Let There Be Light! We now finish our introduction to elec- - - PowerPoint PPT Presentation

Let There Be Light!

We now finish our introduction to elec- tromagnetism by completing the list of Maxwell’s equations and deriving the properties of light.

• 1. Show that Maxwell’s Equations in

a charge-free vacuum give rise to separate differential equations for

• E and

B.

• 2. Show the solutions to those differ-

ential equations are waves.

• 3. What is the speed of those waves?
• 4. What constraints are imposed by

Maxwell’s equations?

∇ · D = ρ ∇ · B = 0 ∇ × E = − ∂ B ∂t ∇ × H = J + ∂ D ∂t

Light – p.1/21

Faraday’s Law

A conducting bar moves on two frictionless, paral- lel rails in a uniform magnetic field directed into the

• plane. The bar has length l and initial velocity

vi to the right. What is the electric potential across the bar using the magnetic force law Fmag = Q v × B? Faraday’s Law states that E = − dΦ dt where E is the electromotive force or voltage and Φ is the magnetic flux. Does Faraday’s Law agree with the result for the potential?

Light – p.2/21

Faraday’s Law

A conducting bar moves on two frictionless, paral- lel rails in a uniform magnetic field directed into the

• plane. The bar has length l and initial velocity

vi to the right. What is the electric potential across the bar using the magnetic force law Fmag = Q v × B? Faraday’s Law states that E = − dΦ dt where E is the electromotive force or voltage and Φ is the magnetic flux. Does Faraday’s Law agree with the result for the potential?

What happens if you change the B field?

Light – p.2/21

Maxwell’s Equations (so far)

∇ · E = ρ ǫ0 ∇ × B = µ0 J ∇ × E = −∂ B ∂t ∇ · B = 0

Gauss’s Law Ampere’s Law Faraday’s Law Martha’s Law

Light – p.3/21

Vector Identities from Griffith’s Inside Cover

• A · (

B × C) = B · ( C × A) = C · ( A × B) (1)

• A × (

B × C) = B( A · C) − C( A · B) (2) ∇(fg) = f∇g + g∇f (3) ∇( A · B) = A × (∇ × B) + B × (∇ × A) + ( A · ∇) B + ( B · ∇) A (4) ∇ · (f A) = f(∇ · A) + ( A · (∇f) (5) ∇ · ( A × B) = B · (∇ × A) − A · (∇ × B) (6) ∇ × (f A) = f(∇ × A) − A × (∇f) (7) ∇ × ( A × B) = ( B · ∇) A − ( A · ∇) B + A(∇ · B) − B(∇ · A) (8) ∇ · (∇ × A) = 0 (9) ∇ × (∇f) = 0 (10) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A (11)

Light – p.4/21

Fixing Ampere’s Law

v

d

E positive negative

+

Area = A i d

surface for current z x

Light – p.5/21

Fixing Ampere’s Law

v

d

E positive negative

+

Area = A i d

volume for current z x

Light – p.6/21

Maxwell’s Equations (so far)

∇ · E = ρ ǫ0 ∇ × B = µ0 J ∇ × E = −∂ B ∂t ∇ · B = 0

Gauss’s Law Ampere’s Law Faraday’s Law Martha’s Law

Light – p.7/21

Evidence for Faraday’s Law

Lenz’s Law. Measurement by R. C. Nicklin, Am. J. Phys. 54, 422 (1986).

Pulse from 2−cm−long magnet falling through a 0.79−cm−radius coil. Points calculated with a dipole model.

Light – p.8/21

Maxwell’s Equations

∇ · E = ρ ǫ0 ∇ × B = µ0 J + µ0ǫ0 ∂ E ∂t ∇ × E = −∂ B ∂t ∇ · B = 0

Gauss’s Law Ampere’s Law Faraday’s Law Martha’s Law

Light – p.9/21

Vector Identities from Griffith’s Inside Cover

• A · (

B × C) = B · ( C × A) = C · ( A × B) (1)

• A × (

B × C) = B( A · C) − C( A · B) (2) ∇(fg) = f∇g + g∇f (3) ∇( A · B) = A × (∇ × B) + B × (∇ × A) + ( A · ∇) B + ( B · ∇) A (4) ∇ · (f A) = f(∇ · A) + ( A · (∇f) (5) ∇ · ( A × B) = B · (∇ × A) − A · (∇ × B) (6) ∇ × (f A) = f(∇ × A) − A × (∇f) (7) ∇ × ( A × B) = ( B · ∇) A − ( A · ∇) B + A(∇ · B) − B(∇ · A) (8) ∇ · (∇ × A) = 0 (9) ∇ × (∇f) = 0 (10) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A (11)

Light – p.10/21

Traveling Waves

10 5 5 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x fx Offset 0.

Light – p.11/21

Traveling Waves

10 5 5 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x fx Offset 5.

Light – p.12/21

Traveling Waves

10 5 5 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x fx Offset 10.

Light – p.13/21

Traveling Waves

0.0 5.0 10.0 10 5 5 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x fx Offset 0.

Light – p.14/21

Electromagnetic Waves

Light – p.15/21

Electromagnetic Waves

Light – p.15/21

What Happens to Electromagnetic Waves in Linear Media?

Light – p.16/21

What Happens to Electromagnetic Waves in Linear Media?

