{\bf Angular momentum of EM fields} \[ @@ -1630,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1619,7 +1614,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
-+
{\bf Continuity equation} \[ @@ -1660,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index c7260a5..56f7902 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1612,7 +1607,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
This is thus simply a conservation law for momentum, with
-+{\bf Momentum density in the EM fields} \[ @@ -1624,7 +1619,7 @@ This is thus simply a conservation law for momentum, with
In a region in which the mechanical momentum is not changing due to external influences, we then have the
-+{\bf Continuity equation for EM momentum} \[ @@ -1653,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index af4312d..732c909 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1654,7 +1649,7 @@ and similarly for \({\boldsymbol B}\). We thus get
This expression can be greatly simplified by introducing the
-+{\bf Maxwell stress tensor} \[ @@ -1677,7 +1672,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
We then obtain
-+{\bf EM force per unit volume} \[ @@ -1689,7 +1684,7 @@ We then obtain
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
-+{\bf Total force on charges in volume} \[ @@ -1718,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index a97c82a..01f82a7 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1666,7 +1661,7 @@ so we get
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
we obtain
-+
{\bf Poynting's theorem} \[ @@ -1691,7 +1686,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
Energy per unit time, per unit area carried by EM fields:
-+{\bf Poynting vector} \[ @@ -1704,7 +1699,7 @@ Energy per unit time, per unit area carried by EM fields:
We can thus express Poynting's theorem more compactly:
-+{\bf Poynting's theorem} \[ @@ -1717,7 +1712,7 @@ We can thus express Poynting's theorem more compactly:
where we have defined the total
-+{\bf Energy in electromagnetic fields} \[ @@ -1740,7 +1735,7 @@ Then, \] so we get the
-+{\bf Poynting theorem (differential form)} \[ @@ -1757,7 +1752,7 @@ and has a similar for to the continuity equation -
+\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting. Assuming that the field is uniform, the electric field parallel to the wire is @@ -1802,7 +1797,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_emw.html b/build/emd_emw.html index 681b284..52d85ec 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1598,8 +1593,8 @@ Table of contents
emd.emw
--
+
+-Prerequisites
-
@@ -1608,8 +1603,8 @@ Prerequisites
-+
+Objectives
-
@@ -1650,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index 689e7b4..2c05365 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1628,7 +1623,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
-+
{\bf Poynting vector of a monochromatic EM wave} \[ @@ -1644,7 +1639,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
Similary, we get the
-+{\bf Momentum density of a monochromatic EM wave} \[ @@ -1695,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index dc497bd..51f16db 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1629,7 +1624,7 @@ 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
-+
{\bf E and B fields for a monochromatic EM plane wave} \[ @@ -1673,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index a9f9433..177d261 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1626,7 +1621,7 @@ 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
-+
{\bf Wave equations for electric and magnetic fields in vacuum} \[ @@ -1682,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emdm.html b/build/emdm.html index ea5a728..4956e9e 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
- @@ -1622,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 62e26ed..82a370b 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
- @@ -1622,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.
-diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 39214c7..7f3f484 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - +Created: 2022-02-17 Thu 08:42
+Created: 2022-02-21 Mon 10:33
Pre-Quantum Electrodynamics @@ -602,11 +602,11 @@ Table of contents- -Point Chargeems.ms.lf.pc +Point Chargesems.ms.lf.pc
- -Currentsems.ms.lf.c +Steady Currentsems.ms.lf.sc @@ -614,21 +614,12 @@ Table of contents
-
+Charge Conservation and the Continuity Equationems.ms.ce
--
+
+ -
Steady Currents: the Biot-Savart Lawems.ms.BS
-
-
-
-
-
- -The Magnetic Field issuing from a Steady Currentems.ms.BS.sc - - - -
-
@@ -640,11 +631,15 @@ Table of contents
- -Straight-line Currentsems.ms.dcB.sc +Simplistic case: infinite wireems.ms.dcB.iw
- -Divergence and Curl of \({\bf B}\) from Biot-Savartems.ms.dcB.BS +Divergence of \({\bf B}\) from Biot-Savartems.ms.dcB.d + + +
- +Curl of \({\bf B}\) from Biot-Savart; Ampère's Lawems.ms.dcB.c @@ -661,6 +656,10 @@ Table of contents
- +Definition; Gauge Choicesems.ms.vp.A + + +
- Magnetic Boundary Conditionsems.ms.vp.mbc @@ -698,10 +697,6 @@ Table of contents
- -A proper definition of "statics"emsm.esm.s - - -
-
@@ -1435,7 +1430,7 @@ Table of contents
- -Product Rulesc.m.dc.pr +Product argumentsc.m.dc.pr
-
@@ -1633,7 +1628,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
-+
{\bf Polarization current density} \[ @@ -1651,7 +1646,7 @@ the polarization current is the result of linear motion of charge when polarization changes). We can check consistency with the continuity equation associated to the conservation of bound charges:
-