Abstract
In this paper, we will find in terms of Fourier integrals an exact expression for the magnetic and electric fields generated by an infinite solenoid where an arbitrary current density flows. Considering the central region of an infinite solenoid, we will obtain an exact expression relating the flowing current and the magnetic and electric fields as functions of the time. Being an exact expression, it will allow us to go beyond the quasistatic approximation. The result can apply in theoretical problems and experimental setups where the flowing current may change abruptly in time, such as pulsed or stochastic currents.
1. Introduction
In physics and engineering uniform magnetic fields are ubiquitous. Many studies are devoted about how to generate them. For this purpose, Helmholtz coils are widely used but new designs to create uniform fields are present in literature [1–3]. For example, uniform fields are used in magneto-hydrodynamic studies [4–8], in ferrofluid flow studies [9], in new analytical approaches to the studies of micropolar conducting ferrofluids with applications to new engineering materials [10, 11], in magnetic resonance spectroscopy [12], in technological applications [13, 14], in testing devices [15], or in measuring instruments such as magnetometers. Strictly related to the previous topic we also consider time-varying uniform magnetic fields that are widely used in theoretical and experimental works such as magnetic field susceptibility tests or biomedical studies [16–18]. The literature on this topic is quite vast, and we limit ourselves to a few exemplary applicative papers [19–22]. The typical approach is based on the quasistatic approximation where the magnetic field is given, in the SI system of units, by the flowing current density , i.e., . If we consider arbitrary time-varying currents, then things start to be a little more complicated. Via Maxwell’s equations, we know that an electric field is created, according to the Lenz’s law [23, 24]. We focus on which kind of magnetic and electric fields we have to consider when the first order approximation is a uniform magnetic field varying arbitrarily in time. This can be realized by studying the electromagnetic field generated by an infinite solenoid where an arbitrary time-varying current density flows. To the best of our knowledge, from an analytical point of view, the generation of an electromagnetic field by an infinite solenoid is not completely solved except for particular cases [25, 26]. The knowledge of the central values of the fields will allow us to go beyond the quasistatic approximation and we may apply the results to actual laboratory setups. For example, circuits with abruptly changing currents, such as an impulse current generator circuit, are currently used in electrical engineering. Another important case, where the quasistatic approximation loses validity due to the abrupt changes of the current, is a magnetic field generated by a stochastic current, a well-known phenomenon since a long time ([27] and references therein). With respect to these cases, the first-order approximation of a uniform magnetic field proportional to the flowing current is not sufficient since the magnetic and the induced electric fields are related to the derivatives of the flowing current.
Let us clarify this crucial point considering the set of Maxwell’s equations [23, 24] in the empty spacewhere is the electric field, is the electric displacement field related to the electric field via the relationship , and is the magnetic induction, related to the magnetic field via the relationship . The constant farads/meter is the vacuum permittivity in the SI system of units. The constant henries/meter is the vacuum permeability in the SI system of units.
If we solve iteratively the Maxwell equations we may assume that a slow time-varying current generates a magnetic induction field and an electric field where is the uniform field associated with the current density, i.e., , and , are the corrections. At first order, in the vacuum space, for (1) and (2) we haveFrom (5) and (6) we deduce that iterating this approach, we end up into time derivatives of the fields of arbitrary order. This fact can invalidate the above approach in presence of an arbitrary time-varying current . Indeed considering the case when the current changes abruptly, then the iterative approach would give rise to very large derivatives of the fields, forcing us to take into account all subsequent derivative orders. A simple and realistic example is a dichotomously fluctuating current. The derivatives of the fields would produce Dirac’s delta functions and its derivatives. These considerations imply that a simple perturbative-iterative approach is not feasible for the problem that we are studying.
The paper is organized as follows: In Section 2 we will evaluate an exact expression, in terms of Fourier transform, for the magnetic and electric fields generated by an arbitrary current density flowing on the surface of a solenoid. In Section 3 we will apply the result to a harmonic current density flowing on the surface of the solenoid obtaining an exact expression for the fields inside and outside of the solenoid. In Section 4 we will evaluate the exact expression of magnetic and electric fields, in time representation, at the center of the solenoid. The obtained expressions take into account the correction to the quasistatic approach giving an analytical expression for the fields generated by an arbitrary time-varying current density. Finally in Section 5 we will draw the final remarks.
