Angular momentum of EM fields
-Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
Continuity equation conteq \[ @@ -1683,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
This is thus simply a conservation law for momentum, with
-Momentum density in the EM fields
-In a region in which the mechanical momentum is not changing due to external influences, we then have the
-Continuity equation for EM momentum
-Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
This expression can be greatly simplified by introducing the
-Maxwell stress tensor
--The element \(T_{ij}\) represents the force per unit area in the $i$th direction acting on a surface element oriented in the $j$th direction. Diagonal elements are pressures, off-diagonal elements are shears. +The element \(T_{ij}\) represents the force per unit area in the \(i\) direction acting on a surface element oriented in the \(j\) direction. Diagonal elements are pressures, off-diagonal elements are shears.
@@ -1707,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th directionWe then obtain the
-EM force per unit volume
-where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
-Total force on charges in volume
-Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
- Gr (8.6)
Poynting's theorem
-- Gr (8.9)
Energy per unit time, per unit area carried by EM fields: given by the
-Poynting vector
-- Gr (8.10)
We can thus express Poynting's theorem more compactly:
-Poynting's theorem (integral form)
-- Gr (8.11)
where we have defined the total
-Energy in electromagnetic fields
-- Gr (8.5)
Poynting theorem (differential form)
-- Gr (8.14)
Example: Joule heating
@@ -1895,7 +1887,7 @@ and points radially inwards. Energy per unit time passing surface of wire: \[ \int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I \] -where the minus sign means energy is flowing {\it in} (the wire heats up), +where the minus sign means energy is flowing in (the wire heats up), and the value is as expected. @@ -1920,7 +1912,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
+
+
Prerequisites
@@ -1632,8 +1624,8 @@ Prerequisites
-
-
+
+
Objectives
@@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Prerequisites
-
@@ -1632,8 +1624,8 @@ Prerequisites
+
+
Objectives
@@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Objectives
-
@@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
+Poynting vector of a monochromatic EM wave +
+-{\bf Poynting vector of a monochromatic EM wave} \[ - {\boldsymbol S} = c u ~\hat{\boldsymbol k} - \] +{\boldsymbol S} = c u ~\hat{\boldsymbol k} +\tag{Poynting_mpw}\label{Poyting_mpw} +\]
Similary, we get the
--{\bf Momentum density of a monochromatic EM wave} +Momentum density of a monochromatic EM wave \[ - {\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k} - \] +{\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k} +\]
-The average power per unit time per unit area transported by an EM wave is called the {\bf Intensity} +The average power per unit time per unit area transported by an EM wave is called the Intensity \[ I \equiv \langle S \rangle = \frac{c\varepsilon_0}{2} E_0^2 \]
-The {\it radiation pressure} is the momentum transfer per unit area per unit of time +The radiation pressure is the momentum transfer per unit area per unit of time \[ P = \frac{1}{A}\frac{\Delta p}{\Delta t} = \frac{\langle g \rangle A c \Delta t}{A \Delta t} = \frac{\varepsilon_0}{2} E_0^2 = \frac{I}{c}. \] @@ -1719,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
-A {\it monochromatic} wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have +A monochromatic wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have \[ {\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm} {\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)} @@ -1632,7 +1624,7 @@ Maxwell's equations impose constraints. Since \({\boldsymbol \nabla} \cdot {\bf (E_0)_z = 0 = (B_0)_z \label{Gr(9.44)} \] -so {\bf electromagnetic waves are transverse}. +so electromagnetic waves are transverse.
@@ -1643,9 +1635,26 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia \label{Gr(9.46)} \] so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and +
+ +\[ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. -\label{Gr(9.47)} +\tag{EBmpw}\label{EBmpw} \]
@@ -1653,19 +1662,37 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. Generalizing to propagation in the direction of an arbitrary wavevector \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the -+E and B fields for a monochromatic EM plane wave +
+-{\bf E and B fields for a monochromatic EM plane wave} \[ - {\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n}, - \hspace{10mm} - {\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n} - = \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t) - \] -with the transversality condition +{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n}, +\hspace{10mm} +{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n} += \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t) +\tag{mpw}\label{mpw} +\] +with the transversality condition \[ - \hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0 - \] +\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0 +\]
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
-\subsubsection*{The wave equation for \({\bf E}\) and \({\bf B}\)} Take Maxwell's equations in vacuum:
\begin{align} @@ -1650,15 +1641,32 @@ These take the form of coupled first-order partial differential equations for \( Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), we get the -+Wave equations for electric and magnetic fields in vacuum +
+-{\bf Wave equations for electric and magnetic fields in vacuum} \[ - {\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2}, - \hspace{1cm} - {\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}. - \label{Gr(9.41)} - \] +{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2}, +\hspace{1cm} +{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}. +\tag{WaveEq}\label{WaveEq} +\]
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
Created: 2022-03-02 Wed 15:45
+Created: 2022-03-07 Mon 20:38
-From static case: electric polarization \({\bf P}\) produces bound charge density (\ref{Gr(4.12)}) +From static case: electric polarization \({\bf P}\) produces bound charge density rhob \[ \rho_b = -{\boldsymbol \nabla} \cdot {\bf P} \label{Gr(7.46)} \] -and magnetization \({\bf M}\) produces bound current density (\ref{Gr(6.13)}) +and magnetization \({\bf M}\) produces bound current density JbcurlM \[ {\bf J}_b = {\boldsymbol \nabla} \times {\bf M} \label{Gr(7.47)} @@ -1657,13 +1649,30 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the
-