Lecture 2 Maxwell's Equations in Free Space Co-Ordinate Systems And

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Lecture 2

Maxwell’s Equations in Free Space

In this lecture you will learn:

• Co-ordinate Systems and Course Notations

• Maxwell’s Equations in Differential and Integral Forms

• Electrostatics and Magnetostatics

• Electroquasistatics and Magnetoquasistatics

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Co-ordinate Systems and Vectors

Cartesian Coordinate System z

(xo, yo ,zo )

zo y xo

yo x

Vectors in Cartesian Coordinate System r A = Ax xˆ + Ay yˆ + Az zˆ All mean exactly the same thing … r ˆ ˆ ˆ just a different notation for the unit A = Ax i + Ay j + Az k vectors r ˆ ˆ ˆ A = Ax ix + Ay iy + Az iz The first one will be used in this course

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

1 Co-ordinate Systems and Vectors

Cylindrical Coordinate System z

(ro,φo,zo )

y φo zo ro x

Vectors in Cylindrical Coordinate System r A = A rˆ + A φˆ + A zˆ Both mean exactly the same thing … r φ z just a different notation for the unit vectors r ˆ ˆ ˆ A = Ar ir + Aφ iφ + Az iz The first one will be used in this course

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Co-ordinate Systems and Vectors

Spherical Coordinate System z

(r ,φ ,θ ) r o o o θo o y φo

Vectors in Spherical Coordinate System

r ˆ ˆ Both mean exactly the same thing … A = Ar rˆ + Aφ φ + Aθ θ just a different notation for the unit vectors r ˆ ˆ ˆ A = Ar ir + Aφ iφ + Aθ iθ The first one will be used in this course

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

2 Vector Fields

In layman terms, a vector field implies a vector associated with every point is space:

Examples: r r Electric Field: E()x, y,z,t or E(rr,t )

r r Magnetic Field: H()x,y,z,t or H(rr,t )

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Maxwell’s Equations in Free Space

Integral Form Differential Form

r r r (1) ∫∫ εo E . da = ∫∫∫ ρ dV Gauss’ Law ∇. εo E = ρ

r r r (2) ∫∫ µo H . da = 0 Gauss’ Law ∇. µo H = 0

∂ r r r r r r ∂ µoH (3) ∫ E . ds = − ∫∫ µoH. da Faraday’s Law ∇ × E = − ∂t ∂t

r r r r r ∂ r r r r ∂ ε E (4) ∫ H . ds = ∫∫ J .da + ∫∫ εoE . da Ampere’s Law ∇ × H = J + o ∂t ∂t

Lorentz Force Law

r r r r Lorentz Law describes the effect of F = q ( E + v × µoH ) electromagnetic fields upon charges

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

3 Physical Quantities, Values, and SI Units

Quantity Value/Units r E Electric Field Volts/m r H Magnetic Field Amps/m

εo Permittivity of Free Space 8.85x10-12 Farads/m

µo Permeability of Free Space 4πx10-7 Henry/m

q Electronic Unit of Charge 1.6x10-19 Coulombs

ρ Volume Charge Density Coulombs/m3 r J Current Density Amps/m2

r r 2 D = εoE Electric Flux Density Coulombs/m r r B = µoH Magnetic Flux Density Tesla

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Gauss’ Law – Integral Form r r r r ∫∫ εo E . da = ∫∫∫ ρ dV or ∫∫ D . da = ∫∫∫ ρ dV

What is this law saying ……?? A closed surface of arbitrary shape surrounding a ρ(x,y,z) charge distribution

Carl F. Gauss D (1777-1855)

Gauss’ Law: The total electric flux coming out of a closed surface is equal to the total charge enclosed by that closed surface (irrespective of the shape of the closed surface) Points to Note: This law establishes charges as the “sources” or “sinks” of the electric field (i.e. charges produce or terminate electric field lines).

