Geometry of Light
Refath Bari
Purpose
- Lorentz Transformation from Maxwell's Equations
- Time Dilation from Point Charge and Infinite Sheet
- Displacement Current from Biot-Savart Law
Illustrate the relationship between Electromagnetism & Special Relativity
Properties of Light
Spoiler!
Light is lazy
Property I
Light travels straight
Property II
Reflects off equal angles
Goal: Get from A to B if you have to hit the metal
A
B
A
B
D
A
B
D
A
B
D
E
AE+BE
Minimize
A
B
D
E
C
B'
A
B
D
E
C
B'
A
B
D
E
C
B'
F
A
B
D
E
C
B'
F
BF = BF'
A
B
D
E
C
B'
F
\angle BFC = 90
BF = BF'
A
B
D
E
C
B'
F
BF = BF'
\angle BFC = 90
A
B
D
E
C
B'
F
BF = BF'
\angle BFC = 90
AE+BE
Minimize
A
B
D
E
C
B'
F
BF = BF'
\angle BFC = 90
AE+BE
Minimize
AE+B'E
A
B
E
B'
F
BF = BF'
\angle BFC = 90
AE+BE
Minimize
AE+B'E
A
B
E
A
B
E
A
B
E
{\theta}_{i}
{\theta}_{r}
A
B
E
{\theta}_{i}
{\theta}_{r}
{\theta}_{i} = {\theta}_{r}
{\theta}_{i} = {\theta}_{r}
Property III
Light refracts in different mediums
{\theta}_{i}
{\theta}_{r}
{\theta}_{i}
{\theta}_{r}
{n}_{1}
{n}_{2}
{\theta}_{i}
{\theta}_{r}
{n}_{1}
{n}_{2}
Properties
- Light travels straight
- Light reflects in equal angles
- Light refracts in different medius
Properties
+
+
Properties
+
+
Properties
+
+
Properties
+
+
Brachistochrone Problem
Brachistochrone Problem
4th Property?
\nabla \cdot E=\frac { \rho }{ \varepsilon _{ 0 } }
\nabla \cdot B= 0
\nabla \times E=-\frac { \partial B }{ \partial t }
\nabla \times B=\mu _{ 0 }j+\frac { 1 }{ { c }^{ 2 } } \frac { \partial E }{ \partial t }
v
c
v+c
c
Error: Galilean Relativity
Correction: Special Relativity
Lorentz Transformation
from Maxwell's Equations
Time Dilation
from Point Charge and Infinite Sheet
v
v
Q
v
v
Q
v
v
{ v }_{ y }^{ ' }(t)=\frac { QEt' }{ m }
{ v }_{ y }(t)=\frac { QEt' }{ m } \left( 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } \right)
Displacement Current
from Biot-Savart Law and Special Relativity
\frac { \partial E }{ \partial t } =\frac { \partial E }{ \partial x } \cdot \frac { \partial x }{ \partial t }
E_{ \parallel }=E'_{ \parallel }\\ B_{ \parallel }=0\\ E_{ \perp }=\gamma E'_{ \perp }\\ B=+\gamma \frac { v }{ c^{ 2 } } \times E'
\frac { \partial B }{ \partial t } =\gamma { \mu }_{ 0 }v\times ({ \varepsilon }_{ 0 }\frac { \partial E }{ \partial t } )
Summary
- Lorentz Transformations from Maxwell's Equations
- Time Dilation from Electromagnetic Scenario
- Displacement Current from Biot-Savart Law
Thank You ...
Dr. Kabat,
My teachers,
Mom, Dad, Little Brother
Storage
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