Download
# Module I Electromagnetic waves Lecture Time dependent EM elds relaxation propagation Amol Dighe TIFR Mumbai Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave Dec PDF document - DocSlides

calandra-battersby | 2014-12-14 | General

### Presentations text content in Module I Electromagnetic waves Lecture Time dependent EM elds relaxation propagation Amol Dighe TIFR Mumbai Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave Dec

Show

Page 1

Module I: Electromagnetic waves Lecture 3: Time dependent EM ﬁelds: relaxation, propagation Amol Dighe TIFR, Mumbai

Page 2

Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 3

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 4

Stationary and non-stationary states Stationary state, by deﬁnition, means that the currents are steady and there is no net charge movement, i.e. ∇· or 0 (1) These statements are equivalent, due to continuity. If the initial distribution of charges and currents does not satisfy the above criteria, they will redistribute themselves so that a stationary state is reached. This process of “relaxation” happens over a time scale that is characteristic of the medium, called the relaxation time.

Page 5

Relaxation time The continuity equation, combining with ∇· , gives ∇· −∇· (2) Using and ∇· 0 (3) The solution to this differential equation is +( / (4) where is the initial current distribution / is the relaxation time −∇· / , etc. relax at the same rate.

Page 6

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 7

Time-dependent electric ﬁeld No free charges, no external EMF sources. Maxwell ∇× ∇× )= ∇× (5) ∇· fr (6) This gives the second order partial differential equation µ µ 0 (7) Depending on whether the / term dominates or the / one, we’ll get two different extremes of behaviour. The former will lead to a propagating wave , the latter will lead to a diffusion equation , corresponding to a decaying wave

Page 8

Looking for solution of the form The differential equation becomes µ 0 (8) There are two time scales here: 1 / and / µ 0 (9) When τω>> µ (10) which is a wave propagating with speed µ When τω<< (11) which is the equation for diffusion. In the context of EM waves, this will lead to a decaying solution. we shall explore these behaviours in detail now.

Page 9

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 10

Propagating (plane wave) solution for ωτ >> displacement current dominates over conduction current µ ωµ 0 (12) Plane wave: all ﬁelds are functions of the distance of a plane from the origin. is the normal to this plane. ∂/ Maxwell’s equations in this language: (13) (14)

Page 11

Longitudinal components of and : longitudinal component of / equation and dot product of with the / equation 0 (15) For non-conducting media (e.g. vacuum), is a constant. : longitudinal component of / equation and dot product of with the / equation 0 (16) Only stationary longitudinal component of is possible, i.e. is constant (note: we have taken

Page 12

Transverse components of and Combining the two equations: µ )= 0 (17) Differential equation for General solution: ut )+ ut )] If is sinusoidal: (18) Direction of propagation (19) Using / )= / (20)

Page 13

Propagating wave in short and are constants in space and time, hence not interesting for wave propagation and can have dependence, with =( / and ﬁelds are transverse to the direction of motion, and also orthogonal to each other.

Page 14

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 15

Decaying plane wave When ωτ << 1, conduction current dominates over displacement current µ ωµ 0 (21) The solution of the form kx implies π/ (22) π/ (23) This gives Re Im (24) The wave then decays with a Im dependence inside the conducting medium.

Page 16

Skin depth in metals For metals, 10 14 sec. So for ω< 10 14 , conduction current dominates. A wave incident on a metallic surface will decay as / (25) where, from the last page, (check factor of 2) (26) Within a distance from the surface of the metal, the wave would decrease in magnitude by a factor 1 . This is the “skin depth of the metal. The surface currents will ﬂow within this width. “Ideal” conductor 0.

