Supplemental Material:

Alternative formulae for J and P as derived in [8], [9] and
direct calculation of P using Eq. 10.
For the sake of completeness we sketch briefly in SM1 how Schäfer and Heiden [Ref.

8] derived their formulae for the current distribution. They obtain formulae for J
numerically equivalent, but different from the ones (Eq. 8a,b,c) obtained by us. They
did not give an integrated expression for the power dissipation P, a result obtained
later by Groh and Dixon [Ref. 9] that we reproduce here. We also derive in SM2 a
third expression for P obtained by integration of Eq. 10. Altogether we have three
different expressions for P whose equivalency we have checked numerically. The
most compact and most easy to handle formula is the one given by Eq. 11; however
the two other ones can be useful to derive approximations valid in particular cases
(see Appendix A).
SM1. The approach of Schäfer and Heiden and its extension
by Groh and Dixon

These authors use for the exciting field E ' Eq. 3a and not Eq. 3b as we have done. As
mentioned in the main text this leads to a different surface charge distribution that

creates a different electric field Ech' . At the end of the calculation one should find the
 
same total field E '  Ech' , and thus the same current distribution.

Calculation of J
 


We introduce the reduced fields e '  E ' / B0r sin(  )   yuz   sin(  )uz and


ech'  Ech' / B0r sin(  ) .

ech' derives from a potential vch'  Vch' / B0r sin(  ) that obeys Laplace’s equation and
satisfies the following boundary conditions (BCs) that replace those of the main text:
vch'
BC1
 0,   a1 , z   b, b,   0,2 

vch'
BC2
 0,   a2 , z   b, b,   0,2 

vch'
   sin(  ), z  b,   a1 , a2 ,   0,2  BC3
z
BC3 implies that the solution varies as sin(φ) and is an odd function of z. We may
express it as:
sinh(mz)
vch' = å[ Am J1 (mr ) + BmY1 (mr )]
sin(f ) A 1
cosh(mb)
m
1
where J1 and Y1 are first order Bessel functions of the first and second kind. The term
cosh(mb) has been introduced to simplify the equations that follow. The coefficients
Am and Bm, as well as the non integer index m are determined from the BC1,2,3
which yield respectively:
Am J 0 (ma1 )  J 2 (ma1 )  Bm Y0 (ma1 )  Y2 (ma1 )  0 A 2
Am J 0 (ma2 )  J 2 (ma2 )  Bm Y0 (ma2 )  Y2 (ma2 )  0 A 3
 mA J (m )  B Y (m )  
m 1
A4
m 1
m
The first two equations have non zero solutions only if their determinant is zero, that
is if:
J 0 (ma1 )  J 2 (ma1 )Y0 (ma2 )  Y2 (ma2 )  Y0 (ma1 )  Y2 (ma1 )J 0 (ma2 )  J 2 (ma2 )
This equation allows numerical determination of the parameters m1, m2,…
Eq A 4 involves radial functions Rn,m (mr ) = Am Jn (mr )+ BmYn (mr ) (with n=1), which
are solutions of a Bessel equation, which according to A2 and A3 verify:
R'n,m (a1 ) = R'n,m (a2 ) = 0 . This is a Sturm-Liouville problem: the solutions are
orthogonal with the weight P(  )   appropriate for a Bessel equation1, i.e. we have:
ò
a2
a1
r Rn,m (r )Rn,m (r )d r = 0
i
j
for i≠j; the normalisation factor is equal to1
2
a2
2
2
1
2
2
ò a1 r éëRn,mi (r )ùû d r = 2m2 éë( mi a2 ) - n2 ùûéëRn,mi (a2 )ùû - éë( mi a1 ) - n2 ùûéëRn,mi (a1 )ùû A 5
i
Multiplying both side of A 4 by  Am J1 (m )  BmY1 (m ) and integrating between a1
and a2, we obtain:
{
}
m ò r [ Am J1 (mr )+ BmY1 (mr )] d r = - ò r 2 [ Am J1 (mr )+ BmY1 (mr )] d r
a2
a2
2
a1
a1
A6
The integral of the left side of A 6 is deduced from A 5, and for the right side we use
the relations:
ò x J (x)dx = x J (x) A 7
ò x Y (x)dx = x Y (x) A7bis
2
2
1
2
2
2
1
2
from which we obtain:
(ma )  1A J (ma )  B Y (ma )  2a A J (ma )  B Y (ma )
 (ma )  1A J (ma )  B Y (ma )  2a  A J (ma )  B Y (ma )
2
2
2
m 1
2
m 1
2
2
1
2
2
2
m 1
1
m 1
m
2
1
1
2
m
2
2
m 2
1
m 2
2
1
B
Introducing  m  m (given by A 2 or A 3) and C1, 2 (m )  J1, 2 (m )   mY1, 2 (m ) ,
Am
one finds after some algebra:
2a22C2 (ma2 )  2a12C2 (ma1 )
Am 
A8
(ma1 ) 2  1 C12 (ma1 )  (ma2 ) 2  1 C12 (ma2 )
This result, apart from a slightly different definition of Am and Bm, is identical to the
one given in the Groh and Dixon reference.


