Latest revision as of 15:01, July 14, 2016

The Maxwell Equations

One the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism. When we set $\mathbf {E} \$ as the electric field, $\mathbf {B} \$ as the magnetic field, $\mathbf {D} \$ as the electric displacement field, and $\mathbf {H} \$ as the magnetizing field, then the equations describing these quantities are

$\nabla \cdot \mathbf {D} =\rho$

$\nabla \cdot \mathbf {B} =0$

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}$

We can integrate all these equations to get the integral forms:

$\oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =\int _{V}\rho \,\mathrm {d} V$

$\oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$

$\oint _{C}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\int _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

$\oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} +\int _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the Maxwell equations, after James Clerk Maxwell.

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+=The Maxwell Equations=
-=Generalized Integration=
+One the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism.  When we set <math>\mathbf{E} \ </math> as the [[electric field]],  <math>\mathbf{B} \ </math> as the [[magnetic field]], <math>\mathbf{D} \ </math> as the [[electric displacement field]], and <math>\mathbf{H} \ </math> as the [[magnetizing field]], then the equations describing these quantities are
-=The Line Integral=
+<math>\nabla \cdot \mathbf{D} = \rho</math>
+<math>\nabla \cdot \mathbf{B} = 0</math>
-We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space.  But lines, and curves, can be contained in the plane, and surfaces can be contained in space.  Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains?
+<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
+<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
-Well, because it is simpler, and useful to have under our belts before we expand our repetoire.  But there is no reason we can't integrate a function over a region in its domain which is a lower dimension than the domain itself.
+We can integrate all these equations to get the '''integral forms''':
-We'll start with the line integral.  Technically, it should be called the curve integral, but let's let that slide for now and get to the definition.  We're trying to integrate over a curve, say, ''C''.  At first, we'll let this curve lie in the ''xy'' plane - it's not difficult to generalize to three dimensions.  So we proceed as before: we divide this region into ''n'' pieces <math>C_k, 1\leq k \leq n</math>, and we pick a point <math>(x_k,y_k)\in C_k</math> for each piece.  Letting the length of <math>C_k</math> be <math>L_k</math>.  The definition follows as expected:
+<math>\oint_S  \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math>
-<math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math>
+<math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
-=Green's Theorem in the Plane=
+<math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}  = -  \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A} </math>
+<math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
+ \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
+The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the [[Maxwell equations]], after [[James Clerk Maxwell]].
-=The Surface Integral=
 {{MVCalc}}

Difference between revisions of "Calc3.8"

Latest revision as of 15:01, July 14, 2016

The Maxwell Equations