Signals induced on electrodes by moving charges, a general theorem for Maxwell's equations based on Lorentz-reciprocity

This report discusses a signal theorem for charged particle detectors where the finite propagation time of the electromagnetic waves produced by a moving charge cannot be neglected. While the original Ramo-Shockley theorem and related extensions are all based on electrostatic or quasi-electrostatic approximations, the theorem presented in this report is based on the full extent of Maxwell's equations and does account for all electrodynamic effects. It is therefore applicable to all devices that detect fields and radiation from charged particles.


Introduction
Ever since the publication of Shockley [1] and Ramo [2] discussing the theorems for induction of currents on grounded electrodes by moving charges, there have been efforts to extend these relations to more general situations: geometries including space-charge [3], signals on electrodes connected with impedance elements [4], formulation of equivalent circuits [5], inclusion of permittivity and non-linear materials [6] as well as general geometries that contain finite resistivity [8,9]. All of these extensions are based on static or quasi-static approximations of Maxwell's equations together with Green's reciprocity theorem relating electro-static potentials and charge distributions. They apply to situations where the finite propagation velocity of electromagnetic waves, and indeed all radiation phenomena, can be neglected. However, e.g. for detectors with long readout electrodes, the finite propagation velocity of signals plays a significant role. This situation is typically treated by calculating the induced signal using the Ramo-Shockley theorem and then placing the signal as an ideal current source on the transmission line at the place of the charge movement [11]. While 'intuitively' this is the correct solution, it is interesting to investigate whether more general reciprocity theorems that are valid for the full extent of Maxwell's equations can be directly applied to this situation. We can also view antennas that detect the radiation of moving charges to be 'electrodes' on which the signal is induced. A generalized theorem might also be applied to this situation. It turns out that the Lorentz reciprocity theorem [12] can indeed be used to derive such a very general signal theorem, where a weighting field E w (x, t) of the electrode in question is used to calculate the signal. This report is structured as follows. We first outline the Lorentz reciprocity theorem and two immediate consequences, namely the network reciprocity theorem and the antenna reciprocity theorem. Then we derive the generalized signal theorem and apply it to three specific examples, namely transmission lines, synchrotron radiation from spiralling electrons and signals in beam current transformers. Two Appendices show the explicit equivalence of the direct signal calculation from the Lienard-Wiechert potentials and the calculation using weighting fields, for the cases of infinitesimal electric and magnetic dipole antennas.

Lorentz reciprocity theorem
We assume the most general form of Maxwell's equations for a linear anisotropic material of positionand frequency-dependent permittivity matrixε(x, ω), permeability matrixμ(x, ω) and conductivity matrixσ(x, ω). These 3 × 3 matrices relate the vector fields The source of the fields is an externally impressed current density J e (x, ω). In the Fourier domain, Maxwell's equations then read as Let us now look at the situation where two different externally impressed current densities J e and J e are placed on the same material distribution, as shown in Fig. 1. The current density J e will cause fields E and H and the current density J e will cause fields E and H. We can relate these quantities using the a) following expression, which holds for two general vector fields F and G: We can therefore write Also, for any two vectors F and G it holds that By subtracting the two expressions from above and assumingε,μ andσ to be symmetric we get Integrating over a volume V enclosed by surface A and applying Gauss' theorem we have If the sources are compact i.e. all sources are contained in a finite region of space, the left hand side evaluates to zero. This is true even in the presence of electromagnetic radiation propagating towards spatial infinity [13]. We thus get the Lorentz reciprocity theorem: It is important to appreciate that J e (x, ω) refers to the 'externally impressed' currents. In case the conductivity matrixσ(x) is different from zero there will be additional currents according to J =σE. The above relation holds only for J e = 0 and J e = 0. If one of the two currents is equal to zero the relation simplifies to 0 = 0 and has therefore no useful application.

Network reciprocity a)
Z8(ω) The fact that the Lorentz reciprocity theorem holds for arbitraryε(x, ω),μ(x, ω),σ(x, ω) allows us to assume that a general linear reactive network made from discrete elements and connected with wires is just a specific realization of these material distributions, as shown in Fig. 2. In Fig. 2a) we connect external localized 'point-current' sources I n (ω) to the system. The resulting electric fields between the two open terminals are related to the measured voltages V n (ω) = E n (ω)ds. In the second situation in Fig. 2b) we use a different set of currents I n (ω) resulting in voltages V n (ω). Using the reciprocity theorem with the current sources J n (x, ω) = I n (ω)δ(x − x n )ds etc. and resulting voltages V n (ω) = E(x, ω)ds we find This expression is the counterpart of the electrostatic reciprocity theorem, where different sets of charges and voltages on electrodes are related by We now allow only a single current source to deliver a nonvanishing current I 1 (ω), while all other sources are switched off. Likewise, we place only a single current I 10 (ω) and no currents across the other sources. We then get This is the network reciprocity theorem: The voltage across an impedance element Z m (ω) due to a current I(ω) on a different element Z n (ω) is equal to the voltage across Z n (ω) for the same current I(ω) on Z m (ω).

