← (nm06) 電磁場和電磁波 EM-Fields & Waves →

Key Points

• Connection between B and E (not independent)
• Maxwell's 4 Equations: Complete description of B and E fields
• Electromagnetic Waves

Transformations relate B & E (Observer B vs. Observer A)

Galilean	Lorentz (next course)
valid only if $$ \mathbf{ \vec{ \beta}} = \mathbf{ \vec{ v}} / c << 1 $$	No conditions $$ \gamma = { (1 - \beta^2)}^ {-1/2} $$
$$ \mathbf{ \vec{ E}}_B \approx \mathbf{ \vec{ E}}_A \, + \, \mathbf{ \vec{ v}}_{BA} \times \mathbf{ \vec{ B}}_A $$	$$ \mathbf{ \vec{ E}}_B = \gamma ( \mathbf{ \vec{ E}}_A \, + \, \mathbf{ \vec{ \beta}} \times \mathbf{ \vec{ B}}_A ) \, - \, {\gamma^2 \over { \gamma + 1}} \vec {\beta} \, ( \vec {\beta} \cdot \mathbf{ \vec{ E}}_A ) $$
$$ \mathbf{ \vec{ B}}_B \approx \mathbf{ \vec{ B}}_A \, - \, \mathbf{ \vec{ v}}_{BA}/c^2 \times \mathbf{ \vec{ E}}_A $$	$$ \mathbf{ \vec{ B}}_B = \gamma ( \mathbf{ \vec{ B}}_A \, - \, \mathbf{ \vec{ \beta}} \times \mathbf{ \vec{ E}}_A ) \, - \, {\gamma^2 \over { \gamma + 1}} \vec {\beta} \, ( \vec {\beta} \cdot \mathbf{ \vec{ B}}_A )$$
Note that:	$$ c = { (\mu_o \epsilon_o) }^{-1/2} $$

Maxwell's Equations

Integral Form
$$ \bigcirc \!\!\!\!\!\!\!\!\!\iint_S \mathbf {\vec E} \, \cdot d \mathbf {\vec A} = \Phi_e = {Q \over \epsilon} $$	$$ \bigcirc \!\!\!\!\!\!\!\!\!\iint_S \mathbf {\vec B} \, \cdot d \mathbf {\vec A} = \Phi_m = 0 $$
$$ \oint_C \mathbf {\vec B} \, \cdot \, d \, \mathbf {\vec s} = {\mu_o I} + \color{fuchsia} { \epsilon_o \mu_o {{d \Phi_e} \over {dt}} } $$	$$ \oint_C {\mathbf {\vec E}}(t) \, \cdot \, d \, \mathbf {\vec s} = - { {d \Phi_m} \over {dt}} = - {d \over dt} \bigcirc \!\!\!\!\!\!\!\!\!\iint_S { \mathbf {\vec B}}(t) \, \cdot d \mathbf {\vec A}(t)$$

Differential Form
$$\nabla \cdot \mathbf { \vec{E}} = \frac{\rho_0}{\varepsilon_0} $$	$$ \nabla \cdot \mathbf {\vec{B}} = 0 $$
$$ \nabla \times \mathbf {\vec{B}} = \mu_0 \mathbf {\vec{J}} + \color{fuchsia} {\mu_0 \varepsilon_0\frac{\partial \mathbf {\vec{E}}}{\partial t}} $$	$$ \nabla \times \mathbf {\vec{E}} = -\frac{\partial \vec{B}}{\partial t} $$

where

• E, B are the electric and magnetic fields
• ρ₀ is called the charge density
• J is called the current density
• ε₀ is a constant called the permittivity of free space
• μ₀ is a constant called the permeability of free space
• The speed of light, c, is calculated from these equations using the formula $$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$

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$$ \mathbf a = { \sum \mathbf F \over m_e} = {1 \over m_e} \left ( m_g \mathbf {\vec G} + q \mathbf {\vec E} + q \mathbf {\vec v} \times \mathbf {\vec B} \right ) $$

Lecture Power Points for Knight Chapter 34