On the Ideal Ampère Solenoid under AC

Gerhard W. Bruhn, Darmstadt University of Technology

Oct 26, 2008
revised and supplemented on Nov 11, 2008

1. Basics

1.1 Geometry
Let (r,θ,z) denote cylinder coordinates around the z-axis of an Euclidean 3-D space, the basic vectors of which are
        x = xi +yj + zk,     r = x − zk = xi +yj = r r°    
and the radial and azimuthal unit vectors r° and r°×k respectively, where r = |r| = |x×k| = |r×k|.

1.2 Geometry of the solenoid
Axis = z-axis, radius R , length infinite.
The region r < R is called inner domain, the region r > R is called outer domain.

1.3 Physical data of the solenoid
σ = conductivity per z-length unit, λ = inductivity per z-length unit.

1.4 Physics around the solenoid using SI units
(1): The homogeneous Maxwell equations to be satisfied inside and outside the solenoid, i.e. separately for r < R and for r > R.
− No sources of E and B
(1.1)                 div E = 0, div B = 0 ,
− Faraday's law
(1.2)                 curl E = − B_t ,
− Ampère's law
(1.3)                 curl B = ¹/_c² E_t     (+ μj     if r=R)
(2): The Maxwell equations imply the wave equations
(1.4)                 ΔE = ¹/_c² E_tt ,         ΔB = ¹/_c² B_tt         for r ≠ R.
(3): All fields are oscillating ~ e^−iωt at constant angular frequency ω (converts the wave eqs. to Helmholtz eqs.).
(4): E in azimuthal direction, B in direction of the axis of the solenoid, both spatially only depending on r. j flowing azimuthally at r=R.
(1.5)                 B = B k ,         E = E r° × k ,         j = j r° × k .
(5): Prescribed coupling conditions at r=R:
      (a) E passing continously.
      (b) B passing with a jump µj (according to Ampère's law), where j is the current density within the surface r=R.
(6): Alternating emf given by circulation of E around the solenoid: ∫_r=R E·dr = 2πR E_o e^−iωt.

Exercise: Evaluate div B und div E for B = B(r,t) k und E = E(r,t) r°×k .

2. The stationary case

All time derivatives and the angular frequency ω are vanishing. The current density is time independent: j_o = σE azimuthal on the surface of the solenoid.
This implies the boundary conditions

(2.1)         E = j_o/σ for r → R from outside, and

(2.2)         E → 0 for r → ∞ .

The unique solution of this outer domain problem is well-known:

(2.3)         E = E_o ^R/_r r°×k , where E_o = j_o/σ.

B satifies div B = 0 and curl B = 0 and is | | k. Due to B = B k this yields grad B = 0, and hence B = const. Supposed B ≠ 0 we would have infinite flux of B in the outer domain. Thus, B=0 for r > R.

I the inner domain the constant B must be chosen such that the jump condition (5b) of Sect.1,B(R+o)−B(R-o) = μ j_o, is fulfilled. Then r < R yields B(r) = B(R−o) = − μ j_o
N.B. Orientation of j questionable, to be checked.

3. The AC case

At the solenoid at r = R an azimuthal alternating current is assumed:

(3.1)                 j = j_o e^−iωt = j_o r°×k e^−iωt ,

where j_o is a complex constant. For more about the correlation with the effective electrical field E see Sect. 6.

The effective electrical field E is assumed

to be a solution of the wave equation (1.4-1) for r ≠ R. This yields the Helmholtz equation

(3.3)                 E'' + ¹/_r E' + (κ² − ¹/_r²) E = 0         where ' = ^d/_dr and κ = ^ω/_c.

This is a Bessel type differential equation with well-known solutions:

E(r) is a linear combination of the (complex-valued) Hankel functions H₁⁽¹⁾, H₁⁽²⁾ of order 1:

(3.4)                 E(r) = c₁H₁⁽¹⁾(κr) + c₂H₁⁽²⁾(κr)

Likewise the ansatz (see (1.5))

(3.5)                 B = B(r) e^−iωt k

for the wave equation (1.4-2) yields the Helmholtz equation

(3.6)                 B'' + ¹/_r B' + κ² B = 0 ,

the solutions of which are Hankel functions of order 0:

(3.7)                 B(r) = C₁H_o⁽¹⁾(κr) + C₂H_o⁽²⁾(κr)

There is a coupling between B(r) and E(r) by the Maxwell equations (1.2) and (1.3), s. Eq. (3.9) below.

The still free constants are to be determined from boundary and transmission conditions as will be shown in Sect. 4.

Knowing the field E we can calculate the field B by means of the Faraday law (1.2)

(3.8)                 B_t = − curl E

to obtain

                        curl E = grad [rE(r)] e^−iωt × (r°×k) / r = (rE)' grad r × e^−iωt (r°×k) / r
(3.9)
                                  = ¹/_r(rE)' e^−iωt r° × (r°×k) = − ¹/_r(rE)' e^−iωt k ,

using the fact that the field (r°×k) / r is irrotational. Hence

(3.10)                 B_t = ¹/_r (rE)' e^−iωt k ,

and

(3.11)                 B = B(r) e^−iωt k = ⁱ/_ωr (rE)' e^−iωt k ,

where a time-independent summand was ommitted.