The current J  B0r sin(  ) j is calculated from:
  
 
j  e '  v'ch   yu z    v'ch
which leads to:




2
1
sinh( mz)
mAm ( J 0 (m )  J 2 (m ))  Bm (Y0 (m )  Y2 (m ))
sin(  ) A 9 a

2 m
cosh( mb)
1
Am J 1 (m )  BmY1 (m ) sinh( mz) cos( ) A 9b
j 

 m
cosh( mb)
cosh( mz)
j z    sin(  )   mAm J 1 (m )  BmY1 (m )
sin(  ) A 9c
cosh( mb)
m
with Am given by A 8, and Bm by A 2 or A 3
Calculation of P
Groh and Dixon used a practical shortcut for the calculation of P: it is provided by the


relation P  r .u z between P and the braking torque  with respect to the rotation

axis.  results from Laplace’s forces and is equal to:
j 
(r × B0 )J dv - éë òòò (r × j )dvùûB0 = òòò (r × B0 )J dv
V
V
 
The last equality results from the fact that  (r  j )dv  0 as shown now; starting
V
 
from   J  0 we can write :
 


 
 
 
0   r 2  Jdv     (r 2 J )dv   J  (  r 2 )dv   (r 2 J )  nds  2 (r  J )dv
V
V
V
S
V

The surface integral is zero because of the boundary conditions verified by J ,
 
whence the nullity of the volumic integral of r  J
G=
òòò
V
r ´ (J ´ B0 )dv =
òòò

V


As in chapter 4 we can discard the longitudinal component of B0 and we are left
with :
dV '
G.uz = òòò (r × B0 sin(q )uy )J ×uz dv = -s B0 sin(q ) òòò y(B0wr y + ch )dv
V
V
dz
dVch '
The integral involving
can be transformed into a surface integral using relation
dz

dVch '



V dv  Snds applied to   y dz and projected along u z . One obtains:
G.uz = -s B0 sin(q )éëB0wr òòò y 2 dv + òò Vch ' yuz .n dsùû
V
S
The two integrals are easily evaluated using A 1 for Vch'  vch' B0r sin(  ) and making
use of A7; we obtain:
a2


2
1
2 2
2
4
4
P  r .u z  2r B0 sin ( ) b(a2  a1 )   tanh( mb) Am J 2 (m )  BmY2 (m )
4
m
m

a1

In the case of a plain cylinder (a1=0, a2=a) it is easily shown that Bm=0, that the
1
2a 2
1
solutions ma of A 3 are the roots of J’1(ma), and that Am  2 2 2 2
.
m a m a  1 J 2 (ma)
This yields:

b 

tanh (ma)  
 1 b

a 

P  r .u z  4r2 B02 sin 2 ( )a 5 


3
2
 8 a ( ma ) (ma) (ma)  1 




3





SM2. Calculation of P by direct integration of Eq. 10.
We come back to the approach followed in the main text which leads to currents
given by Eq. 8a,b,c with coefficients Am and Bm given by Eq. 6a,b. We wish to
calculate P as given by Eq. 10.
Integration over φ is straightforward; for the integration over z we use the following
simple results:
b
b
b
0 sin( m1z ) sin( m2 z )dz  0 cos(m1z ) cos(m2 z )dz  2  n1n2
n
b

 1
0 z sin( mz)dz  m2
(2n  1)
m
, n  0,1,...
2b
We are left with the integration of a function of ρ, which except for one term is easily
integrated, yielding:
P
2 3 2

b a2  a12
2
3
 B0r sin(  ) 


bAm2 I1 (u )uI 0 (u )  I1 (u )  Bm2 K1 (u )uK 0 (u )  K1 (u )
 
n 4
  1 3 uAm I1 (u )  Bm K1 (u )
m
m
u  ma1
u  ma21
2

 I (u ) K1 (u )  I 0 (u ) K1 (u )  I1 (u ) K 0 (u )
 2b Am Bm   u 1
du
ma1
m


 uI 0 (u ) K 0 (u )  I1 (u ) K1 (u )
The remaining integral is more complex to evaluate, but using the properties of Bessel
functions (see chapters 9.6 and 11.6 of Ref [2]), one obtains:
2

  u I1 (u) K1 (u)  I 0 (u) K1 (u)  I1 (u) K 0 (u)  uI 0 (u) K 0 (u)  I1 (u) K1 (u)du
1
 uI 0 (u ) K1 (u )  I 1 (u ) K 0 (u )  I 1 (u ) K1 (u )
2
ma2
We finally get a fully integrated expression for P:
P
2 3 2

b a2  a12
2
3
 B0r sin(  ) 




b Am2 I1 (u )uI 0 (u )  I1 (u )  Bm2 K1 (u )uK 0 (u )  K1 (u )
n 4
 
  1 3 uAm I1 (u )  Bm K1 (u )
m
m
 bAm Bm uI 0 (u ) K1 (u )  I1 (u ) K 0 (u )  2 I1 (u ) K1 (u )
u  ma2
A 10
u  ma1
For a plain cylinder (a2=a, a1=0) one can show that A 10 reduces to:
 (2n  1) a 
I1 

P
2 64
2
b 




 ( B0r sin(  )) 2 a 2b 3
3  3 n 0
 (2n  1) a  (2n  1) a 
 (2n  1) a  
(2n  1) 4 
I0 
 I1 


2
b 
2
b
2
b  


4
1
E. Durand, “Electrostatique Tome 2”, Masson et Cie , Paris, 1966.
M. Abramovitz and I. A. Stegun, “Handbook of Mathematical Functions”,
Ninth Printing. New York: Dover Publications Inc., 1970.
2
5