Cross field

Content

1 Equations linking transversal field components $({\vec {E}}_{T},\;{\vec {H}}_{T})$ {\ displaystyle ({\ vec {E}} _ {T}, \; {\ vec {H}} _ {T})} ${\ displaystyle ({\ vec {E}} _ {T}, \; {\ vec {H}} _ {T})}$ with components along the direction of radiation ${\vec {E}}_{Z}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ and ${\vec {H}}_{Z}$ ${\ displaystyle {\ vec {H}} _ {Z}}$ ${\ displaystyle {\ vec {H}} _ {Z}}$
2 Transverse electric field
3 Transverse magnetic field
4 Transverse electromagnetic field
5 Equivalent circuit

Equations linking transversal field components $({\vec {E}}_{T},\;{\vec {H}}_{T})$ ${\ displaystyle ({\ vec {E}} _ {T}, \; {\ vec {H}} _ {T})}$ ${\ displaystyle ({\ vec {E}} _ {T}, \; {\ vec {H}} _ {T})}$ with components along the direction of radiation ${\vec {E}}_{Z}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ and ${\vec {H}}_{Z}$ ${\ displaystyle {\ vec {H}} _ {Z}}$ ${\ displaystyle {\ vec {H}} _ {Z}}$

Let the electromagnetic field propagating in the direction ${\vec {z}}$ ${\ displaystyle {\ vec {z}}}$ , and $\omega$ ${\ displaystyle \ omega}$ - cyclic frequency of the harmonic signal.

${\vec {E}}_{T}$ ${\ displaystyle {\ vec {E}} _ {T}}$ associated with ${\vec {E}}_{Z}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ and ${\vec {H}}_{Z}$ ${\ displaystyle {\ vec {H}} _ {Z}}$ the following relation:

{\vec {E}}_{T}={\frac {-\gamma \nabla _{T}E_{Z}-j\omega \mu _{0}(\nabla _{T}H_{Z})\times {\vec {z}}}{\gamma ^{2}+k_{0}^{2}}}

{\ displaystyle {\ vec {E}} _ {T} = {\ frac {- \ gamma \ nabla _ {T} E_ {Z} -j \ omega \ mu _ {0} (\ nabla _ {T} H_ {Z}) \ times {\ vec {z}}} {\ gamma ^ {2} + k_ {0} ^ {2}}}}

${\vec {H}}_{T}$ ${\ displaystyle {\ vec {H}} _ {T}}$ associated with ${\vec {E}}_{Z}$ ${\ displaystyle {\ vec {E}} _ {Z}}$ and ${\vec {H}}_{Z}$ ${\ displaystyle {\ vec {H}} _ {Z}}$ the following relation:

{\vec {H}}_{T}={\frac {-\gamma \nabla _{T}H_{Z}+j\omega \varepsilon _{0}(\nabla _{T}E_{Z})\times {\vec {z}}}{\gamma ^{2}+k_{0}^{2}}}

{\ displaystyle {\ vec {H}} _ {T} = {\ frac {- \ gamma \ nabla _ {T} H_ {Z} + j \ omega \ varepsilon _ {0} (\ nabla _ {T} E_ {Z}) \ times {\ vec {z}}} {\ gamma ^ {2} + k_ {0} ^ {2}}}}

Where

$\nabla _{T}$ ${\ displaystyle \ nabla _ {T}}$ - transversal Laplace operator ;
$k_{0}={\frac {\omega }{c}};$ ${\ displaystyle k_ {0} = {\ frac {\ omega} {c}};}$
$\gamma ^{2}=k_{c}^{2}-k_{0}^{2};$ ${\ displaystyle \ gamma ^ {2} = k_ {c} ^ {2} -k_ {0} ^ {2};}$
$\varepsilon$ ${\ displaystyle \ varepsilon}$ - dielectric constant ;
$\mu$ ${\ displaystyle \ mu}$ - magnetic permeability .

Transverse Electric Field

When ${\vec {E}}_{Z}={\vec {0}}$ ${\ displaystyle {\ vec {E}} _ {Z} = {\ vec {0}}}$ and ${\vec {H}}_{Z}\neq {\vec {0}}$ ${\ displaystyle {\ vec {H}} _ {Z} \ neq {\ vec {0}}}$ , then the previous formulas take the form:

{\vec {E}}_{T}={\frac {-j\omega \mu _{0}}{\gamma ^{2}+k_{0}^{2}}}(\nabla _{T}H_{Z})\times {\vec {z}}

{\ displaystyle {\ vec {E}} _ {T} = {\ frac {-j \ omega \ mu _ {0}} {\ gamma ^ {2} + k_ {0} ^ {2}}} (\ nabla _ {T} H_ {Z}) \ times {\ vec {z}}}

{\vec {H}}_{T}={\frac {-\gamma }{\gamma ^{2}+k_{0}^{2}}}\nabla _{T}H_{Z}

{\ displaystyle {\ vec {H}} _ {T} = {\ frac {- \ gamma} {\ gamma ^ {2} + k_ {0} ^ {2}}} \ nabla _ {T} H_ {Z }}

