now, curl equations:

( abla imes ec{E}=-rac{partial ec{B}}{partial t})

Thus,

(left({1 over ho}{partial E_z over partial phi}    - {partial E_phi over partial z} ight)=-{partial B_{ ho} over partial ho})

(left({partial E_ ho over partial z} - {partial E_z over partial ho} ight)=-{partial B_{phi} over partial phi})

(left({partial left( ho E_phi ight) over partial ho} - {partial E_ ho over partial phi} ight)=-{partial B_{z} over partial z})

Also,

( abla imes ec{B}=mu_0 ec{J}+mu_0 u_0rac{partial ec{E}}{partial t})

Thus,

(left({1 over ho}{partial B_z over partial phi}    - {partial B_phi over partial z} ight)=mu_0 J_ ho+mu_0 u_0{partial E_{ ho} over partial ho})

(left({partial B_ ho over partial z} - {partial B_z over partial ho} ight)=mu_0 J_phi+mu_0 u_0{partial E_{phi} over partial phi})

(left({partial left( ho B_phi ight) over partial ho} - {partial B_ ho over partial phi} ight)=mu_0 J_z+mu_0 u_0{partial E_{z} over partial z})

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