Primer in Special Relativity and Electromagnetic Equations (Lecture 13)

Transcription

1 Primer in Special Relativity and Electromagnetic Equations (Lecture 13) January 29, /441

2 Lecture outline We will review the relativistic transformation for time-space coordinates, frequency, and electromagnetic field. 213/441

3 Maxwell s equations Classical electrodynamics in vacuum is governed by the Maxwell equations. In the SI system of units, the equations are D = ρ B = 0 E = B t H = j + D t where ρ is the charge density, j is the current density, with (13.1) D = ɛ 0 E, H = B/µ 0. B is called the magnetic induction, and H is called the magnetic field. The equations are linear: the sum of two solutions, E 1, B 1 and E 2, B 2, is also a solution corresponding to the sum of densities ρ 1 + ρ 2, j 1 + j /441

4 Point charge For a point charge moving along a trajectory r = r 0 (t), ρ(r, t) = qδ(r r 0 (t)), j(r, t) = qv(t)δ(r r 0 (t)), (13.2) with v(t) = dr 0 (t)/dt. Proper boundary conditions should be specified in each particular case. On a surface of a good conducting metal the boundary condition requires that the tangential component of the electric field is equal to zero, E t S = /441

5 SI units We will use the SI system of units (sometimes energy in ev). To convert an equation written in SI variables to the corresponding equation in Gaussian variables, replace according to the following table (from Jackson s book): Quantity SI Gaussian Velocity of light (µ 0 ɛ 0 ) 1/2 c Electric field, potential E, φ E 4πɛ0, φ 4πɛ0 Charge density, current q, ρ, j q 4πɛ 0, ρ 4πɛ 0, j 4πɛ 0 Magnetic induction B B µ0 4π We introduce the vacuum impedance Z 0, µ0 Z 0 = 377 Ohm. (13.3) ɛ 0 In CGS units Z 0 = 4π/c. 216/441

6 Wave equations In free space with no charges and currents field components satisfy the wave equation 1 2 f c 2 t 2 2 f x 2 2 f y 2 2 f z 2 = 0. (13.4) A particular solution of this equation is a sinusoidal wave characterized by frequency ω and wave number k and propagating in the direction of unit vector n: where A is a constant and ω = ck. f = A sin(ωt kn r), (13.5) 217/441

7 Vector and scalar potentials It is often convenient to express the fields in terms of the vector potential A and the scalar potential φ: E = φ A t B = A (13.6) Substituting these equations into Maxwell s equations, we find that the second and the third equations are satisfied identically. We only need to take care of the first and the fourth equations. 218/441

8 Energy balance and the Poynting theorem The electromagnetic field has an energy and momentum associated with it. The energy density of the field (energy per unit volume) is The Poynting vector u = 1 2 (E D + H B) = ɛ 0 2 (E 2 + c 2 B 2 ). (13.7) S = E H (13.8) gives the energy flow (energy per unit area per unit time) in the electromagnetic field. 219/441

9 Energy balance and the Poynting theorem A n V Consider charges that move inside a volume V enclosed by a surface A. The Poynting theorem states t V udv = V j EdV n SdA, A (13.9) where n is the unit vector normal to the surface and directed outward. The LHS of this equation is the rate of change of the electromagnetic energy. The first term on the right hand side is the work done by the electric field. The second term describes the electromagnetic energy flow from the volume through the enclosing surface. 220/441

10 Photons The quantum view on the radiation is that the electromagnetic field is represented by photons. Each photon carries the energy hω and the momentum hk, where the vector k is the wave number which points to the direction of propagation of the radiation, h = J sec is the Planck constant divided by 2π, and k = ω/c. 221/441

11 Lorentz transformation and matrices Consider two coordinate systems, K and K. The system K is moving with velocity v in the z direction relative to the system K. The coordinates of an event in both systems are related by the Lorentz transformation x = x, y = y, z = γ(z + βct ), t = γ(t + βz /c), (13.10) where β = v/c, and γ = 1/ 1 β 2. The vector (ct, r) = (ct, x, y, z) is called a 4-vector, and the above transformation is valid for any 4-vector quantity. 222/441

12 Lorentz transformation and matrices We will often deal with ultrarelativistic particles, which means that γ 1. In this limit, a useful approximation is β = 1 1 γ γ 2. (13.11) The Lorentz transformation (13.10) can also be written in the matrix notation x x x y z = y 0 0 γ cβγ z = L y z t 0 0 γ t t βγ c. (13.12) 223/441

