e.m. field of charge in rectilinear motion I Start with Maxwells - - PowerPoint PPT Presentation

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Oct 23, 2023 254 likes •413 views

e.m. field of charge in rectilinear motion I Start with Maxwells equations Lets work out our way to an equation for A and P. Piot, PHYS 571 Fall 2007 e.m. field of charge in rectilinear motion II Using we get:

SLIDE 1

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion I

Start with Maxwell’s equations
Let’s work out our way to an equation for A and Φ

SLIDE 2

P. Piot, PHYS 571 – Fall 2007
Using

we get:

In the Lorenz’ Gauge:
Thus
Using

gives

e.m. field of charge in rectilinear motion II

[JDJ 6.16] [JDJ 6.15] Inhomogeneous wave equations

SLIDE 3

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion III

So the problem is to solve the equation for A and Φ given the form
f ρ and J
For a moving charge distribution with velocity v

      − − =       Α Φ       ∂ ∂ − ∇ J t r r µ µε

ε ρ 2 2 2

SLIDE 4

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion IV

For both A and Φ we have to solve an inhomogeneous d’Alembert

equation of the form

Consider
Then
So

with

SLIDE 5

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion V

….
Then our d’Alembert equation has the form
Consider a point charge
Vector potential is along z
And we have to solve

SLIDE 6

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion VI

The equation

is solve by inspection from

The results are

SLIDE 7

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion VII

The E-field can be calculated from
Which gives

SLIDE 8

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion VIII

In spherical coordinate
So
and
In vacuum

SLIDE 9

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion IX

Consider
Note that

the field line are squashed long the direction of motion.

E-field line associated to a moving charge with Lorentz factor γ

SLIDE 10

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion X

B- field can also be computed

SLIDE 11

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion XI

Further reduction (try to introduce an impact parameter b)
introducing

b x

SLIDE 12

P. Piot, PHYS 571 – Fall 2007

e.m. field of charge in rectilinear motion XII

Example of E-field associated to a charge at rest (γ=1)and moving

with γ=10.

SLIDE 13

P. Piot, PHYS 571 – Fall 2007

Space charge effects I

Let’s consider the interaction of two particle moving at parallel to

each other and consider the force experience by the particle of charge q0 from the other particle

SLIDE 14

P. Piot, PHYS 571 – Fall 2007

Space charge effects II

Let’s consider the fields are generated by a source particle of unit