Fields, Potentials and Equations of Motion
59
recursively allows the calculation of coefficients. Indeed, the terms a k,0 (s),
a k,1 (s) can be chosen freely, and all others are uniquely determined through
them.
To study the significance of the free terms, let us consider the electric and
magnetic case separately. In the electric field, in order to ensure that orbits
that were in the plane stay there, there must not be any field components in
the y-direction in the plane corresponding to y = 0. Computing the gradient
of the potential, we have
E x (x, y = 0) = −
k
a k,0
x
k−1
(k − 1)!
, E y (x, y = 0) = −
k
a k,1
x
k
k!
= 0,
and looking at E y , we conclude that a k,1 = 0 for all k. So the terms a k,0 alone
specify the field. Looking at E x , we see that these are just the coefficients that
specify the field within the plane, and so the midplane field determines
the entire field. Furthermore, looking at the details of the recursion relation
(3.8), it becomes apparent that all second indices are either l or l + 2. This
entails that as long as a k,1 terms do not appear, also a k,3 , a k,5 , . . . terms do not
appear. Indeed, the resulting potential is fully symmetric around the plane,
and the resulting field lines above and below the plane are mirror images.
In the magnetic field, the argument is rather similar. Considering the
fields in the plane, we have
B y (x, y = 0) = −
k
a k,1
x
k
k!
, B x (x, y = 0) = −
k
a k,0
x
k−1
(k − 1)!
= 0.
In order for particles in the midplane to stay there, we must have that B x
vanishes in the midplane, which entails a k,0 = 0. So in the magnetic case, the
coefficients a k,1 specify everything. These coefficients, however, again describe
the shape of the field in plane, and so again the midplane field determines
the entire field. In the magnetic case, the potential is fully antisymmetric
around the plane, and again the resulting field lines are mirror images of each
other.
To summarize the findings,
Electric field:
a k,1 = 0 for all k, a k,0 specify everything.
Magnetic field: a k,0 = 0 for all k, a k,1 specify everything.
To conclude, we note that it is possible to extend the entire discussion also to
cases where the motion is not confined to a simple midplane. The derivations
connected to this most general case become exceedingly complicated [49, 48]
and go beyond what is appropriate for this book.
59
recursively allows the calculation of coefficients. Indeed, the terms a k,0 (s),
a k,1 (s) can be chosen freely, and all others are uniquely determined through
them.
To study the significance of the free terms, let us consider the electric and
magnetic case separately. In the electric field, in order to ensure that orbits
that were in the plane stay there, there must not be any field components in
the y-direction in the plane corresponding to y = 0. Computing the gradient
of the potential, we have
E x (x, y = 0) = −
k
a k,0
x
k−1
(k − 1)!
, E y (x, y = 0) = −
k
a k,1
x
k
k!
= 0,
and looking at E y , we conclude that a k,1 = 0 for all k. So the terms a k,0 alone
specify the field. Looking at E x , we see that these are just the coefficients that
specify the field within the plane, and so the midplane field determines
the entire field. Furthermore, looking at the details of the recursion relation
(3.8), it becomes apparent that all second indices are either l or l + 2. This
entails that as long as a k,1 terms do not appear, also a k,3 , a k,5 , . . . terms do not
appear. Indeed, the resulting potential is fully symmetric around the plane,
and the resulting field lines above and below the plane are mirror images.
In the magnetic field, the argument is rather similar. Considering the
fields in the plane, we have
B y (x, y = 0) = −
k
a k,1
x
k
k!
, B x (x, y = 0) = −
k
a k,0
x
k−1
(k − 1)!
= 0.
In order for particles in the midplane to stay there, we must have that B x
vanishes in the midplane, which entails a k,0 = 0. So in the magnetic case, the
coefficients a k,1 specify everything. These coefficients, however, again describe
the shape of the field in plane, and so again the midplane field determines
the entire field. In the magnetic case, the potential is fully antisymmetric
around the plane, and again the resulting field lines are mirror images of each
other.
To summarize the findings,
Electric field:
a k,1 = 0 for all k, a k,0 specify everything.
Magnetic field: a k,0 = 0 for all k, a k,1 specify everything.
To conclude, we note that it is possible to extend the entire discussion also to
cases where the motion is not confined to a simple midplane. The derivations
connected to this most general case become exceedingly complicated [49, 48]
and go beyond what is appropriate for this book.