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Chapter 3: E and B Field Transformations

A pure electric field in one frame appears as a combination of electric and magnetic fields in another. This unification explains magnetism as a relativistic effect of moving charges.

Field Transformation Formulas

For a boost in the x-direction at velocity v:

Electric Field

\( E'_x = E_x \) (parallel unchanged)
\( E'_y = \gamma(E_y - vB_z) \)
\( E'_z = \gamma(E_z + vB_y) \)

Magnetic Field

\( B'_x = B_x \) (parallel unchanged)
\( B'_y = \gamma(B_y + vE_z/c^2) \)
\( B'_z = \gamma(B_z - vE_y/c^2) \)

Magnetism as a Relativistic Effect

Consider a current-carrying wire and a moving charge. In the wire's frame, there's only a magnetic force. In the charge's rest frame, length contraction changes charge densities, creating an electric force. Both give the same physical force—just different descriptions!

Key insight: Magnetism is electrostatics + special relativity. There's no independent magnetic phenomenon—just electric fields viewed from different frames.

Interactive Simulations

EM Field Transformation Under Lorentz Boost

Python

script.py52 lines

import numpy as np

print("=" * 70)
print("ELECTROMAGNETIC FIELD LORENTZ TRANSFORMATION")
print("Boost along x-axis with velocity v")
print("=" * 70)

def transform_fields(Ex, Ey, Ez, Bx, By, Bz, beta):
    """Transform E and B fields under Lorentz boost along x."""
    gamma = 1.0 / np.sqrt(1.0 - beta**2)
    # Parallel components unchanged
    Ex_p = Ex
    Bx_p = Bx
    # Perpendicular components mix
    Ey_p = gamma * (Ey - beta * Bz)
    Ez_p = gamma * (Ez + beta * By)
    By_p = gamma * (By + beta * Ez)
    Bz_p = gamma * (Bz - beta * Ey)
    return Ex_p, Ey_p, Ez_p, Bx_p, By_p, Bz_p

# Case 1: Pure electric field -> generates magnetic field
print("\n--- Case 1: Pure E field, E = (0, 1, 0) ---")
print(f"{'v/c':>6} {'Ey_p':>8} {'Ez_p':>8} {'By_p':>8} {'Bz_p':>8} {'B^2-E^2':>10}")
print("-" * 52)
for beta in [0.0, 0.1, 0.3, 0.5, 0.7, 0.9, 0.99]:
    Ep = transform_fields(0, 1, 0, 0, 0, 0, beta)
    E2 = Ep[0]**2 + Ep[1]**2 + Ep[2]**2
    B2 = Ep[3]**2 + Ep[4]**2 + Ep[5]**2
    print(f"{beta:6.2f} {Ep[1]:8.4f} {Ep[2]:8.4f} {Ep[4]:8.4f} {Ep[5]:8.4f} {B2-E2:10.6f}")

# Case 2: Pure magnetic field -> generates electric field
print("\n--- Case 2: Pure B field, B = (0, 0, 1) ---")
print(f"{'v/c':>6} {'Ey_p':>8} {'Ez_p':>8} {'By_p':>8} {'Bz_p':>8} {'B^2-E^2':>10}")
print("-" * 52)
for beta in [0.0, 0.1, 0.3, 0.5, 0.7, 0.9, 0.99]:
    Ep = transform_fields(0, 0, 0, 0, 0, 1, beta)
    E2 = Ep[0]**2 + Ep[1]**2 + Ep[2]**2
    B2 = Ep[3]**2 + Ep[4]**2 + Ep[5]**2
    print(f"{beta:6.2f} {Ep[1]:8.4f} {Ep[2]:8.4f} {Ep[4]:8.4f} {Ep[5]:8.4f} {B2-E2:10.6f}")

# Case 3: EM wave (E perp B, |E|=|B|)
print("\n--- Case 3: EM wave, E=(0,1,0), B=(0,0,1) ---")
print("B^2 - E^2 = 0 (invariant for radiation)")
print(f"{'v/c':>6} {'|E_p|':>10} {'|B_p|':>10} {'B^2-E^2':>10}")
print("-" * 40)
for beta in [0.0, 0.3, 0.5, 0.8, 0.9]:
    Ep = transform_fields(0, 1, 0, 0, 0, 1, beta)
    E2 = Ep[0]**2 + Ep[1]**2 + Ep[2]**2
    B2 = Ep[3]**2 + Ep[4]**2 + Ep[5]**2
    print(f"{beta:6.2f} {np.sqrt(E2):10.4f} {np.sqrt(B2):10.4f} {B2-E2:10.6f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: EM Field Boost Transformation

Fortran

program.f9066 lines

program field_transformations
  implicit none
  double precision :: beta, gamma_val
  double precision :: Ex, Ey, Ez, Bx, By, Bz
  double precision :: Exp, Eyp, Ezp, Bxp, Byp, Bzp
  double precision :: E2, B2, E2p, B2p
  double precision :: betas(7)
  integer :: i

betas = (/0.0d0, 0.1d0, 0.3d0, 0.5d0, 0.7d0, 0.9d0, 0.99d0/)

print '(A)', repeat('=', 65)
  print '(A)', 'EM FIELD LORENTZ TRANSFORMATION (boost along x)'
  print '(A)', repeat('=', 65)

! Case: Static charge -> current + magnetic field
  ! In rest frame: pure radial E field
  Ex = 0.0d0; Ey = 1.0d0; Ez = 0.0d0
  Bx = 0.0d0; By = 0.0d0; Bz = 0.0d0

print '(A)', 'Original: E = (0, 1, 0), B = (0, 0, 0)'
  print '(A)', '(Pure electric field of a stationary charge)'
  print '(A)', repeat('-', 65)
  print '(A6, A10, A10, A10, A10, A12)', &
    'v/c', 'Ey''', 'Ez''', 'By''', 'Bz''', 'B''^2-E''^2'
  print '(A)', repeat('-', 65)

do i = 1, 7
    beta = betas(i)
    if (beta < 1.0d-15) then
      gamma_val = 1.0d0
    else
      gamma_val = 1.0d0 / sqrt(1.0d0 - beta**2)
    end if

! Transform
    Exp = Ex
    Bxp = Bx
    Eyp = gamma_val * (Ey - beta * Bz)
    Ezp = gamma_val * (Ez + beta * By)
    Byp = gamma_val * (By + beta * Ez)
    Bzp = gamma_val * (Bz - beta * Ey)

E2p = Exp**2 + Eyp**2 + Ezp**2
    B2p = Bxp**2 + Byp**2 + Bzp**2

print '(F6.2, F10.4, F10.4, F10.4, F10.4, F12.6)', &
      beta, Eyp, Ezp, Byp, Bzp, B2p - E2p
  end do

E2 = Ex**2 + Ey**2 + Ez**2
  B2 = Bx**2 + By**2 + Bz**2
  print '(/,A,F10.4)', 'Original B^2 - E^2 = ', B2 - E2
  print '(A)', 'Invariant B^2 - E^2 is preserved in all frames!'

! Show that a moving charge creates a magnetic field
  print '(/,A)', repeat('=', 65)
  print '(A)', 'KEY INSIGHT:'
  print '(A)', 'A stationary charge has only E field.'
  print '(A)', 'A moving charge has BOTH E and B fields!'
  print '(A)', 'Magnetism is a relativistic effect of electricity.'
  print '(A)', repeat('=', 65)

end program field_transformations

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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