Recall Clausius-Mossotti (CM) ǫr = 1 + 2Nα

3ǫ0

1 − Nα

3ǫ0

= 1 + χe

• D = ǫ0ǫr

E = ǫ E and the magnetic susceptibility χm = − 1 1 + 16πmeRE

2µ0e2

= µ µ0 − 1

• H = 1

µ

• B

Light – p.16/21

What Happens to Electromagnetic Waves in Linear Media?

Recall Clausius-Mossotti (CM) ǫr = 1 + 2Nα

3ǫ0

1 − Nα

3ǫ0

= 1 + χe

• D = ǫ0ǫr

E = ǫ E and the magnetic susceptibility χm = − 1 1 + 16πmeRE

2µ0e2

= µ µ0 − 1

• H = 1

µ

• B

In free space with no charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µ0ǫ0 ∂

E ∂t

Light – p.16/21

What Happens to Electromagnetic Waves in Linear Media?

Recall Clausius-Mossotti (CM) ǫr = 1 + 2Nα

3ǫ0

1 − Nα

3ǫ0

= 1 + χe

• D = ǫ0ǫr

E = ǫ E and the magnetic susceptibility χm = − 1 1 + 16πmeRE

2µ0e2

= µ µ0 − 1

• H = 1

µ

• B

In free space with no charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µ0ǫ0 ∂

E ∂t

In a medium with no free charge: ∇ · D = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × H = ∂

D ∂t

Light – p.16/21

What Happens to Electromagnetic Waves in Linear Media?

Recall Clausius-Mossotti (CM) ǫr = 1 + 2Nα

3ǫ0

1 − Nα

3ǫ0

= 1 + χe

• D = ǫ0ǫr

E = ǫ E and the magnetic susceptibility χm = − 1 1 + 16πmeRE

2µ0e2

= µ µ0 − 1

• H = 1

µ

• B

In free space with no charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µ0ǫ0 ∂

E ∂t

In a medium with no free charge: ∇ · D = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × H = ∂

D ∂t

In a linear medium with no free charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µǫ ∂

B ∂t

Light – p.16/21

What Happens to Electromagnetic Waves in Linear Media?

Recall Clausius-Mossotti (CM) ǫr = 1 + 2Nα

3ǫ0

1 − Nα

3ǫ0

= 1 + χe

• D = ǫ0ǫr

E = ǫ E and the magnetic susceptibility χm = − 1 1 + 16πmeRE

2µ0e2

= µ µ0 − 1

• H = 1

µ

• B

In free space with no charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µ0ǫ0 ∂

E ∂t

In a medium with no free charge: ∇ · D = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × H = ∂

D ∂t

In a linear medium with no free charge: ∇ · E = 0 ∇ × E = − ∂

B ∂t

∇ · B = 0 ∇ × B = µǫ ∂

B ∂t

For nitrogen gas (N2): n(CM) = √ǫr = 1.0002939 n(measured) = 1.0002982 → 1.0003012

Light – p.16/21

The Mote in God’s Eye

Described by Robert Heinlein as the finest Science Fiction book ever, The Mote in God’s Eye by Larry Niven and Jerry Pournelle is a story about our First Con- tact with an alien civilization. The Moties use a laser cannon to shine light on a solar sail and push it across a distance of 35 light-years.

Light – p.17/21

The Mote in God’s Eye

Described by Robert Heinlein as the finest Science Fiction book ever, The Mote in God’s Eye by Larry Niven and Jerry Pournelle is a story about our First Con- tact with an alien civilization. The Moties use a laser cannon to shine light on a solar sail and push it across a distance of 35 light-years. 1. What is the expression for the energy per time per area transported by EM waves in vacuum? 2. What is the expression for the pressure of the electromagnetic waves? 3. Assume the sail is 3000 km in diameter, the total mass is m = 4.5 × 105 kg, and the craft accelerates uniformly for 75 years to reach the halfway point. What is the mini- mum laser power required? 4. What is the minimum pressure of the light?

Light – p.17/21

Vector Identities from Griffith’s Inside Cover

• A · (

B × C) = B · ( C × A) = C · ( A × B) (1)

• A × (

B × C) = B( A · C) − C( A · B) (2) ∇(fg) = f∇g + g∇f (3) ∇( A · B) = A × (∇ × B) + B × (∇ × A) + ( A · ∇) B + ( B · ∇) A (4) ∇ · (f A) = f(∇ · A) + ( A · (∇f) (5) ∇ · ( A × B) = B · (∇ × A) − A · (∇ × B) (6) ∇ × (f A) = f(∇ × A) − A × (∇f) (7) ∇ × ( A × B) = ( B · ∇) A − ( A · ∇) B + A(∇ · B) − B(∇ · A) (8) ∇ · (∇ × A) = 0 (9) ∇ × (∇f) = 0 (10) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A (11)

Light – p.18/21

Maxwell’s Equations

∇ · E = ρ ǫ0 ∇ × B = µ0 J + µ0ǫ0 ∂ E ∂t ∇ × E = −∂ B ∂t ∇ · B = 0

Gauss’s Law Ampere’s Law Faraday’s Law Martha’s Law

Light – p.19/21

Light Pressure

n z x y dz a b E B J

Light – p.20/21

Time Averaging

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 xΩt cos2kxΩ t

Light – p.21/21