13 Lorentz transformation and matrices The advantage of using matrices is that they can be consecutively applied in several steps. Here is an example: we want to generate a matrix which corresponds to a moving coordinate system along the x axis. 224/441

14 Lorentz transformation and matrices Let us rotate K system by 90 degrees around the y axis, in such a way that the new x axis is equal to the old z. The rotated frame is denoted by K and the y y y y x x V V M L M -1 rot rot x z z x z z coordinate transformation from K to K is given by x = z, z = x, or in matrix notation M rot = (13.13) /441

15 Lorentz transformation and matrices A new frame K is moving along along the z axis and we then use the Lorentz transformation L to transform from K to K. Finally, we transform from K to the lab frame K using the matrix Mrot 1. The sequence of these transformations is given by the product (M rot ) 1 L M rot = γ 0 0 γβc γβ/c 0 0 γ. (13.14) This result, of course, can be easily obtained directly from the original transformation by exchanging x z. 226/441

16 Lorentz contraction and time dilation Two events occurring in the moving frame at the same point and separated by the time interval t will be measured by the lab observes as separated by t, t = γ t. (13.15) This is the effect of relativistic time dilation. An object of length l aligned in the moving frame with the z axis will have the length l in the lab frame: l = l γ. (13.16) This is the effect of relativistic contraction. The length in the direction transverse to the motion is not changed. 227/441

17 Doppler effect Consider a wave propagating in a moving frame K. It has the time-space dependence: cos(ω t k r ), (13.17) where ω is the frequency and k is the wavenumber of the wave in the moving frame. What kind of time-space dependence an observer in the frame K would see? We need to make a Lorentz transformation of coordinates and time to get cos(ω t k r ) = cos(ω γ(t βz/c) k x x k y y k z γ(z βct)) = cos(γ(ω + k z βc)t k x x k y y γ(k z + ω β/c)z). (13.18) 228/441

18 Doppler effect We see that in the K frame this process is also a wave with the frequency and wavenumber cos(ωt kr), (13.19) k x = k x, k y = k y, The object (ω, ck) is a 4-vector. k z = γ(k z + βω /c), ω = γ(ω + βck z ). (13.20) The result ω = γ(ω + k z βc) matches E = γ(e + p z βc), and similarly for k z and p z. 229/441

19 Doppler effect The above transformation is valid for any type of waves (electromagnetic, acoustic, plasma waves, etc.) Now let us apply it to electromagnetic waves in vacuum. For those waves we know that ω = ck. (13.21) Assume that an electromagnetic wave propagates at angle θ in the frame K cos θ = k z k, (13.22) and has a frequency ω in that frame. What is the angle θ and the frequency ω of this wave in the lab frame? 230/441

20 Doppler effect We can always choose the coordinate system such that k = (0, k y, k z ), then tan θ = k y k z = k y γ(k z + βω /c) = sin θ γ(cos θ + β). (13.23) In the limit γ 1 almost all angles θ (except for those very close to π) are transformed to angles θ 1/γ. This explains why radiation of an ultrarelativistic beams goes mostly in the forward direction, within an angle of the order of 1/γ. 231/441

21 Doppler effect For the frequency, a convenient formula relates ω with ω and θ (not θ ). To derive it, we use first the inverse Lorentz transformation which gives ω = γ(ω βck z ) = γ(ω βck cos θ), (13.24) ω = ω γ(1 β cos θ). (13.25) Using β 1 1/2γ 2 and cos θ = 1 θ 2 /2, we obtain ω = 2γω 1 + γ 2 θ 2. (13.26) The radiation in the forward direction (θ = 0) gets a factor 2γ in the frequency. 232/441

22 Lorentz transformation of fields The electromagnetic field is transformed from K to K according to following equations E z = E z, E = γ ( E v B ), ( B z = B z, B = γ B + 1 ) c 2 v E, (13.27) where E and B are the components of the electric and magnetic fields perpendicular to the velocity v: E = (E x, E y ), B = (B x, B y ). 233/441

23 Lorentz transformation of fields The electromagnetic potentials (φ/c, A) are transformed exactly as the 4-vector (ct, r): A x = A x, A y = A y, ( A z = γ A z + v c 2 φ ), φ = γ(φ + va z), (13.28) 234/441

24 Lorentz transformation and photons It is often convenient, even in classical electrodynamics, to consider electromagnetic radiation as a collection of photons. How do we transform photon properties from K to K? The answer is simple: the wavevector k and the frequency of each photon ω is transformed as described above. The number of photons is a relativistic invariant it is the same in all frames. 235/441