2. Infinite Solenoid Field
In this section, we will evaluate the exact expression for the magnetic and electric field generated by an arbitrary current density flowing on the surface of an infinite solenoid of radius (see Figure 1). In presence of arbitrary currents and zero charge density, the general solution of (1)-(4) is given in terms of the potential vector (Coulomb gauge [24])where is the volume element at , the current density is a given function of space and time, and the time is given by the relation [24]We may rewrite (7) asKnowing the potential vector we may evaluate the magnetic induction and electric fields via the equationsConsidering as system an infinite solenoid, we have two empty spaces separated by the surface of the solenoid where the current is flowing. If we consider the Fourier representation of (7), we may write thatwhere we definedTaking into account the fact that is a two-dimensional current density, we have that where is the solenoid radius. For the sake of simplicity, we consider a uniform current density in the angular direction, i.e., , and we obtainSetting we may rewrite (15) asTo compact the notation, we define the following quantities:and we will dropped the space and frequency dependance of the field . Performing the integration on the -variable and projecting along the axes we obtainwhere and are the zero order Bessel functions of the first and second kind, respectively, while is the function that gives the sign of . Using Graf’s addition theorem for Bessel functions [28, 29] we havewhere is an arbitrary complex number, and , is any of the Bessel functions or the Hankel functions of the first and second kind, and , respectively. Let us focus on the solution inside to the solenoid. Then and ; thus we haveor in polar coordinatesExploiting the relationship we obtain the magnetic field in the inner region of the solenoid via the Fourier transform of (10), i.e.,Similarly the electric field, via the Fourier transform of (11), isGoing back to the time representation we may writeIn a similar way, we can evaluate the magnetic and electric fields in the region outside of the solenoid. In this case we find
3. Exact Case: Harmonic Current Density
As an exact example, we will consider the case of a harmonic current density flowing on the solenoid surface along the angular direction . We assume thatTaking the Fourier transform we haveFrom (28) and (29) it follows thatWe shall focus now on the internal magnetic field, i.e., . For and consequently , we may approximate the Bessel functions asThus the current density is a slow time-varying function and we recover the well-known resultIn the opposite limit, i.e., , we may distinguish two cases: on the axis and on the solenoid surface. On the axis we havewhile on the solenoid surface we haveAs we can infer from (34), there are frequencies at which the magnetic field evaluated on the surface vanishes (see Figures 2 and 3). Such frequencies are given by the zeros of that in the high-frequency limit areOn the other hand the amplitude of the central value of the magnetic field grows as . In a similar way, from (35) we deduce that the frequencies, at which the electric field evaluated on the surface vanishes, are given by the zero of , i.e.,
Analogous calculation are performed for the external fields. Using (30) and (31), for the magnetic and electric fields we haveAn easy check shows that for we recover the quasistatic limit . Finally using the Bessel function identity , it is straightforward to verify that satisfies to the boundary condition
A remarkable property of (43) and (44) is that there is a series of frequencies at which both magnetic and electric fields vanish outside of the solenoid. Indeed having the expression of the fields the common factor , we find that the outside fields vanish whenThe first exact frequency at which the fields and vanish is given by the first zero of corresponding to . The results are shown visually in Figures 4 and 5.
4. Axial Fields
In this section, we will evaluate the magnetic and electric field at the center of the solenoid. The central region of the solenoid is the region where the magnetic field can be considered uniform. Being evaluated using an exact expression, we will have the field values that are consistent with Maxwell’s equations. For the sake of simplicity, we will set the origin of time at . In other words, there are no currents flowing for . We will find a generic expression for the field and . To do that we consider (28) evaluated at . We haveApplying the properties of Fourier transform we may rewrite the above expression aswhere, exploiting the symmetries of the integral, we define the function asThe integral can be solved analytically [29] and giveswhere is the step function. If admits a finite limit for we can integrate by parts (48); otherwise we must consider carefully its expression. With the assumption that such a limit exists then we may rewrite (48) aswhere is the limit for of and is the derivative of the current . As first example we turn on a constant current at , i.e.,Applying (51) we obtainThe two step functions make vanish the first term on the right side of (53). It is straightforward to obtain the well-known result for . Next, we consider two cases where the corrections to the quasistatic approximation play an important role. For this purpose we study the case of a strong current pulse that we can approximate with a Dirac’s delta, i.e.,where is a constant with current density dimension and is a constant with the time dimension. Applying directly (48) we obtainThe first term represents the pulse arriving at the center of the solenoid while the correction shows that, after the pulse, still it is present a magnetic field decaying as . Finally we will study a solenoid where a sinusoidal current is switched on at t=0After straightforward algebra we obtainIn the asymptotic limit, , we haveTaking the limit for we recover the well-known quasistatic resultTaking the limit for we obtain the resultThe last two examples show that if we deal with fields that are not quasistatic, we can still consider a region where the magnetic field is uniform. On the other hand, to be consistent with Maxwell’s equations we have to use the expression in (51). From expression (58) we infer that we may change the phase between flowing current and magnetic field. Selecting, for example, the frequency in such a way that we have a magnetic field dephased of with respect to the flowing current. In the quasistatic regime while for high frequencies the amplitude of the magnetic field grows as and magnetic field and current density are not anymore in phase (see Figures 6 and 7). With respect to the evaluation of the electric field expression near the center of the solenoid, we may proceed in a similar way such as for the magnetic field. At the first nonvanishing order, we haveIn the time representation we haveFormally (62) coincides with the integration in cylindrical coordinates of (5), as in the quasistatic approximation, but the magnetic field is given by (48) that in general is not proportional to the flowing current as shown by the two last examples of this section.
5. Results and Discussion
In this paper, we found an exact expression for the magnetic field generated by an infinite solenoid where an arbitrary current density is flowing. It has been shown that abrupt changes in time of the flowing current generate magnetic and electric fields that, in general, differ from the fields generated by the quasistatic approximation. This result takes in account all the corrections to the quasistatic approximation, so giving an exact expression for the uniform magnetic field generated by an arbitrary time-dependent current. Consequently, also the exact expression for the electric field associated with the time-varying magnetic field is given. The fields so found are consistent with Maxwell’s equation at all time derivative order. This result can be used in several relevant applications where the current abruptly changes in time, such as pulsed current or stochastic fluctuation of the current value.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
Mauro Bologna acknowledges financial support from UTA Mayor Project no. 8765-17 and the Phy.C.A for the logistical support.