Spring PHYS4202/ E&M II (Dr. Andrei Galiautdinov, UGA) Part 3: Lectures Special Relativity

Transcription

1 Spring PHYS4202/ E&M II (Dr. Andrei Galiautdinov, UGA) Part 3: Lectures Special Relativity 0

2 Lecture 37 (Wednesday, Apr. 15/2015) A bit of Special Relativity Special Relativity as a Theory of Space and Time Inertial reference frames, properties of space & time, relativity principle 1

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4 Lecture 38 (Friday, Apr. 17/2015) A bit of Special Relativity Derivation of Lorentz transformation (without Einstein s 2 nd Postulate) 3

5 From properties of Space & Time to Lorentz Transformations 1. Galileo s Law of Inertia for freely moving point masses: Linearity 2. Motion of O relative to K & homogeneity of space: Direct transformation is 3. Inversion of x and y axes: α v = α v, k v = k v, δ v = δ v, γ v = γ v 4. K moves relative to K with ( v): Inverse transformation is 5. Inverse transformation applied to y and z : k v = ±1, choose Direct transformation applied to motion of O relative to K : x O t O γ v 7. Represent δ v = vv v 2 γ v : Direct transformation is x = α(v)(x + vv ) y = k v y z = k v z t = δ v x + γ v t = v = v α v γ v x = γ(v)(x vv) y = y z = z t = γ v vv v 2 x + t x = α(v)(x vv) y = k v y z = k v z t = δ v x + γ v t, and thus α v = 8. Group properties 1 & 2: Fix γ v and f, and lead to the Law of Addition of Velocity along x-axis 9. FINAL RESULTS 4

6 From properties of Space & Time to Lorentz Transformations 1. Galileo s Law of Inertia for freely moving point masses: Linearity 2. Motion of O relative to K & homogeneity of space: Direct transformation is 3. Inversion of x and y axes: α v = α v, k v = k v, δ v = δ v, γ v = γ v 4. K moves relative to K with ( v): Inverse transformation is 5. Inverse transformation applied to y and z : k v = ±1, choose Direct transformation applied to motion of O relative to K : x O t O γ v 7. Represent δ v = vv v 2 γ v : Direct transformation is x = α(v)(x + vv ) y = k v y z = k v z t = δ v x + γ v t = v = v α v γ v x = γ(v)(x vv) y = y z = z t = γ v vv v 2 x + t x = α(v)(x vv) y = k v y z = k v z t = δ v x + γ v t, and thus α v = 8. Group properties 1 & 2: Fix γ v and f, and lead to the Law of Addition of Velocity along x-axis 9. FINAL RESULTS 5

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8 k k Notice that v was the velocity of K relative to K, so it was measured by K using unprimed coordinates (x,t). The velocity of K relative to K is measured by K using primed coordinates (x,t ), so there is no guarantee that it should be equal to (-v). Need a proof (use Relativity Principle): Argument 1: Two spring guns. Argument 2: A third reference frame. Given K, construct K & K moving symmetrically w.r.t. K. By RP, extend the existence of such K to any two frames K & K. Then show that K moves relative to K with (-v). 7

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10 Lecture 39 (Monday, Apr. 20/2015) A bit of Special Relativity Derivation of Lorentz transformation (without Einstein s 2 nd Postulate cont.) Limiting speed 9

11 From properties of Space & Time to Lorentz Transformations 1. Galileo s Law of Inertia for freely moving point masses: Linearity 2. Motion of O relative to K & homogeneity of space: Direct transformation is 3. Inversion of x and y axes: α v = α v, k v = k v, δ v = δ v, γ v = γ v 4. K moves relative to K with ( v): Inverse transformation is 5. Inverse transformation applied to y and z : k v = ±1, choose Direct transformation applied to motion of O relative to K : x O t O γ v 7. Represent δ v = vv v 2 γ v : Direct transformation is x = α(v)(x + vv ) y = k v y z = k v z t = δ v x + γ v t = v = v α v γ v x = γ(v)(x vv) y = y z = z t = γ v vv v 2 x + t x = α(v)(x vv) y = k v y z = k v z t = δ v x + γ v t, and thus α v = 8. Group properties 1 & 2: Fix γ v and f, and lead to the Law of Addition of Velocity along x-axis 9. FINAL RESULTS 10

12 We choose the plus sign, k(v)=+1, to preserve the mutual orientation of x & x axes. 11

13 From properties of Space & Time to Lorentz Transformations 1. Galileo s Law of Inertia for freely moving point masses: Linearity 2. Motion of O relative to K & homogeneity of space: Direct transformation is 3. Inversion of x and y axes: α v = α v, k v = k v, δ v = δ v, γ v = γ v 4. K moves relative to K with ( v): Inverse transformation is 5. Inverse transformation applied to y and z : k v = ±1, choose Direct transformation applied to motion of O relative to K : x O t O γ v 7. Represent δ v = vv v 2 γ v : Direct transformation is x = α(v)(x + vv ) y = k v y z = k v z t = δ v x + γ v t = v = v α v γ v x = γ(v)(x vv) y = y z = z t = γ v vv v 2 x + t x = α(v)(x vv) y = k v y z = k v z t = δ v x + γ v t, and thus α v = 8. Group properties 1 & 2: Fix γ v and f, and lead to the Law of Addition of Velocity along x-axis 9. FINAL RESULTS 12

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15 Lecture 40 (Wednesday, Apr. 22/2015) A bit of Special Relativity Limiting speed Velocity addition formula (along the x-axes) Invariance of the limiting speed Speed of light 14

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17 therefore, speed of light is c (the limiting speed!) 16

18 Lecture 41 (Friday, Apr. 24/2015) A bit of Special Relativity Relativity of simultaneity Length contraction Time dilation (at home) Velocity addition formula (if time permits) 17

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20 Lecture 42 (Monday, Apr. 27/2015) A bit of Special Relativity Time dilation Review of Final Exam 19

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23 End of Lectures 22

24 A bit of history ancient Greeks: 1. Amber (by wool) + feather 2. Magnetite (Fe 2 O 3 ) + iron William Gilbert: Electrification is not limited to amber; it s a general phenomenon Charles Dufay (King of France s gardener): Electrically charged objects can also repel each other Benjamin Franklin: 1. + and electricity 2. Likes repel, opposites attract Charles Coulomb: Inverse-square force law for electricity Hans Oersted: Connection b/w electricity and magnetism (compass needle is deflected by current) Michael Faraday: 1. Concept of E & M fields 2. EM Induction (changing magnetic field produces current in a circuit) James Clerk Maxwell: 1. Laws of E&M in modern form 2. Existence of EM waves 3. Light is an EM wave to 1873 Heinrich Hertz: Produced EM waves in the lab Alexander Popov Guglielmo Marconi: Radio Joseph Thomson: Discovery of the electron P. N. Lebedev: Light pressure E. Rutherford: planetary model of atom Niels Bohr: (semi-) quantum model of atom

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26 End of Part 3 25