This yields the following coupling between E(r) and B(r)

There exist quite similar couplings between the Hankel functions H₁(s) and H_o(s):

(3.13)                 ¹/_s ^d/_ds[sH₁(s)] = H_o(s) .

which lead to couplings between the coefficients c und C in the Bessel representations (3.4) of E(r) and (3.7) of B(r):

(3.14)                 C_k = ⁱ/_c c_k         (k = 1, 2)

Rechenkontrolle (kann übergangen werden)

mit der Ampère-Gleichung (1.3) im Innen- und Außenraum (j = 0):

(3.15)                curl B = ¹/_c² E_t

(3.11) ergibt

(3.16)                 curl B = ⁱ/_ω [¹/_r (rE)']' e^−iωt grad r × k
                                      = ⁱ/_ω [¹/_r (rE)']' e^−iωt r°×k
                                      = ⁱ/_ω (E'' + ¹/_r E' − ¹/_r² E) e^−iωt r°×k
                                      = − ⁱ/_ω κ² E e^−iωt r°×k
                                      = − ^iω/_c² E e^−iωt r°×k .

(3.2) liefert andererseits

(3.17)                 ¹/_c² E_t = − ^iω/_c² E e^−iωt r°×k

somit ist (3.15) in der Tat erfüllt.

4. The transmission between inner and outer domain

The field functions for the inner (−) and the outer (+) domain are given by

(4.1+)                 B⁺(r) = C₁⁺ H_o⁽¹⁾(κr) + C₂⁺ H_o⁽²⁾(κr)

and

(4.2+)                 E⁺(r) = c₁⁺ H₁⁽¹⁾(κr) + c₂⁺ H₁⁽²⁾(κr) .

As a consequence of the Maxwell equations we have equation (3.14), which applied to the present situation yields

(4.3+)                 C_k⁺ = ⁱ/_c c_k⁺ .

The coupling between inner and outer domain is given by the prescribed jump of B at the current density j at the surface r=R:

Integration of B·dx = B dz along a closed path inside and outside the discontinuity surface r=R (where B is almost everywhere | | dx = kdz) due to the Ampère law (1.3) yields the following jump condition (since the identical factors e^iωt of B and j can be reduced):

(4.4)                 [B] = B(R+0) − B(R−0) = μ j_o .

The limits B(R+0) are given by (3.7):

(4.5)                 B(R+0) = C₁⁺ H_o⁽¹⁾(κR) + C₂⁺ H_o⁽²⁾(κR) ,

hence (4.4) yields the 1st jump relation:

where

(4.7)                 C_k := C_k⁺ − C_k⁻ (k=1,2).

The azimuthal field E, hence also E(r), must pass continuously through r=R:

(4.8)                 c₁ H₁⁽¹⁾(κR) + c₂ H₁⁽²⁾(κR) = [E(R+0)] = 0

where c_k := [c_k⁺] (k=1,2). Due to the proportionality (4.3) we obtain the 2nd jump relation

Since the Hankel functions H_o^(1)/(2), H₁^(1)/(2) constitute a fundamental system of the Bessel de. the determinant of the linear system (4.6+9) is well-known to fulfil the identity

(4.10)                 D := ( H_o⁽¹⁾ H₁⁽²⁾ − H_o⁽²⁾ H₁⁽¹⁾ ) (κR)     =     ^{4 i}/_πκR .

Therefore the linear system (4.7+10) can be solved explicitly:

Due to (4.7) the cofficient C_k are the jumps C_k⁺ − C_k⁻. Therefore we have to look for further conditions to be fulfilled by C_k⁺ in the inner and the outer domain respectively.

The outer domain: Here the first term in equ. (4.1+) represents an out-running waves ~ H_k⁽¹⁾(κr) e^−iωt while the second summand ~ H_k⁽²⁾(κr) e^−iωt represents an incoming waves as can be seen by the asymptotic behavior of both summands (see Sect. 5, Discussion). Due to the causality principle no incoming waves can be admitted in the outer domain:

(4.12+)                 C₂⁺ = 0     and therefore     C₂⁻ = − C₂ = − ¼ i μj_o πκR H₁⁽¹⁾(κR).

The inner domain: Here the regularity of B is required. However, the (opposite) imaginary parts of the Hankels H_o^(1)/(2) in (4.1−) have logarithmic singularities at r=0. Therefore the singularities have to be compensated by the condition

(4.12−)                 C₁⁻ = C₂⁻.

So, by (4.12+) the coefficient C₁⁻ is determined as well to obtain:

These are the field functions for the inner domain. For the outer domain we know C₂⁺ = 0 due to the causality principle, and C₁⁺ = C₁ + C₁⁻ where the right hand side is completely known now due to (4.11) and (4.12+).

Therefore the emf induced in the circular coil r = r₁ > R is

(4.14)                 U(r₁) = 2π r₁ E(r₁) e^−iωt         with E(r₁) from (4.13+).

In addition and as response to a contradictory claim in the Abstract of the paper [3] we'll here calculate the emf induced in a coil ∂Ω: Due to the Stokes integral theorem we have

(4.15)                 ∫_∂Ω E ·dx = ∫_Ω curl E · do .

Here we can substitute curl E by −^∂B/_∂t due to the Faraday law that is fulfilled by construction of E and B (see (3.8)) to obtain

5. Discussion