Page 17

Recap of topics covered in this lecture Relaxation to stationary state, relaxation time Electromagnetic wave: displacement current and conduction current Transverse electromagnetic ﬁeld solutions for a propagating wave Decay of EM waves in a condunctor, skin depth

Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave brPage 4br Stationary and nonstationary states Stationary state by de64257nition means that the currents are steady and there is no net charge movement ID: 24088

- Views :
**193**

**Direct Link:**- Link:https://www.docslides.com/calandra-battersby/module-i-electromagnetic-waves
**Embed code:**

Download this pdf

DownloadNote - The PPT/PDF document "Module I Electromagnetic waves Lecture ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

Module I: Electromagnetic waves Lecture 3: Time dependent EM ﬁelds: relaxation, propagation Amol Dighe TIFR, Mumbai

Page 2

Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 3

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 4

Stationary and non-stationary states Stationary state, by deﬁnition, means that the currents are steady and there is no net charge movement, i.e. ∇· or 0 (1) These statements are equivalent, due to continuity. If the initial distribution of charges and currents does not satisfy the above criteria, they will redistribute themselves so that a stationary state is reached. This process of “relaxation” happens over a time scale that is characteristic of the medium, called the relaxation time.

Page 5

Relaxation time The continuity equation, combining with ∇· , gives ∇· −∇· (2) Using and ∇· 0 (3) The solution to this differential equation is +( / (4) where is the initial current distribution / is the relaxation time −∇· / , etc. relax at the same rate.

Page 6

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 7

Time-dependent electric ﬁeld No free charges, no external EMF sources. Maxwell ∇× ∇× )= ∇× (5) ∇· fr (6) This gives the second order partial differential equation µ µ 0 (7) Depending on whether the / term dominates or the / one, we’ll get two different extremes of behaviour. The former will lead to a propagating wave , the latter will lead to a diffusion equation , corresponding to a decaying wave

Page 8

Looking for solution of the form The differential equation becomes µ 0 (8) There are two time scales here: 1 / and / µ 0 (9) When τω>> µ (10) which is a wave propagating with speed µ When τω<< (11) which is the equation for diffusion. In the context of EM waves, this will lead to a decaying solution. we shall explore these behaviours in detail now.

Page 9

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 10

Propagating (plane wave) solution for ωτ >> displacement current dominates over conduction current µ ωµ 0 (12) Plane wave: all ﬁelds are functions of the distance of a plane from the origin. is the normal to this plane. ∂/ Maxwell’s equations in this language: (13) (14)

Page 11

Longitudinal components of and : longitudinal component of / equation and dot product of with the / equation 0 (15) For non-conducting media (e.g. vacuum), is a constant. : longitudinal component of / equation and dot product of with the / equation 0 (16) Only stationary longitudinal component of is possible, i.e. is constant (note: we have taken

Page 12

Transverse components of and Combining the two equations: µ )= 0 (17) Differential equation for General solution: ut )+ ut )] If is sinusoidal: (18) Direction of propagation (19) Using / )= / (20)

Page 13

Propagating wave in short and are constants in space and time, hence not interesting for wave propagation and can have dependence, with =( / and ﬁelds are transverse to the direction of motion, and also orthogonal to each other.

Page 14

Coming up... Relaxation to a stationary state Electromagnetic waves Propagating plane wave Decaying plane wave

Page 15

Decaying plane wave When ωτ << 1, conduction current dominates over displacement current µ ωµ 0 (21) The solution of the form kx implies π/ (22) π/ (23) This gives Re Im (24) The wave then decays with a Im dependence inside the conducting medium.

Page 16

Skin depth in metals For metals, 10 14 sec. So for ω< 10 14 , conduction current dominates. A wave incident on a metallic surface will decay as / (25) where, from the last page, (check factor of 2) (26) Within a distance from the surface of the metal, the wave would decrease in magnitude by a factor 1 . This is the “skin depth of the metal. The surface currents will ﬂow within this width. “Ideal” conductor 0.

Page 17

Recap of topics covered in this lecture Relaxation to stationary state, relaxation time Electromagnetic wave: displacement current and conduction current Transverse electromagnetic ﬁeld solutions for a propagating wave Decay of EM waves in a condunctor, skin depth

Today's Top Docs

Related Slides