# OpenStax University Physics/E&M/Electromagnetic Waves

## Chapter 16

#### Electromagnetic Waves

Displacement current ${\displaystyle I_{d}=\varepsilon _{0}{\tfrac {d\Phi _{E}}{dt}}}$  where ${\displaystyle \Phi _{E}=\int {\vec {E}}\cdot d{\vec {A}}}$  is the electric flux.

Maxwell's equations {\displaystyle {\begin{aligned}\oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}&={\frac {1}{\epsilon _{0}}}Q_{in}\qquad &\oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {A}}&=0\\\oint _{C}{\vec {E}}\cdot \mathrm {d} {\vec {\ell }}&=-\int _{S}{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}\qquad &\oint _{C}{\vec {B}}\cdot \mathrm {d} {\vec {\ell }}&=\mu _{0}I+\epsilon _{0}\mu _{0}{\frac {\mathrm {d} \Phi _{E}}{\mathrm {d} t}}\end{aligned}}}
▭ Plane EM wave equation ${\displaystyle {\frac {\partial ^{2}E_{y}}{\partial x^{2}}}=\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}}$  where ${\displaystyle c={\tfrac {1}{\sqrt {\varepsilon _{0}\mu }}}}$  is the speed of light
▭ The ratio of peak electric to magnetic field is ${\displaystyle {\tfrac {E_{0}}{B_{0}}}=c}$  and the Poynting vector ${\displaystyle {\vec {S}}={\tfrac {1}{\mu _{0}}}{\vec {E}}\times {\vec {B}}}$  represents the energy flux
▭ Average intensity ${\displaystyle I=S_{ave}={\tfrac {c\varepsilon _{0}}{2}}E_{0}^{2}={\tfrac {c}{2\mu _{0}}}B_{0}^{2}={\tfrac {1}{2\mu _{0}}}E_{0}B_{0}}$
▭ Radiation pressure ${\displaystyle p=I/c}$  (perfect absorber) and ${\displaystyle p=2I/c}$  (perfect reflector).

#### For quiz at QB/d_cp2.16

Displacement current ${\displaystyle I_{d}=\varepsilon _{0}{\tfrac {d\Phi _{E}}{dt}}}$  where ${\displaystyle \Phi _{E}=\int {\vec {E}}\cdot d{\vec {A}}}$  is the electric flux.

Maxwell's equations: ${\displaystyle \epsilon _{0}\mu _{0}=1/c^{2}}$
${\displaystyle \oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}={\frac {1}{\epsilon _{0}}}Q_{in}\qquad }$
${\displaystyle \oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {A}}=0}$
${\displaystyle \oint _{C}{\vec {E}}\cdot \mathrm {d} {\vec {\ell }}=-\int _{S}{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}}$
${\displaystyle \oint _{C}{\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}I+\epsilon _{0}\mu _{0}{\frac {\mathrm {d} \Phi _{E}}{\mathrm {d} t}}}$

${\displaystyle {\frac {\partial ^{2}E_{y}}{\partial x^{2}}}=\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}}$  and ${\displaystyle {\tfrac {E_{0}}{B_{0}}}=c}$

Poynting vector ${\displaystyle {\vec {S}}={\tfrac {1}{\mu _{0}}}{\vec {E}}\times {\vec {B}}}$ =energy flux

Average intensity ${\displaystyle I=S_{ave}={\tfrac {c\varepsilon _{0}}{2}}E_{0}^{2}={\tfrac {c}{2\mu _{0}}}B_{0}^{2}={\tfrac {1}{2\mu _{0}}}E_{0}B_{0}}$

Radiation pressure ${\displaystyle p=I/c}$  (perfect absorber) and ${\displaystyle p=2I/c}$  (perfect reflector).