Transformation of EM Fields 1 1 Electrostatics & Transformation of EM Fields Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

2 2 Electrostatics 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 2 2 2 2 2 2

Electrostatics Assume charges are static, - There are no time-varying E & B fields. In a vacuum, the equations for E then become: This is an irrotational field, like the classical gravitational force (plus negative charges). 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 3 3 3 3

Calculating the Electric Field Using the scalar potential Φ, we can make the calculation easier: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 4 4 4 4

Divergence Theorem and Gauss’s Law 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 5 5 5 5

Electric Multipoles The same procedure for magnetic multipole work for electric. Writing the scalar potential as a polynomial expansion we have: And finding the Laplacian of this potential yields a recurrence relation: Which means all the polynomial coefficients can be written as 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 6 6 6 6

List of Multipoles Horizontal Dipoles: Regular Quadrupoles: Skew Quadrupoles: 7 11/21/2018 7 7 7 7

List of Multipoles Regular Sextupoles: Skew Sextupoles: Regular Octupoles: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 8 8 8 8

Complications from Electric Focusing Complication 1: Electrostatic focusing is only effective for semi-relativistic particles. Complication 2: Bending due to dipole element can be elliptic rather than circular. Complication 3: Electric fields accelerate/decelerate the beam during transverse focusing. Applications: Electrostatic kickers and septums. Storage rings that must be magnetic field-free. Storage rings for semi-relativistic ions. Electron models of proton accelerators. 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 9 9 9 9

Electromagnetic Fields 10 10 Transformation of Electromagnetic Fields 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 10 10 10 10 10 10

Charged Static Wire Consider a charged wire with no current. The E-field is given by: Now Boost with velocity v along the direction of the wire z. Due to length contraction of the wire, charge density: λ  γλ. Due to the velocity of the charge, there is a current flow: I = γλv. Our new frame-shifted E & B fields are given by: Magnetic fields are just electric fields in a different reference frame! 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 11 11 11 11

Neutral Current-Carrying Wire Consider a neutral wire with a current. The B-field is given by: The force on a charge q, with velocity v, in direction z is given by: Shifting the force into the rest frame of the particle we expect: But if the velocity of the particle is zero, this must be an E-field! 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 12 12 12 12

Neutral Current-Carrying Wire (cont.) How do we explain this E-field in terms of charges? A neutral current-carrying wire consists of ions and electrons moving with respect to each other: In a velocity shifted frame, Lorentz contraction impacts these two sets of charges differently and the charges no longer cancel: 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 13 13 13 13

Transformation of EM Fields Putting together these thought experiments, boost in x1 direction: Can also be derived by the transformation of the EM potentials: 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 14 14 14 14