Hence, ${\vec {E}}_{T}={\frac {j\omega \mu _{0}}{\gamma }}{\vec {H}}_{T}\times {\vec {z}}=Z_{m}{\vec {H}}_{T}\times {\vec {z}}.$ ${\ displaystyle {\ vec {E}} _ {T} = {\ frac {j \ omega \ mu _ {0}} {\ gamma}} {\ vec {H}} _ {T} \ times {\ vec {z}} = Z_ {m} {\ vec {H}} _ {T} \ times {\ vec {z}}.}$
Where: $Z_{m}={\frac {j\omega \mu _{0}}{\gamma }}$ ${\ displaystyle Z_ {m} = {\ frac {j \ omega \ mu _ {0}} {\ gamma}}}$
This expression is usually written as follows:

{\vec {H}}_{T}={\frac {{\vec {z}}\times {\vec {E}}_{T}}{Z_{m}}}

{\ displaystyle {\ vec {H}} _ {T} = {\ frac {{\ vec {z}} \ times {\ vec {E}} _ {T}} {Z_ {m}}}}

Transverse Magnetic Field

When ${\vec {H}}_{Z}={\vec {0}}$ ${\ displaystyle {\ vec {H}} _ {Z} = {\ vec {0}}}$ and ${\vec {E}}_{Z}\neq {\vec {0}}$ ${\ displaystyle {\ vec {E}} _ {Z} \ neq {\ vec {0}}}$ , then the previous formulas take the form:

{\vec {E}}_{T}={\frac {-\gamma }{\gamma ^{2}+k_{0}^{2}}}\nabla _{T}E_{Z}

{\ displaystyle {\ vec {E}} _ {T} = {\ frac {- \ gamma} {\ gamma ^ {2} + k_ {0} ^ {2}}} \ nabla _ {T} E_ {Z }}

{\vec {H}}_{T}={\frac {j\omega \varepsilon _{0}}{\gamma ^{2}+k_{0}^{2}}}(\nabla _{T}E_{Z})\times {\vec {z}}

{\ displaystyle {\ vec {H}} _ {T} = {\ frac {j \ omega \ varepsilon _ {0}} {\ gamma ^ {2} + k_ {0} ^ {2}}} (\ nabla _ {T} E_ {Z}) \ times {\ vec {z}}}

Hence,
${\vec {H}}_{T}={\frac {{\vec {z}}\times {\vec {E}}_{T}}{Z_{m}}},$ ${\ displaystyle {\ vec {H}} _ {T} = {\ frac {{\ vec {z}} \ times {\ vec {E}} _ {T}} {Z_ {m}}},}$
Where: $Z_{m}={\frac {\gamma }{j\omega \varepsilon _{0}}}$ ${\ displaystyle Z_ {m} = {\ frac {\ gamma} {j \ omega \ varepsilon _ {0}}}}$

Transverse electromagnetic field

When ${\vec {E}}_{Z}={\vec {H}}_{Z}={\vec {0}}$ ${\ displaystyle {\ vec {E}} _ {Z} = {\ vec {H}} _ {Z} = {\ vec {0}}}$ electromagnetic field propagating in the direction ${\vec {z}}$ ${\ displaystyle {\ vec {z}}}$ is called transversal .

In a vacuum

{\vec {H}}

{\ displaystyle {\ vec {H}}}

associated with

{\vec {E}}

{\ displaystyle {\ vec {E}}}

the following relation:

{\vec {H}}={\frac {{\vec {z}}\times {\vec {E}}}{Z_{0}}}

{\ displaystyle {\ vec {H}} = {\ frac {{\ vec {z}} \ times {\ vec {E}}} {Z_ {0}}}}

Where: $Z_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}$ ${\ displaystyle Z_ {0} = {\ sqrt {\ frac {\ mu _ {0}} {\ varepsilon _ {0}}}}}$ - vacuum impedance.

Equivalent circuit

Let be $Z={\sqrt {\frac {\mu }{\varepsilon }}}Z_{0}$ ${\ displaystyle Z = {\ sqrt {\ frac {\ mu} {\ varepsilon}}} Z_ {0}}$ - wave resistance of the medium in which the electromagnetic wave propagates in the direction $\mathbf {z}$ ${\ displaystyle \ mathbf {z}}$ . Insofar as $Z\mathbf {H} =\mathbf {z} \times \mathbf {E} ,$ ${\ displaystyle Z \ mathbf {H} = \ mathbf {z} \ times \ mathbf {E},}$
$\mathbf {E} =Z\mathbf {J} ,$ ${\ displaystyle \ mathbf {E} = Z \ mathbf {J},}$ Where $\mathbf {J} =\mathbf {H} \times \mathbf {z}$ ${\ displaystyle \ mathbf {J} = \ mathbf {H} \ times \ mathbf {z}}$ - "current density".