If the total flux through a closed surface is positive, then the total charge enclosed is positive. If the total flux is negative, then the total charge enclosed is negative

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

4 Gauss’ Law – Differential Form

Divergence Theorem: r r r For any vector field: ∫∫ A . da = ∫∫∫ ∇ . A dV The flux of a vector through a closed surface is equal to the integral of the divergence of the vector taken over the volume enclosed by that closed surface

Using the Divergence Theorem with Gauss’ Law in Integral Form:

r r ∫∫ D . da = ∫∫∫ ρ dV

r D ⇒ ∫∫∫ ∇ .D dV = ∫∫∫ ρ dV

r r ρ(x,y,z) ⇒ ∇ .D = ρ or ∇ .εo E = ρ

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Gauss’ Law for the Magnetic Field – Integral Form r r r r ∫∫ µo H . da = 0 or ∫∫ B . da = 0

What is this law saying ……?? A closed surface of arbitrary shape

Gauss’ Law for the Magnetic Fields: The total magnetic flux coming out of a closed surface is always zero.

Points to Note: This law implies that there are no such things as “magnetic charges” that can emanate or terminate magnetic field lines.

If magnetic field is non-zero, then the flux into any closed surface must equal the flux out of it - so that the net flux coming out is zero.

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

5 Gauss’ Law – Differential Form

Using the Divergence Theorem with Gauss’ Law for the Magnetic Field in Integral form: r r r r ∫∫ B . da = 0 (Remember B = µo H ) B r ⇒ ∫∫∫ ∇ .B dV = 0

r r ⇒ ∇ .B = 0 or ∇ .µo H = 0

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Faraday’s Law – Integral Form

r r ∂ r r r r ∂ r r ∫ E . ds = − ∫∫ µ H . da or ∫ E . ds = − ∫∫ B . da ∂t o ∂t What is this law saying ……??

Michael Faraday A closed contour (1791-1867)

Faraday’s Law: The line integral of electric field over a closed contour is equal to –ve of the time rate of change of the total magnetic flux that goes through any arbitrary surface that is bounded by the closed contour

Points to Note: This law says that time changing magnetic fields can also generate electric fields

The positive directions for the surface normal vector and of the contour are related by the right hand rule

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

6 Faraday’s Law – Differential Form

Stokes Theorem: r r r For any vector field: ∫ A.ds = ∫∫ (∇ × A). da The line integral of a vector over a closed contour is equal to the surface integral of the curl of that vector over any arbitrary surface that is bounded by the closed contour

Using the Stokes Theorem with Farady’s Law in Integral Form:

r r ∂ r r ∫ E . ds = − ∫∫ B . da ∂t r r ∂ r r ⇒ ∫∫ ()∇ × E .da = − ∫∫ B . da ∂t r r r ∂ B r ∂ µ H ⇒ ∇ × E = − or ∇ × E = − o ∂t ∂t

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Ampere’s Law – Integral Form

r r r r ∂ r r r r r r ∂ r r ∫ H . ds = ∫∫ J .da + ∫∫ ε E . da or ∫ H . ds = ∫∫ J .da + ∫∫ D.da ∂t o ∂t What is this law saying ……?? electric flux density

J D electric current density

A. M. Ampere (1775-1836) Ampere’s Law: The line integral of magnetic field over a closed contour is equal to the total current plus the time rate of change of the total electric flux that goes through any arbitrary surface that is bounded by the closed contour

Points to Note: This law says that electrical currents and time changing electric fields can generate magnetic fields. Since there are no magnetic charges, this is the only known way to generate magnetic fields

The positive directions for the surface normal vector and of the contour are related by the right hand rule

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

7 Ampere’s Law – Differential Form

Using the Stokes Theorem with Ampere’s Law in Integral Form:

r r r r ∂ r r ∫ H . ds = ∫∫ J .da + ∫∫ D.da ∂t r r r r ∂ r r ⇒ ∫∫ ()∇ × H .da = ∫∫ J .da + ∫∫ D.da ∂t r r r r ∂ D r r ∂ ε E ⇒ ∇ × H = J + or ∇ × H = J + o ∂t ∂t

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Maxwell’s Equations and Light – Coupling of E and H Fields r r ∂ µ H r ∇ × E = − o ∇. εo E = ρ ∂t

r r ∇. µo H = 0 r r ∂ ε E ∇ × H = J + o ∂t Time varying electric and magnetic fields are coupled - this coupling is responsible for the propagation of electromagnetic waves

