## 2 Methods
The second term \(I^{*}U^{*}\) is the power to produce excited neutrals; \(I^{*}\) represents the neutral excitation rate, and \(U^{*}\) represents the average excitation potential. The other terms of the equation represent the power lost to the electrodes; at walls, ions recombine and electrons are lost. \(I_{s}\) represents the ion current to the screen grid, \(I_{k}\) the ion current back to the cathode, \(I_{ia}\) the ion current lost to the anode, \(I_{a}\) the electron current from the plasma going to the anode, \(\epsilon_{e}\) the plasma electron energy lost to the wall, and \(I_{L}\) the primary electron current lost directly to the anode. Figure 5 shows a representation of the currents and potentials in the discharge chamber, defined above. For what concerns the current balances, Equations (12)-(15) have been exploited. Figure 3:6  
Figure 3:7  
Figure 3:8 Currents and potential distribution in the discharge chamber of a gridded ion thruster. Figure 3:9  
Figure 2:0  
Equation (12) expresses the conservation of the particles flowing to the anode, Equation (13) the current emitted from the hollow cathode, and Equations (14) and (15) the ion production rate. In this tool, the term \(I_{k}\) has been neglected for simplicity, due to the negligible values it generally assumes. Due to the number of equations and parameters needed to express all of the currents of the above expressions, it was decided, in this manuscript, to show only the terms in which the magnetic field is involved; more details about each of the parameters and related equations can be found in Ref. [13]. The currents \(I_{L}\), \(I_{a}\), and \(I_{ia}\), described above, are influenced by the magnetic field in the discharge chamber. In Equation (16), the primary electron current lost to the anode wall \(I_{L}\) is shown. Figure 2:1  
In Equation (16), \(n_{p}\) represents the primary-electron density, \(v_{p}\) the primary-electron velocity, and \(A_{p}\) the loss area that can be re-written as a function of the Larmor radius \(r_{p}\) and the total length of the magnetic cusps \(I_{c}\) (Equation (17))  
Figure 2:2  
This loss, due to the magnetic field configuration, occurs at the cusps of the magnetic field because in these locations the magnetic field lines are perpendicular to the anode wall. The value of the magnetic field strength at the cusps on the anode wall \(B_{cusp}\) influences the Larmor radius \(r_{p}\) as shown In Equation (18) and, therefore, the loss area. In this equation, \(m\) is the electron mass. Figure 2:3  
In Equation (19), the secondary electron current (plasma's electrons) lost to the anode wall at the magnetic cusps \(I_{a}\) is shown. Figure 2:4  
In Equation (19), \(k\) represents the Boltzmann's constant; \(T_{e}\) the electron temperature; \(n_{e}\) the electron density in the plasma; and \(A_{a}\) the hybrid anode area, a function of the ions and plasma electrons Larmor radius (\(r_{i}\) and \(r_{e}\)), due to the influence of ions in the electron motion, as shown In Equation (20). Figure 2:5  
As for the primary electrons Larmor radius \(r_{p}\), the Larmor radii \(r_{i}\) and \(r_{e}\) depend on the magnetic field strength at the cusps on the anode wall \(B_{cusp}\). For what concerns the ion current to the anode wall \(I_{ia}\), it can be expressed as shown in Equation (21). Figure 2:6  
In Equation (21), \(A_{as}\) represents the anode area exposed to plasma; \(n_{i}\) the ion density; and \(f_{c}\) is the confinement factor, a scaling factor related to the presence of magnetized electrons that influence the ion motion, causing a reduction in the Bohm current to the wall. Without a magnetic field, the ion current that flows to the anode is the Bohm current; if a magnetic field is introduced in the chamber, only a fraction of the Bohm ion current is lost to the anode. The main magnetic parameter that influences this factor is the maximum transverse magnetic field between the cusps, as shown in Equation (22).