The Cryogenic Neutron EDM Experiment:
Calculation of the Magnetic Field in the vicinity of the Ramsey Cell
(with trim coils)
Introduction
The “B0” field in the cryogenic neutron EDM experiment is maintained by a
superconducting solenoid wound inside the cylindrical helium tank. It is clearly useful
for us to be able to predict both the magnitude and direction of this field throughout
the volume of the Ramsey cell, so that we can make any modifications necessary to
ensure that it complies with the various experimental constraints.
The chief constraint is that the geometric phase induced false EDM signal should be
no greater than the overall error that the experiment is aiming to achieve. This signal
is caused by the presence in the neutron’s rest frame of a magnetic field component
due to the relativistic “E x v” effect, which combines with any radial component of
field to produce a rotating magnetic field and hence a false EDM.1
Inside a solenoid of finite length, there will indeed be a small radial component of the
magnetic field everywhere except on the axis. As the size of the Ramsey Cell will be
of the same order of magnitude as the solenoid, we cannot approximate the field to its
on-axis value, and therefore need to investigate this radial component at all points in
the volume occupied by the cell. In terms of actually measuring it, however, there is a
problem: since the radial component in our solenoid is several orders of magnitude
smaller than the axial component, in any orientation of the probe that is not entirely
along the radius vector, the former will be swamped by the latter; and even if the
probe could be aligned precisely along the radius vector, its finite width would
probably still cause problems.
Luckily, the radial component is related to the field gradient in the axial direction.
Writing the equation div B = 0 in cylindrical polar co-ordinates, we have
1 
rBr   1 B  Bz  0
r r
r 
z
Assuming azimuthal symmetry we can omit the middle term, and expanding the first
term this therefore becomes
Br Br Bz


0
r
r
z
If we can further assume a linear dependence of Br on r so that
this becomes
2 Br Bz

0
r
z
whence
Br Br

,
r
r
Br 

r Bz
2 z
Using this relation, Pendlebury et al.1 derive a maximum permissible value of 1nT/m
B z
for the axial field gradient
.
z
Calculating the Off-Axis Field
The standard formula for the field at a point on the axis of a finite solenoid
Bz 
0 NI
2l
cos 1  cos  2 
where ,  are the angles subtended at the point in question by the radii at the ends,
is clearly of no use when the point is not on the axis. Instead we will have to derive a
formula from first principles.
Consider a small current element dl at a point P(xP,yP,zP) on a circular loop of wire:
y
P
x

I
According to the Biot-Savart Law, the magnetic field due to this element at a point Q
(xQ,yQ,zQ) is given by the formula
dB 
0 I  dl  PQ 
4  PQ 3 
(1)
If the radius of the circular loop is a, the length of the element dl is ad and so in
cartesian co-ordinates the vector dl is given by
dl   a sin d i  a cos d  j
To find the components of the vector PQ , we consider the projection of the circular
loop in the plane of the point Q:
P'
Q
R
The vector PP' is clearly zQ  z P k , so that the vector PQ can be written
PQ  PP'  P' R  RQ  zQ  zP .k  a sin  . j  a cos   xQ .i
Hence

 
dl  PQ   a sin d i  a cos d  j  zQ  z P k  a sin   j  a cos   xQ i

 zQ  z P a sin d  j  a 2 sin 2 d k  zQ  z P a cos d i  a cos d a cos   xQ k
 zQ  z P a cos d i  zQ  z P a sin d  j  a 2  axQ cos  d k
(2)
Furthermore,
PQ 2  PP'2  P' R 2  RQ 2  zQ  z P 2  a sin  2  a cos   xQ 2
 zQ  z P   a 2  2axQ cos   xQ2
2
Substituting from (2) and (3) into (1), therefore, we obtain
(3)