A widespread assertion has existed for a long time, believing the external field of an infinitely long solenoid should be zero, but it is proofed to be wrong in this work. The components of magnetic flux density of current-carrying, closely wound cylindrical solenoids are calculated. At a distant field point, the external field definitely has a nonzero component, being equal to that of a straight wire of equal length. Since this equivalence is length-independent, it still holds true for ideal solenoids having infinite length. Hence the incorrect and still spreading inference about long solenoids should be rectified. Furthermore, theoretical and experimental discussions involving solenoids should be reviewed again carefully.
Scientific research related to magnetic field not only has a long history [
This paper is devoted to correcting this long-standing and widespread fallacy about long solenoids which is still spreading in textbooks [
A simple transforming process could explain intuitively why the external field of current-carrying solenoids might not be zero. For simplicity, the values are calculated and discussed without any other magnetic media nearby. As shown in
with total length L. Then the winding turns per unit length of this solenoid is, n = 1/H0. The axis of the cylinder is set as the Z axis. The distance from a field point, P, to Z-axis is d. The perpendicular line from P to Z-axis is parallel to the Y axis.
As shown in
In general, the detailed characteristics of the wire-shape localized within the scale of R, should not play an important role in determining the magnetic field at a far away pointP ( d ≫ R ). The overall magnetic field distribution of a very thin solenoid should be very close to that of a simple straight wire. On the other hand, when a straight wire changes its structure (such as bending or twisting) around a certain spot localized within the scale of R, the corresponding change of its field could not be reflected significantly at those distant points.
Theoretically, each elementary current Idl of the wire could be decomposed into three components, dlz (paralleling to Z-axis), dlT (along the tangential direction of the cross-section circumference), and dlR (along the radial direction of the circle). d l = d l z + d l τ + d l R . Since the radius R is fixed here for solenoids, and the diameter of the wire D0 is negligible compared to R ( D 0 < H 0 ≪ R ), dlR equals zero. Then we have
d l = d l z + d l τ (1)
The infinitesimal, dB, of the field generated by Idl at a point P is given by the Biot-Savart Law,
d B = ( μ 0 4 π ) I d l × r r 3 (2a)
Then the magnetic flux density B generated by the current-carrying wire, would be the integration of dB along the wire,
B = ∫ d B = ( μ 0 4 π ) I ∫ d l z × r r 3 + ( μ 0 4 π ) I ∫ d l τ × r r 3 (2b)
At a distant point P, the first term is approximately equal to the field of a straight wire with the same length of the solenoid, while the second term is approximately equal to that of a series of circular currents with radius R, arranged at equal intervals along the Z axis. If the total length L is infinite, then the second term is zero, while the first term is not. Therefore, the result of B at P is approximately equal to that generated by an infinitely long straight wire.
As for toroids, a similar assertion exists, believing that their external magnetic field is zero [
In the following section, the magnetic field distribution of cylindrical solenoids will be discussed with detailed calculations and analyses.
Let the expression of the helical wire be
x = R cos ( α ) y = R sin ( α ) z = ( H 0 2 π ) α (3)
where α is an independent variable for determining the position of each elementary current, Idl. An Idl could be expressed by the differentials of the coordinates,
I d l = I ( d x , d y , d z ) = ( − R sin ( α ) , R cos ( α ) , H 0 2 π ) I d α (4)
The position of the field point P is at (xP, yP, zP). Set xP = 0 for simplicity. As mentioned earlier, the distance from P to Z-axis, d, equals yP. Then we have
x P = 0 y P = d z P = z 0
where z0 is another variable for determining the position of P.
The following calculations are based on the Biot-Savart Law. In Equation (2a), I is the current intensity in the wire, r is the position vector pointing from elementary current Idl to P, and r is the module of r.
r = r P − r I d l = ( − R cos ( α ) , d − R sin ( α ) , z 0 − H 0 2 π α )
r = | r | = [ − R cos ( α ) ] 2 + [ d − R sin ( α ) ] 2 + ( z 0 − H 0 2 π α ) 2
The result of the outer product could be expanded as,
d l × r = u ( α ) d α = [ i j k ] T [ − H 0 d 2 π − α H 0 R cos ( α ) 2 π + R z 0 cos ( α ) + H 0 R sin ( α ) 2 π − H 0 R cos ( α ) 2 π − α H 0 R sin ( α ) 2 π + R z 0 sin ( α ) R 2 − R d sin ( α ) ] d α (5)
where u(α) is a vector function of variable α. The magnetic flux density at P is given by the integration of dB,
B = ∫ d B = ( μ 0 4 π ) I ∫ d l × r r 3 = ∫ α 1 α 2 f ( α ) d α , (6)
where the vector function f(α) could be expressed as,
f ( α ) = ( μ 0 4 π ) I u ( α ) r ( α ) 3 .