Antenna reciprocity
a)   3 shows an arrangement of three different antennas, driven by current sources I n (ω). These currents result in voltages V n (ω) across the terminals of the antennas, while the currents I n (ω) generate a different set of voltages V n (ω). As before, the Lorentz reciprocity theorem relates these two states and we have If we again assume just the first antenna to be driven by a current I 1 (ω) = I(ω), and in the second state we assume the second antenna to be driven by the same current I 2 (ω) = I(ω), we find that V 2 (ω) = V 1 (ω). Since this relationship holds for arbitrary antenna geometries and arbitrary relative orientations, we deduce the remarkable result that the reception and transmission characteristics of each antenna must be identical. This is called the antenna reciprocity theorem.

Moving point charge, signal theorem
x 1 x 2 Figure 4: a) A moving point charge is creating an electric field and therefore a 'potential difference' between the points x 1 to producing an electric field Ew(x, t), the so called 'weighting field'.
We now turn to the signal created by the movement of a point charge q as shown in Fig 4. Electrodes, wires and other detector elements can be absorbed into the material parametersε(x),μ(x),σ(x). Let us assume that the point charge q moves along an arbitrary trajectory x 0 (t), as shown in Fig 4a). The current density created by this motion is given by We also assume that the detector setup implied by the material distribution delivers its readout signal at the positions x 1 and x 2 . The integral of the electric field along a particular path x s (s) connecting x 1 and x 2 is a quantity related to a potential difference V ind (ω) between these two points, and therefore to our detector signal: In general V (ω) is not independent of the path between x 1 and x 2 because ∇ × E = 0. For the second situation, shown in Fig 4b), we remove the point charge and place a line current I 0 (ω) between the detector terminals at x 1 and x 2 along the chosen path x s (s). This current will create the weighting field E w as solution to Maxwell's equations. Inserting the above expressions into the reciprocity relation we have If we assume the current I 0 to be independent of the frequency ω, it corresponds to a delta-like current I 0 (t) = Q 0 δ(t) in the time domain. We can then perform the inverse Fourier transform of the above expression and are left with This is our desired theorem which has the same form as the one given in [4] and [8], but is now shown to hold for the full extent of Maxwell's equations and is therefore applicable to a very wide range of detector types using signals induced on electrodes by the movement of charges: The potential difference V ind (t) induced between two points x 1 and x 2 by the movement of a point charge q along a trajectory x 0 (t) can be calculated in the following way: the charge q is removed and a delta current pulse Q 0 δ(t) is placed along a path between x 1 and x 2 . The response to this current is the electric weighting field E w (x, t). The detected voltage signal can be calculated by convolving the weighting field with the velocity of the particle according to Eq. 19.
In case we are not dealing with a single charge q that is moving along a trajectory x 0 (t) for −∞ < t < ∞, but we are assuming two charges q, −q that are created at a single point at t = 0 and then move from there along trajectories x 1 (t) and x 2 (t), the signal is given by We are now in the position to answer the question about signals in transmission lines that was outlined in the introduction. Let us assume a coaxial transmission line with wire radius a and tube radius R as shown in Fig. 5a). This setup resembles a wire chamber drift tube. We want to calculate the voltage generated at the amplifier input by a charge moving in radial direction at position z = z 0 . We assume that only TEM field modes are propagating along the transmission line, which is true for frequencies lower than c/R. In this case there are no electric field components along the z-direction. The potential difference between the conductors can then be defined in a unique way by V (z, t) = R a E(r, z, t)dr along any path in the x−y plane [14]. We therefore consider the situation in Fig. 5b) and place a delta current source on the electrode to determine the weighting field E w (r, z, t) inside the transmission line. The TEM approximation allows this electric field to be written as

Signals in transmission lines
The voltage V (z, t) is determined by the transmission line equations for the given stimulus and the potential ϕ(x, y) corresponds to the two dimensional static weighting potential of the central conductor.
Through Eq. 19 we have The current I 0 (t) corresponds to the induced current on the grounded electrode calculated with the Ramo-Shockley theorem. Writing this expression in the frequency domain gives The impedance Z(ω) relates the voltage V ind (ω) to the current I 0 (ω) at position z 0 . Due to the network reciprocity from Section 2 we have therefore proven the following theorem: The voltage induced on a transmission line by a charge moving in the x−y plane at position z 0 can be calculated by first calculating the induced current on the grounded electrode with the electrostatic two dimensional weighting field and then placing this current as an ideal current source one the transmission line at position z 0 .
In case the charge movement also has a z−component, it is more practical to first calculate the voltage signal V (z, t) and then to perform the convolution We now move the point x 1 in Fig. 4a) very close to x 2 and realize a dipole antenna by connecting two metal cylinders to these points, as shown in Fig. 6a). The signal corresponding to the voltage between the points x 1 and x 2 is calculated from the field as V ind (ω) = E 1 (x 1 , ω)ds. As an example, we use this antenna to detect the radiation from an electron that is gyrating in a static homogeneous magnetic field. Following our signal theorem, this voltage signal can be calculated by using the situation in Fig. 6b). An ideal current source is connected to the two terminals of the antenna and a current pulse I(t) = Q 0 δ(t) is applied. The weighting field E w is then given as the solution to Maxwell's equations arising from this current distribution. We assume here that the antenna is short, which allows us to treat it as an infinitesimal dipole. In practise this means that we are dealing with wavelengths that are much longer than the size of the antenna. We orient the antenna along the z-direction and assume the entire space to be filled with a homogeneous and isotropic medium. The weighting field in spherical coordinates then takes the form and where Θ(x) is the Heaviside step function and δ (x) is the distributional derivative of the Dirac delta distribution. The propagation speed of the shock front is given by c = 1/ √ µ 0 ε 0 . Note that for t > r c the weighting field corresponds to the field generated by a static electric dipole with dipole moment p = Q 0 ds situated at the origin.

