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 . ∇ · � D = ρ 2. Show the solutions to those differ- ∇ · � B = 0 ential equations are waves. E = − ∂ � B ∇ × � Let There Be Light! ∂t 3. What is the speed of those waves? J + ∂ � D ∇ × � H = � ∂t 4. What constraints are imposed by Maxwell’s equations? 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 � v i to the right. What is the electric potential across the bar using the magnetic force law � v × � F mag = Q� B ? Faraday’s Law states that E = − d Φ 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 � v i to the right. What is the electric potential across the bar using the magnetic force law � v × � F mag = Q� B ? Faraday’s Law states that E = − d Φ 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 ∇ · � Gauss’s Law ǫ 0 ∇ × � B = µ 0 � J Ampere’s Law E = − ∂� B ∇ × � Faraday’s Law ∂t ∇ · � Martha’s Law B = 0 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 ) − ∇ 2 � ∇ × ( ∇ × � A ) = ∇ ( ∇ · � (11) A Light – p.4/21
Fixing Ampere’s Law positive surface for current negative + x z v Area = A d E i d Light – p.5/21
Fixing Ampere’s Law positive volume for current negative + x z v Area = A d E i d Light – p.6/21
Maxwell’s Equations (so far) E = ρ ∇ · � Gauss’s Law ǫ 0 ∇ × � B = µ 0 � J Ampere’s Law E = − ∂� B ∇ × � Faraday’s Law ∂t ∇ · � Martha’s Law B = 0 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 ∇ · � Gauss’s Law ǫ 0 ∂� E ∇ × � B = µ 0 � J + µ 0 ǫ 0 Ampere’s Law ∂t E = − ∂� B ∇ × � Faraday’s Law ∂t ∇ · � Martha’s Law B = 0 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 ) − ∇ 2 � ∇ × ( ∇ × � A ) = ∇ ( ∇ · � (11) A Light – p.10/21
Traveling Waves Offset � 0. 4.0 3.5 3.0 2.5 f � x � 2.0 1.5 1.0 0.5 0.0 � 10 � 5 0 5 10 x Light – p.11/21
Traveling Waves Offset � 5. 4.0 3.5 3.0 2.5 f � x � 2.0 1.5 1.0 0.5 0.0 � 10 � 5 0 5 10 x Light – p.12/21
Traveling Waves Offset � 10. 4.0 3.5 3.0 2.5 f � x � 2.0 1.5 1.0 0.5 0.0 � 10 � 5 0 5 10 x Light – p.13/21
Traveling Waves Offset � 0. 4.0 3.5 � 10.0 � 5.0 0.0 3.0 2.5 f � x � 2.0 1.5 1.0 0.5 0.0 � 10 � 5 0 5 10 x 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) 1 + 2 Nα 3 ǫ 0 ǫ r = = 1 + χ e 1 − Nα 3 ǫ 0 D = ǫ 0 ǫ r � � E = ǫ � E and the magnetic susceptibility 1 = µ χ m = − − 1 1 + 16 πm e R E µ 0 2 µ 0 e 2 H = 1 � � B µ Light – p.16/21
What Happens to Electromagnetic Waves in Linear Media? Recall Clausius-Mossotti (CM) In free space with no charge: 1 + 2 Nα 3 ǫ 0 ǫ r = = 1 + χ e 1 − Nα E = − ∂ � ∇ · � ∇ × � B E = 0 3 ǫ 0 ∂t B = µ 0 ǫ 0 ∂ � ∇ · � ∇ × � E B = 0 ∂t D = ǫ 0 ǫ r � � E = ǫ � E and the magnetic susceptibility 1 = µ χ m = − − 1 1 + 16 πm e R E µ 0 2 µ 0 e 2 H = 1 � � B µ Light – p.16/21
What Happens to Electromagnetic Waves in Linear Media? Recall Clausius-Mossotti (CM) In free space with no charge: 1 + 2 Nα 3 ǫ 0 ǫ r = = 1 + χ e 1 − Nα E = − ∂ � ∇ · � ∇ × � B E = 0 3 ǫ 0 ∂t B = µ 0 ǫ 0 ∂ � ∇ · � ∇ × � E B = 0 ∂t D = ǫ 0 ǫ r � � E = ǫ � E and the magnetic susceptibility In a medium with no free charge: 1 = µ E = − ∂ � ∇ · � ∇ × � B χ m = − − 1 D = 0 1 + 16 πm e R E ∂t µ 0 H = ∂ � 2 µ 0 e 2 ∇ · � ∇ × � D B = 0 ∂t H = 1 � � B µ Light – p.16/21
What Happens to Electromagnetic Waves in Linear Media? Recall Clausius-Mossotti (CM) In free space with no charge: 1 + 2 Nα 3 ǫ 0 ǫ r = = 1 + χ e 1 − Nα E = − ∂ � ∇ · � ∇ × � B E = 0 3 ǫ 0 ∂t B = µ 0 ǫ 0 ∂ � ∇ · � ∇ × � E B = 0 ∂t D = ǫ 0 ǫ r � � E = ǫ � E and the magnetic susceptibility In a medium with no free charge: 1 = µ E = − ∂ � ∇ · � ∇ × � B χ m = − − 1 D = 0 1 + 16 πm e R E ∂t µ 0 H = ∂ � 2 µ 0 e 2 ∇ · � ∇ × � D B = 0 ∂t H = 1 � � B In a linear medium with no free charge: µ E = − ∂ � ∇ · � ∇ × � B E = 0 ∂t B = µǫ ∂ � ∇ · � ∇ × � B B = 0 ∂t Light – p.16/21
What Happens to Electromagnetic Waves in Linear Media? Recall Clausius-Mossotti (CM) In free space with no charge: 1 + 2 Nα 3 ǫ 0 ǫ r = = 1 + χ e 1 − Nα E = − ∂ � ∇ · � ∇ × � B E = 0 3 ǫ 0 ∂t B = µ 0 ǫ 0 ∂ � ∇ · � ∇ × � E B = 0 ∂t D = ǫ 0 ǫ r � � E = ǫ � E and the magnetic susceptibility In a medium with no free charge: 1 = µ E = − ∂ � ∇ · � ∇ × � B χ m = − − 1 D = 0 1 + 16 πm e R E ∂t µ 0 H = ∂ � 2 µ 0 e 2 ∇ · � ∇ × � D B = 0 ∂t H = 1 � � B In a linear medium with no free charge: µ E = − ∂ � ∇ · � ∇ × � B E = 0 For nitrogen gas ( N 2 ): ∂t B = µǫ ∂ � ∇ · � ∇ × � B B = 0 n ( CM ) = √ ǫ r = 1 . 0002939 ∂t 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 × 10 5 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
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