“…This attraction due to charge polarization is stronger for r=−13 (solid lines), where the smaller sphere is the more positive of the two. The behaviour of the force versus separation in figure 6 is qualitatively similar to that of equal-sized spheres with fixed charges [10]. Figure 6The dimensionless force, f Q , is plotted versus the relative surface separation, s − 1, for r=−13 (solid lines), i.e.…”
Section: Energy and Force At Constant Chargementioning
confidence: 55%
“…is stronger for r = − 1 3 (solid lines), where the smaller sphere is the more positive of the two. The behaviour of the force versus separation in figure 6 is qualitatively similar to that of equal-sized spheres with fixed charges [10].…”
Section: (C) Calculations Of Force F Qmentioning
confidence: 56%
“…This attraction at sufficiently small separation when v ≠ 1 is understood mathematically from equation (4.4)—the leading term is negative and eventually dominates all the other terms. The behaviour of the force versus separation in figure 2 is qualitatively similar to that of equal-sized spheres held at constant voltages [10,27].…”
Section: Energy and Force At Constant Voltagementioning
confidence: 74%
“…Lord Kelvin [1], Poisson et al [2] and Russell [3,4] are among those who have worked on this problem. Due to its non-trivial nature, the problem continues to generate interest to this date [5][6][7][8][9][10][11][12]. The simple and ubiquitous geometry of spheres makes the analytical results relevant in a wide variety of applications where electrostatic forces play an important role such as the interaction of raindrops in clouds [13], dust and powder interactions [14,15], cellular and molecular interactions [16], atomic force microscopy [17,18], etc.…”
We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres
increases
when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal
lowering
of the voltage on the larger sphere at an optimal sphere separation.
“…This attraction due to charge polarization is stronger for r=−13 (solid lines), where the smaller sphere is the more positive of the two. The behaviour of the force versus separation in figure 6 is qualitatively similar to that of equal-sized spheres with fixed charges [10]. Figure 6The dimensionless force, f Q , is plotted versus the relative surface separation, s − 1, for r=−13 (solid lines), i.e.…”
Section: Energy and Force At Constant Chargementioning
confidence: 55%
“…is stronger for r = − 1 3 (solid lines), where the smaller sphere is the more positive of the two. The behaviour of the force versus separation in figure 6 is qualitatively similar to that of equal-sized spheres with fixed charges [10].…”
Section: (C) Calculations Of Force F Qmentioning
confidence: 56%
“…This attraction at sufficiently small separation when v ≠ 1 is understood mathematically from equation (4.4)—the leading term is negative and eventually dominates all the other terms. The behaviour of the force versus separation in figure 2 is qualitatively similar to that of equal-sized spheres held at constant voltages [10,27].…”
Section: Energy and Force At Constant Voltagementioning
confidence: 74%
“…Lord Kelvin [1], Poisson et al [2] and Russell [3,4] are among those who have worked on this problem. Due to its non-trivial nature, the problem continues to generate interest to this date [5][6][7][8][9][10][11][12]. The simple and ubiquitous geometry of spheres makes the analytical results relevant in a wide variety of applications where electrostatic forces play an important role such as the interaction of raindrops in clouds [13], dust and powder interactions [14,15], cellular and molecular interactions [16], atomic force microscopy [17,18], etc.…”
We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres
increases
when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal
lowering
of the voltage on the larger sphere at an optimal sphere separation.
“…As to the key parameters of the capacitor, many researchers have researched the parallel plate capacitors with nonlinear electrostatic, nonlinear cantilever springs (hanger), and squeezed film damping things [20,21,[31][32][33][34][35]. Qiu et al discussed the softening effect of their capacitive device with a strong nonlinear component in the system [20].…”
Mechanical stoppers in MEMS capacitive systems can dramatically affect electrical performances and result in complicated mechanical dynamic responses. This paper introduces electromechanical coupling nonlinear dynamic responses in MEMS variable dual-capacitor with an effect of nonlinear and asymmetrical stoppers. We found that the capacitance in the electrical circuit system related to the first-order derivative of the output voltage on a load resistor, and the variable dual-capacitor was strongly affected by the coupling of up and down superposition instantaneous electrostatic force and limited space by the length of nonlinear stoppers. The numerical calculation results and the experimental results in our analysis based on our system had a good agreement, and the numerical simulation results presented rich nonlinear impacts dynamic responses through the imposed voltage and the height of stoppers in MEMS variable dual-capacitive device. The device in operation cannot reach the 0.6 time's initial gap due to small forcing amplitude (1.026 g). However, we observed that the movable plate and stoppers (across the 0.6 time's initial gap) had fierce impacts due to big forcing amplitude (4 g) on to the device. With asymmetric stopper each impact, we also concluded that the movable plate would experience attenuations of the displacement until the moment to the next impacts. Moreover, the height of stoppers can not only result in complicated dynamic motion of the movable plate, but also can modulate a voltage of the fixed plate with its asymmetry structure.
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