Electric Field in a Double Layer and the Imparted Momentum A. Fruchtman Holon Academic Institute of Technology, 52 Golomb Street, Holon 58102, Israel (Received 27 September 2005; published 14 February 2006) It is shown that the net momentum delivered by the large electric field inside a one-dimensional double layer is zero. This is demonstrated through an analysis of the momentum balance in the double layer at the boundary between the ionosphere and the aurora cavity. For the recently observed double layer in a current-free plasma expanding along a divergent magnetic field, an analysis of the evolution of the radially averaged variables shows that the increase of plasma thrust results from the magnetic-field pressure balancing the plasma pressure in the direction of acceleration, rather than from electrostatic pressure. DOI: 10.1103/PhysRevLett.96.065002 PACS numbers: 52.30.q, 52.75.Di, 95.30.Qd, 96.50.Ci The strong electric fields confined to narrow isolated regions in plasmas, called double layers (DL) [1], have been suggested to accelerate particles in the aurora [2], cosmic rays [3], laser-ablated plasmas [4], laboratory ex- periments [5], and recently in gas discharges [6 – 9]. In this Letter we show that the net force and momentum imparted by these strong electric fields in a one-dimensional (1D) DL are identically zero. We demonstrate this often- overlooked vanishing of force and momentum by analyz- ing the DL located at the boundary between the ionosphere and the aurora cavity [10]. For the DL recently observed in the common configuration of an axially current-free plasma expanding along a divergent magnetic field [6– 8], we show that the increase of plasma thrust results from the magnetic field pressure that balances the plasma pressure in the direction of acceleration, rather than from electrostatic pressure. This unfolding of the mechanism of DL acceleration is important for the development of accel- erators and thrusters and for understanding many phe- nomena in space [11]. In general, an insight into momentum balance is important since evidence of a DL existence in distant astrophysical objects must rely on its global aspects [1,12]. The contribution of the electric field to the momentum is deduced from application of the divergence theorem to the momentum equations. If the electomagnetic pressure ten- sor elements are zero at the domain boundaries, the elec- tromagnetic fields do not affect the total mechanical momentum. In a 1D DL this statement is exhibited very clearly. Multiplying the two sides of the equation that expresses Gauss law by the electric field, we obtain 0 ~ r ~ E ~ E ~ E, where ~ E is the electric field, the net charge density, and 0 the permittivity of free space. If all varia- bles depend on z only, the integral form of this equation becomes 0 2 Ez 2 2 Ez 1 2 Z z 2 z 1 Edz: (1) The total electric force equals the difference between the electrostatic pressures on the two boundaries of the domain (z 1 and z 2 ). In a DL the electric field is zero at the two boundaries. Thus, the total force exerted and the momen- tum delivered by the electric field of a 1D DL are zero. As a concrete example we construct a collisionless- plasma DL configuration in which, for simplicity, the trapped (reflected) particle populations have water-bag distributions [13], while the free particle populations are two cold counter-propagating ion and electron beams. We assume the electric potential z at the two sides of the DL to be z 1 0 and z 1 0 ( 0 > 0). Poisson’s equation is written as 0 e d 2 dz 2 n fi 1 e=" fi p n fe 1 e 0 =" fe p n ti 1 e 0 " ti s n te 1 e " te s ; (2) where the first two terms on the right-hand side (RHS) of the equation represent the densities of the ion and electron beams, while the third and fourth terms represent the densities of the trapped ion and electron populations. The maximal densities of the four particle groups are n fi (free ions), n fe (free electrons), n ti (trapped ions), and n te (trapped electrons). The kinetic energies of the beam par- ticles moving towards the DL are " fi (ions) and " fe (elec- trons) and the maximal energies of the trapped ions and electrons are " ti and " te (both are smaller than e 0 , e being the elementary charge). Upon integrating Eq. (2) from z 1 we obtain an equation [which is Eq. (1) for this case] that expresses the momentum balance: 0 2 d dz 2 2n fi " fi 1 e " fi s 1 2n fe " fe 1 e 0 " fe s 1 e 0 " fe s 2n ti " ti 3 1 e 0 " ti 3=2 2n te " te 3 1 e " te 3=2 1 : (3) The first two terms on the RHS of the equation are the PRL 96, 065002 (2006) PHYSICAL REVIEW LETTERS week ending 17 FEBRUARY 2006 0031-9007= 06=96(6)=065002(4)$23.00 065002-1 2006 The American Physical Society