In a practical calculation, the integration could be replaced by the following summation with a properly selected Δα,
B = ∑ α = α 1 α 2 f ( α ) Δ α . (7)
In the following calculations, the steady current in the wire is known as I. The unit of length is selected as the pitch of the helical wire H0, while the unit of B is μ 0 I 4 π H 0 . Then B is related to several variables.
B = B ( L , d , R , z 0 ) . (8)
The meanings of the variables are listed earlier in Section 2.1. In this section, the relation between B and each variable is discussed respectively.
The BX component of the external field of a solenoid is always compared to that of a straight wire with the same length and current intensity. Since z0 is a variable, it is convenient to set the origin at the middle of the solenoids or the straight wires, without losing generality. The field of a straight wire, BL, only has one non-zero component, BLx, the magnitude of which is
B L = | B L | = | B L x | = ( μ 0 4 π ) I d [ L / 2 + z 0 ( L / 2 + z 0 ) 2 + d 2 + L / 2 − z 0 ( L / 2 − z 0 ) 2 + d 2 ] , (9)
where L is also the length of the wire, while other variables are the same as those mentioned earlier. The theoretical values of BLx are used as the standards to compare with the calculated values of BX of the solenoids. An extremely small relative difference between BX and corresponding BLx would show their close relevance.
It is verified by the calculation that, at a fixed distant field point P ( d = 100 R ≫ R ), the BX of the solenoid is always equal to the BLx of the corresponding straight wire placed along Z-axis with an equal length, L, no matter
what the value of L is. The relative difference between them is less than 1 × 10−3. Moreover, when L increases, the absolute value of BX keeps increasing. As L approaches infinity, |BX| approaches its maximum limit, being exactly the value of the magnetic field of an infinite straight wire,
B L → ∞ = μ 0 I 2 π d = ( μ 0 I 4 π H 0 ) 2 H 0 d = ( μ 0 I 4 π H 0 ) × 0.0002 .
For those long enough solenoids (with L > 150R = 15,000H0), the absolute value of Bz keeps decreasing when L increases. As L approaches infinity, |Bz| approaches zero. This could be expected by the picture of the projected magnetic induction lines on the YOZ-plane of solenoids with finite length.
By approximately remains zero at a point with z0 = 0, when L changes. It would be shown in the following section 2.3.3 that this result is in accord with the symmetric characteristic of the magnetic field.
The value of BX of the solenoid still fits very well to the corresponding BLx. Their relative difference is less than 1 × 10−3. Actually, the value of BX matches
the corresponding BLx quite well with any d > 1.3R, although the general requirement is d ≫ R . This is because that at a field point not so far away, the first term of Equation (2b) is approximately equal to the field of a cylindrical current distribution with radius R, whose external field is exactly the same as a straight wire.
The magnitudes of BX and BZ are both decreasing as the distance d increasing. This could be expected, since the field becomes weaker at a further point, in general. As shown in the inset of
Again, the magnetic induction lines projected on the YOZ-plane of a solenoid indicates that BY = 0 at a point with z0 = 0, since the solenoid is approximately symmetric about the origin.
It is clear that z0 can affect all three components of B significantly. BX and BZ are even functions of z0, while BY is an odd function of z0. Here we can see clearly that BY always equals zero at those points with z0 = 0, because of the symmetry of the field distribution, as mentioned earlier. When z0 = 0, both BX and BZ reach their respective minima, while their magnitudes reach the respective maxima.
Again, at all the distant field points, the values of BX of the solenoid consistently match those of BLx of the equally long straight wire. The relative difference between them is less than 1 × 10−3. The values of BY and BZ, are in conformity with the field distribution on the YOZ-plane illustrated by the projected magnetic induction lines of a finite solenoid.
The fixed parameters are as follows, z0 = 0, L = 20,000H0, and d = 10,000H0. As shown in
the value of BX at a distant point, though BX is closer to the value of a straight wire with a lesser R, obviously. The relative difference between the maximum and the minimum of BX is less than 1 × 10−4.
On the contrary, the radius R of the solenoid affects BZ more significantly, because the field distribution on the YOZ-plane is definitely related to the radius of the solenoid with finite length. In fact, at those points where z0 is not zero, both BY and BZ will be influenced by the value of R, while all the values of BY here approximately equal zero because a symmetric field point with z0 = 0 is chosen.
Since adequate discussions have been made for solenoids with finite length, it is possible to infer the results for a very long solenoid.