Synchrotron radiation from electrons in a B-field
We now use the dipole antenna to detect the synchrotron radiation emitted by an electron moving in the galactic magnetic field, which we take to be oriented along the x-direction. We assume that the electron moves with velocity v in the negative x-direction. Its trajectory and velocity vector are given by where the frequency ω 0 of its gyrating motion is given by the cyclotron frequency ω 0 = e 0 B/m e . Since the distance r to the electron is large, we only keep the 1/r term of the weighting field, and along the x−direction we have For t < 0, when the electron is moving towards the antenna, we have with ω = ω 0 /(1 − β). For galactic magnetic fields on the order of 1 nT, the frequency ω 0 is only 176 Hz. For electrons with a kinetic energy E the observed frequency is larger by a factor 1/(1−β) ≈ 2(E/m e c 2 ) 2 .
For an electron with a kinetic energy of 5 GeV, the frequency is increased by a factor 2×10 6 so we measure radio waves of around 352 MHz. The same expression describes the synchrotron radiation emitted by an electron beam that is passed through an undulator with a wavelength λ 0 and where the emitted radiation has a wavelength of λ = λ 0 (1 − β).
The infinitesimal electric dipole antenna at the origin, pointing in z-direction, measures the z-component of the electric field produced by a moving particle. This can be formally shown by calculating the voltage V (t) for a general movement of the charge q along a trajectory r(t) and by comparing it to the electric field due to the same charge movement from the Lienard-Wiechert potentials at the place of the antenna. This is shown in Appendix A. a) x z dA V ind (t) The signal is bipolar and symmetric around the origin. It has two peaks at and the peak value of the signal is For relativistic particles, the peak value of the signal is proportional to γ 2 of the particle and inversely proportional to the third power of the distance x 0 . The time of the signal peak t peak is proportional to the distance x 0 and inversely proportional to γ of the particle. The signal, shown in Fig. 8, has a universal shape given by If we now assume the beam current transformer to be realized by N such loops around the particle beam (i.e. N windings), the signal V ind just has to be multiplied by N . In case the windings are applied on a ferrite core, the signal must also be multiplied by the permeability µ of the ferrite. It has to be noted that we did not account for the propagation delay of the signal between the different windings. If we want to take the propagation delay and a possible difference in distance of the particle to the different loops into account, we have to sum the signals with the proper delays instead of just multiplying V ind (t) by N .
The infinitesimal magnetic dipole antenna at the origin measures the rate of change of the magnetic flux produced by the moving particle. Like for the electric dipole this can be formally shown by calculating the voltage V ind (t) for a general movement of the charge q along a trajectory r(t) and then comparing it to the magnetic field due to the same charge movement from the Lienard-Wiechert potentials at the place of the antenna. This is shown in Appendix B.

Conclusion
We have presented a theorem that allows the calculation of signals generated by moving charges and is valid for the full extent of Maxwell's equations. While the Ramo-Shockley theorem and extensions based on quasi-static approximations are only applicable to traditional particle detectors where the velocity of the charge movement is much smaller that the speed of light, the present theorem is not restricted to these cases. It therefore allows the calculation of signals in detectors where signal propagation times and radiation effects are not negligible, like transmission lines and antennas.
As examples, the signals generated my moving charges in transmission lines and on electric as well as magnetic dipole antennas were presented. The theorem might have important practical applications when performing numerical simulations of signals in particle detectors and on antennas. Instead of explicitly computing the radiation pattern from the charge movement and then calculating the response of a detector to this radiation, one just needs to calculate the weighting field E w (x, t) once and for all. One can then derive the detector response to the motion of an arbitrary collection of charges simply by performing the convolution according to Eq. 19.

Electric field for a moving point charge from the Lienard-Wiechert potentials
We want to find the electric field for a point charge q moving along trajectory r(t) = (x(t), y(t), z(t)). The scalar potential ϕ(x, t) and vector potential A(x, t) are given by (Lienard-Wiechert potentials) The electric field is therefore The z-component at the origin is given by with r(t ) given by |r(t )| = x(t) 2 + y(t) 2 + z(t) 2 .
we have we find the signal which is identical to the result from the Lienard-Wiechert potentials.