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Adapted from notes by Prof. Stuart A. Long

1

Notes 4

Maxwell's Equations

ECE 3317

Applied Electromagnetic Waves

Prof. David R. Jackson

Fall 2022

2 Here we present an overview of Maxwell's equations. A much more thorough discussion of Maxwell's equations may be found in the text and class notes for ECE 3318: http://courses.egr.uh.edu/ECE/ECE3318

Notes 10: Electric Gauss's law

Notes 18: Faraday's law

Notes 28: Ampere's law

Notes 28: Magnetic Gauss law

Extra reference: D. Fleisch, , Cambridge

University Press, 2008. (This is on reserve in the Library.)

Overview

Electromagnetic Fields

Four vector quantities

electric field [Volt/meter] electric flux density[Coulomb/meter 2 magnetic field [Amp/meter] magnetic flux density[Weber/meter 2 ] or [Tesla]

Each are functions of space and time

e.g. (x,y,z,t) electric current density[Amp/meter 2 v electric charge density[Coulomb/meter 3 3

Reminder:

The HandscriptSF font is used

to denote time-varying vectors.

MKS units

Length -meter [m]

Mass -kilogram [kg]

Time -second [s]

Some common prefixes and the power of ten each represent are listed below femto-f-10 -15 pico-p-10 -12 nano-n-10 -9 micro -ȝ10 -6 milli-m-10 -3 mega -M-10 6 giga -G-10 9 tera -T-10 12 peta -P-10 15 centi -c -10 -2 deci -d -10 -1 deka -da-10 1 hecto -h -10 2 kilo -k-10 3 4 0 v t t

Maxwell's Equations

(Time-varying, differential form) 5

Maxwell

James Clerk Maxwell (1831

-1879) James Clerk Maxwellwas a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations - Maxwell's equations - demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the " ", after the first one carried out by Isaac

Newton.

Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. Finally, in 1864 Maxwell wrote where he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is considered to be one of the greatest advances in physics. (Wikipedia) 6

Maxwell's Equations (cont.)

0 B E D HJ B D

Faraday"s law

Ampere"s law

Magnetic Gauss law

Electric Gauss law

7 Questions: When does a magnetic field produce an electric field? When does an electric field produce a magnetic field? When does a current flow produce a magnetic field? When does a charge density produce an electric field?

Charge Density

8 vV

Q dQxyzV dV

Example:Protons are closer together as we move to the right.Non-uniform cloud of charge density v xyz dV dQ xyz

Current Density Vector

9 J E J

Medium

Current flow is defined to be in the direction that positive charges move in. 2

A/mcurrent density vectorJ

Note:If negative charges are moving, we can pretend that positive charges are moving in the opposite direction.

Current Density Vector (cont.)

10

Material[S/m]

Silver6.310

7

Copper6.010

7

Copper (annealed)5.810

7

Gold4.110

7

Aluminum3.510

7

Zinc1.710

7

Brass1.610

7

Nickel1.410

7

Iron1.010

7

Tin9.210

6

Steel (carbon)7.010

6

Steel (stainless)1.510

6

Ohm's law

11

Current through a tilted surface:

I nS J

Current Density Vector (cont.)

E J S n

Medium

Current Density Vector (cont.)

12 S I ndS J Note:

The direction of the unit normal vector

determines whether the current is measured going up or down through the surface. n J S

I nS J

0 t t t

Law of Conservation of Electric

Charge (Continuity Equation)

v t 13 This is the continuity equation in point or differential form. v

Continuity Equation (cont.)

Apply the divergence theorem:Integrate both sides over an arbitrary volume : J out 14 JJ

Hence:

J (outward normal) (current flowing out of )

Continuity Equation (cont.)

Physical interpretation:

out encl encl out (This assumes that the surface is stationary.) encl in or 15

HenceRight-hand side:

Continuity Equation (cont.)

encl in 16 This implies that charge is never created or destroyed.

It only moves from one place to another!

J encl 0 0 0 v v tt

E DHJ B

BD

EHJ BD

and v

EHE H J

0 v E jB H JjD B D jt 18

Maxwell's Equations (cont.)

Constitutive Relations

The characteristics of the media relate to and to 0 0 0 0 permittivity permeability -12 0 -7 0

8.8541878 10 [ F/m]

= 41 0 [H/ m]()µ exact* 00 1 c (exact value that is defined)Free Space 19 8

2.9979245810 [m /s]c

*Prior to 2019 (since 1983) 20 7 2

2 10N /mwhen1 m

Definition of =1Amp:

I d F x2 # 1 # 2

Two infinite wires carrying DC currents

Definition of the Amp*:

2quotesdbs_dbs18.pdfusesText_24

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