Angular momentum of EM fields
-Created: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
- -QED todayqed.t +Lagrangianqed.L @@ -1635,7 +1635,7 @@ This means that \] Since this is true for any volume, we have (re)derived the -
- -QED todayqed.t +Lagrangianqed.L @@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
- -QED todayqed.t +Lagrangianqed.L @@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
- -QED todayqed.t +Lagrangianqed.L @@ -1636,14 +1636,14 @@ done by EM forces? From Lorentz force law: Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus, -
- Gr (8.6)
- Gr (8.9)
- Gr (8.10)
- Gr (8.11)
- Gr (8.5)
- Gr (8.14)
- -QED todayqed.t +Lagrangianqed.L @@ -1614,8 +1614,8 @@ Table of contents emd.emw
- -QED todayqed.t +Lagrangianqed.L @@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave, \] or more succinctly: -
- Gr (9.57)
- -QED todayqed.t +Lagrangianqed.L @@ -1636,14 +1636,14 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia \] so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and -
- Gr (9.47)
Continuity equation conteq \[ @@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
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: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
This expression can be greatly simplified by introducing the
-Maxwell stress tensor
-We 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: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
Poynting's theorem
-Energy per unit time, per unit area carried by EM fields: given by the
-Poynting vector
-We can thus express Poynting's theorem more compactly:
-Poynting's theorem (integral form)
-where we have defined the total
-Energy in electromagnetic fields
-Poynting theorem (differential form)
-Example: Joule heating
@@ -1911,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.Created: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
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Prerequisites
@@ -1624,8 +1624,8 @@ Prerequisites
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Objectives
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Prerequisites
-
@@ -1624,8 +1624,8 @@ Prerequisites
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+
Objectives
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Objectives
-
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42
Poynting vector of a monochromatic EM wave
-Similary, we get the
-Momentum density of a monochromatic EM wave \[ @@ -1729,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-03-22 Tue 10:52
+Created: 2022-03-24 Thu 08:42