It has been verified that, no matter how long the solenoid is, the BX of its external field always equals that of a straight wire having an equal length and carrying an equal current. This equivalence itself is independent of the value of length L. Therefore, at a distant point, an infinitely long cylindrical solenoid would generate a nonzero external magnetic field, with its BX component being equal to that of an infinite straight wire, i.e.
| B x | = μ 0 I 2 π d (10)
On the other hand, according to the symmetry of the infinite solenoid, BY equals zero. Since the length L is infinity, BZ also equals zero. Therefore, both BY and BZ of the external field are zero. Then the magnitude of the external magnetic flux density of an infinitely long solenoid at a distant field point is
B = | B ext | = | B x | = μ 0 I 2 π d . Naturally, the external Bext satisfies the Ampere’s law,
∮ B ext ⋅ d l = μ 0 I , (11)
where I is exactly the flux of the current density through an arbitrary cross-section of the solenoid, i.e., the current enclosed by the Amperian loop parallel to the XOY-plane.
We have proofed that, no matter how long it is, a solenoid always has an external nonzero BX component. So the problem is finding a substitution of solenoid. The answer is simple. A pair of coaxial but chiral-opposite solenoids can build a field-free area enclosing a nonzero amount of magnetic flux.
As for the Aharonov-Bohm effect [
As shown in
I 1 = I 2 = I . (12)
If the Z-components of their current density are in opposite directions, then their internal fields are mutually reinforcing, while their external fields are canceling each other out. Thus, the structure of an external field-free area where the beams passing through with nonzero magnetic flux enclosed between the two beams is ideally realized.
The device mentioned in Section 3.1 can be used as a substitution of a solenoid in certain cases, such as the experiments related to the Aharonov-Bohm effect. Though the leakage of field of the solenoid has been discussed in other works [
In this section, we don’t need B to be zero on the routes of the beams. On the contrary, we let both B and A exist together in the same area, changing conversely. It is only necessary to distinguish which variable can affect the phase difference, the magnetic flux in the area surrounded by the routes related to vector A, or the fields along the routes related to vector B. Therefore, the existence of the field-free areas is not a necessity, if the changing trend of B is opposite to that of A.
In the first step, let the current intensity in solenoid-1 be zero, I1 = 0, and that in solenoid-2, I2, increase very slowly. Then the magnetic flux increase gradually and may cause a changing trend T1 of the interference fringes.
In the second step, fix I2 and let I1 increase very slowly. The magnetic flux generated by both currents increases gradually, while the field outside solenoid-2 decreases gradually. The changing trend of the interference fringes in this step is noted as T2.
By comparing T1 and T2, it could be distinguished whether the change of interference fringes is caused by the magnetic flux or by the magnetic field itself. In these steps, the changing trends of the flux are the same, while those of the field intensity are opposite. If T1 and T2 always show the same tendency, then the change of flux dominates this process. Otherwise, the change of field intensity may have played an important role in this process.
The assertion that the external field of a very long solenoid is zero has been proved to be wrong. Although it directly violates the Ampere’s law and the Biot-Savart Law, this fallacy is still spreading widely.
The occurrence and spread of this fallacy might be due to the following reasons. A possible theoretical reason might be the improper substitution of a series of coaxial circular currents for the original solenoid, which changes the symmetric properties of the system and totally ignores the contribution of the dlz-term to B in Equation (2b). A possible experimental reason might be the truth that the external field is much weaker than the internal field of a solenoid.
B ext = μ 0 I 2 π d < μ 0 I 2 π R ≪ μ 0 I 2 π H 0 < μ 0 I H 0 = B int
The difference of intensity would become more significant when ferromagnetic medium is inserted inside the solenoid. Besides, only the magnetic induction lines projected on the YOZ-plane of solenoids are concerned and shown in experiments, while those projected on the XOY-plane are ignored.
In this work, the external magnetic field flux density of solenoids is calculated numerically. The results of BX are compared with the BLx of the corresponding straight wires. The universal equivalence between BX and BLx is shown. Therefore, the assertion about long solenoids is rectified. A structure of coaxial pair of solenoids is proposed as a substitution of solenoid to create an external field-free area. Theoretical and experimental discussion related to solenoids and toroids might be revised.
The author would like to thank Professor Bang-jiao Ye for the valuable guidance and insightful comments, and Professor Si-zhao Jin for the long-term, tireless help and support.
The author declares no conflicts of interest regarding the publication of this paper.
Ye, S.W. (2022) The Nonzero External Magnetic Field of Long Solenoids. Journal of Electromagnetic Analysis and Applications, 14, 19-30. https://doi.org/10.4236/jemaa.2022.142002