1. Transformation of EM Fields1 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 2
Electrostatics 2
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3. 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
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4. Calculating the Electric Field
Using the scalar potential Φ, we can make the calculation easier: 4
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5. Divergence Theorem and Gauss’s Law5
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6. 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
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7. List of Multipoles Horizontal Dipoles: Regular Quadrupoles:
Skew Quadrupoles: 7
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8. List of Multipoles Regular Sextupoles: Skew Sextupoles:
Regular Octupoles: 8
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9. 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
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10. Electromagnetic Fields10 10
Transformation of
Electromagnetic Fields 10
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11. 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
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12. 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
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13. 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
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14. 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
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