Class 3: Electromagnetism

Transcription

1 Class 3: Electromagnetism In this class we will apply index notation to the familiar field of electromagnetism, and discuss its deep connection with relativity

2 Class 3: Electromagnetism At the end of this session you should be able to describe how electromagnetism is fundamentally a relativistic phenomenon understand how electromagnetism can be formulated in index notation using the Maxwell Field Tensor use the Lorentz transformations to transform electric and magnetic fields between frames describe electromagnetic energy density and flow in terms of index notation

3 Synthesis of ideas Physics is about the synthesis of ideas: understanding apparently different phenomena as the joint consequences of a deeper reality For example, in 1666 Newton realized that the same gravitational force causes both apples to fall downwards on the Earth, and the Earth to orbit the Sun

4 Synthesis of ideas Electromagnetism represents another great synthesis of ideas. Prior to 1830, electricity and magnetism were considered separate phenomena Electricity Magnetism

5 Synthesis of ideas However, Faraday s experiments demonstrated that a magnet, as well as a battery, can drive a current Electricity and magnetism are connected by deeper principles!

6 Electric field Electric charge generates an electric field E in the surrounding region /wiki/electric_field An electric field is a region of space in which an electric charge feels a force, F = qe

7 Magnetic field Electric current (or moving charge) generates a magnetic field B in the surrounding region A magnetic field is a region of space in which an electric current feels a force, F = I dl B

8 Maxwell s Equations Maxwell s Equations describe how E and B-fields are generated from charge and current

9 Electromagnetic potentials Potentials: E and B-fields can be generated from an electrostatic potential V and magnetic potential A Electrostatics: E = 0 E = V Magnetism:. B = 0 B = A Time-varying: E + 9: 9; = 0 E = V A t

10 Electromagnetic energy Energy: it takes work to establish an E or B-field We may think of this work as being stored in the fields with energy density >? ε AE? or >?B C B? The flux of energy is given by > B C D : E (the Poynting vector )

11 4-vectors for electromagnetism Since E is produced by charges, and B by moving charges, it s natural that E and B are linked by viewing in different reference frames i.e., by a Lorentz transformation When a line of charge moves, it becomes a current! S S λ = charge density x x v

12 4-vectors for electromagnetism Since E is produced by charges, and B by moving charges, it s natural that E and B are linked by viewing in different reference frames i.e., by a Lorentz transformation However, to develop electromagnetism in relativistic notation, we need some 4-vectors! In the last class we saw that the sources of the fields form a 4-vector, J B = (ρc, J P, J Q, J R ) with continuity B J B = 0 In this class we will demonstrate that electromagnetism may be described very nicely if the potentials also form a 4-vector, A B = ( U E, A P, A Q, A R )

13 Maxwell field tensor We ll form a wonderful object known as the Maxwell Field Tensor ( tensor because it has 2 indices, μ and ν) F BX = B A X X A B A B = V c, A P, A Q, A R B = 1 c t, x, y, z μ = 0 μ = 1 μ = 2 μ = 3 If in doubt, write it out ν = 0 ν = 1 ν = 2 ν = 3 A A A A A A A A > > A A A A?? A A A A Y Y A A > A A A A > > A > > A > > A?? A > > A Y Y A >? A A A A?? A > > A?? A?? A?? A Y Y A? Y A A A A Y Y A > > A Y Y A?? A Y Y A Y Y A Y

14 Maxwell field tensor We ll form a wonderful object known as the Maxwell Field Tensor ( tensor because it has 2 indices, μ and ν) F BX = B A X X A B A B = V c, A P, A Q, A R Multiplying this out, B = 1 c t, x, y, z F BX = 0 E P /c E Q /c E R /c E P /c 0 B R B Q E Q /c B R 0 B P E R /c B Q B P 0

15 Maxwell s equations Maxwell s equations. E = ρ ε A E + B t = 0. B = 0 B ε A μ A E t = μ AJ can be nicely written as B F BX = μ A J X a F BX + B F Xa + X F ab = 0

16 Transformation of EM fields We can use the Maxwell Field Tensor to deduce how electromagnetic fields transform between frames: F BX = L B c L X a F ca F BX = Maxwell field tensor 0 E P /c E Q /c E R /c E P /c 0 B R B Q E Q /c B R 0 B P E R /c B Q B P 0 L B X = Lorentz transformation γ vγ/c vγ/c γ

17 Transformation of EM fields Multiplying out each component of this equation: E P = E P E Q = γ E Q vb R E R = γ E R + vb Q B P = B P B Q = γ B Q + ve R /c? B R = γ B R ve Q /c? E and B intermingle under a Lorentz transformation!

18 Another path to relativity? Maxwell s Equations keep the same mathematical form under a Lorentz transformation but not under a Galilean transformation This was already known by Lorentz in the 1890s, before Einstein published Special Relativity In the modern story Einstein s postulates are presented first, but it can all be derived from electromagnetism! H.Lorentz ( )

19 Magnetic field of a current When a line of charge moves, it becomes a current S S λ = charge density x x v First consider the field of the stationary charge in S : E Gauss s Law: E Q e = λ 2πε A y B e = 0

20 Magnetic field of a current What is the magnetic field in S? Lorentz transformation: B R = ghd i j E k = ghaj?lm C Q j E k Length contraction implies that the line density of charge in S is γλ, and the current in S is then I = γλ e v Hence, using y e = y and c? = >, we find B m C B R = B Cn C?lQ Ampere s Law! y z x

21 Transformation of EM fields What are the implications of this formulation of electromagnetism? Magnetism is not an independent phenomenon, but a relativistic effect that occurs when electrostatics are observed in a different frame If the speed of light were infinite, there would be no magnetism!

22 Field of a moving charge What is the electric field of a charge +q moving through S? S S +q x v x First consider the field of a stationary charge in S e : E e = q r 4πε A r? B = 0

23 Field of a moving charge Lorentz transformations: E = E P, γe Q, γe R Evaluate the field at t = 0: r e = γx, y, z After some algebra: r? = γ? r? 1 v sin θ /c? E +q θ P v E = q r 4πε A γ? r Y 1 v sin θ /c? Y/?

24 EM energy-momentum tensor The energy-momentum tensor for electromagnetism is: T BX = 1 μ A F B a F Xa 1 4 ηbx F ca F ca F BX = 0 E P /c E Q /c E R /c E P /c 0 B R B Q E Q /c B R 0 B P E R /c B Q B P 0 η BX = How do we de-code this expression?? F B a = η ax F BX = η ax F BX X F ca = η cb η ax F BX F ca F ca = c a F ca F ca

25 EM energy-momentum tensor The energy density is given by the first component, T AA : T AA = 1 μ A F A a F Aa 1 4 ηaa F ca F ca Evaluating this expression, T AA = >? ε AE? + >?B C B? The flux of energy in the i-direction is given by T A~ : T A~ = 1 μ A F A a F ~a 1 4 ηa~ F ca F ca = 0 Evaluating this expression, T A~ = > B C D : E

26 Force on a moving charge The Lorentz force law can be expressed in index notation as dp B dτ = q FB X u X Equivalent to F = q E + v B

27 Gauges In electrostatics, the zero of potential is arbitrary In general, if A B A B + B f, where f is any function of x, t, the electromagnetic fields E and B are unchanged! (We can show this by substituting in F BX = B A X X A B ) This freedom is known as a gauge choice it s convenient to choose f such that B A B = 0 That s because we then find, B F BX = B B A X Combining with Maxwell s Equation B F BX = μ A J X : 1 c?? t? +? x? +? y? +? z? AB = μ A J B