Electromagnetic Wave Equation in Free Space: r Assume: ρ = J = 0 and take the curl of the Faraday’s Law on both sides: r r r r ⎛ ∂ µ H ⎞ ∂ µ ∇ × H ∂2 E ⎜ o ⎟ o ∇ × ()∇ × E = −∇ × ⎜ ⎟ = − = −µo εo 2 ⎝ ∂t ⎠ ∂t ∂t 2 r 1 r ∂ E c = ≈ 3 ×108 m/s ⇒ ∇ × ()∇ × E = −µo εo 2 ∂t εo µo

2 r r 1 ∂ E Equation for a wave traveling ⇒ ∇ × ()∇ × E = − c2 ∂t 2 at the speed c

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

8 Maxwell’s Equations and Light

r r 1 ∂2 E Equation for a wave traveling ∇ × ()∇ × E = − 2 2 c ∂t at the speed c: 1 c = ≈ 3 ×108 m/s εo µo

In 1865 Maxwell wrote:

“This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws”

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Electrostatics and Magnetostatics

Suppose we restrict ourselves to time-independent situations (i.e. nothing is varying with time – everything is stationary)

We get two sets of equations for electric and magnetic fields that are completely independent and uncoupled:

Equations of Electrostatics Equations of Magnetostatics

r ∇. ε E()rr = ρ ()rr r r o ∇. µo H(r ) = 0 r r r r ∇ × E()r = 0 ∇ × H(rr) = J

• Electric fields are produced by • Magnetic fields are produced by only electric charges only electric currents

• In electrostatics problems one • In magnetostatics problems needs to determine electric field one needs to determine magnetic given some charge distribution field given some current distribution

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

9 Electroquasistatics and Magnetoquasistatics - I • The restriction to completely time-independent situations is too limiting and often unnecessary

• What if things are changing in time but “slowly” ……..(how slowly is “slowly” ?)

• Allowing for slow time variations, one often uses the equations of electroquasistatics and magnetoquasistatics Equations of Electroquasistatics Equations of Magnetoquasistatics r r r r r ∇. εo E(r ,t )()= ρ r ,t ∇. µo H(r ,t ) = 0 r r r ∇ × E()rr,t = 0 ∇ × H(rr,t ) = J(rr,t ) r r r r ∂ εoE r ∂ µ H ∇ × H = J + ∇ × E = − o ∂t ∂t • Electric fields are produced by • Magnetic fields are produced by only electric charges only electric currents

• Once the electric field is • Once the magnetic field is determined, the magnetic field can determined, the electric field can be be found by the last equation found by the last equation

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

Electroquasistatics and Magnetoquasistatics - II

Question from the last slide: How slowly is “slowly” ?

• Electromagnetic waves and signals move at the speed c (speed of light)

Answer: Time variations are considered slow if the time scales over which things are changing are much longer compared to the time taken by light to cover distances equal to the length scales of the problem

Example: 100 MHz Operation • Time scale of the problem = 1/(100 MHz) = 10 ns 3 cm • Length scale of the problem = 3 cm • Time taken by light to travel 3 cm = 0.1 ns

Since 10 ns >> 0.1 ns, quasistatics is a valid means of analysis at 100 MHz

10 GHz Operation • Time scale of the problem = 1/(10 GHz) = 0.1 ns • Length scale of the problem = 3 cm • Time taken by light to travel 3 cm = 0.1 ns An amplifier chip operating from 100 MHz to 10 GHz Quasistatics is not a valid means of analysis at 10 GHz

ECE 303 – Fall 2007 – Farhan Rana – Cornell University

10 Electroquasistatics and Magnetoquasistatics - III

Question (contd..): How slowly is “slowly” ?

Electromagnetic wave frequency f and wavelength λ are related to the speed of the wave c by the relation: f λ = c

Let: L = length scale of the problem

T = time scale of the problem ≈ 1/f

Condition for quasitatic analysis to be valid: L T >> c ⇒ cT >> L c ⇒ >> L f ⇒ λ >> L Quasistatic analysis is valid if the wavelength of electromagnetic wave at the frequency of interest is much longer than the length scales involved in the problem

ECE 303 – Fall 2007 – Farhan Rana – Cornell University