Theoretical Calculation and Application Test of Lift Force for Ideal Electric Asymmetric Capacitor

The asymmetric capacitor’s lift force formula can be obtained on the basis of literature review, which can almost cover all practical forms of asymmetric capacity forms. But there are still some problems we should solve. The ﬁrst and foremost one is whether the formulas are correct and can they be veriﬁed in engineering practices? On the contrary, the parameter q in the formulas is normally unknown in the beginning of calculations, how can we get or reckon up it so as to use the formulas smoothly? In this paper, we set out to solve these questions.


Introduction
How can we solve lift force produced by a lifter formed of an asymmetric capacitor? Based on some hypothetical conditions, a formula was obtained through three methods in an ideal scenario [1,2]. But an unknown parameter q is still contained so that numerical calculations are difficult to carry out. is paper intends to solve this problem by eliminating the unknown factor in the hope that the formula can be effortlessly put into practical application and engineering. Following that, experimental tests and practical estimations are provided to verify its validity.
In former papers [1,2], the same result is acquired through three ways using the following equation of electric lift force of asymmetric capacitor loaded by high voltage in ideal condition: An unknown variable q is still included in the above formula. In order to solve this problem thoroughly, the carried charge q should be figured out. Normally, carried charge q of capacitor is relevant to the voltage U and the capacitance C. e voltage U can be known. But the capacitance C is difficult to calculate when the capacitor is in irregular shape.
Nevertheless, the analysis of hypothetical predetermined conditions verifies that the capacitance C of the asymmetric capacity is calculable. When the small plate of asymmetric capacitor is in a slender cylinder form, its capacitance could be estimated at a cylindrical way. When the small plate of asymmetric capacitor [3][4][5] is in sphere form, its capacitance could be estimated at a spherical way. e result might not be ideal in the case of precision. It can still be applied to estimation in engineering assessment [6,7]. Furthermore, the subsequent test data verified that the estimate result was fairly accurate unexpectedly.

Theoretical Derivation
Regarding the reason why the experimental result is more precise than expected, the analysis of the unique characteristics of the asymmetric capacitor has presented several objective reasons: (1) the distance d between two plates is more larger than the dimension of surface area S 1 of plate 1 (small plate), that is, d ≫ S 1 /l or d ≫ �� S 1 ; (2) the area of plate 2 (large plate) is larger than that of plate 1, that is, S 2 ≫ S 1 ; and (3) the voltage loaded between two plates is below the breakdown voltage that is relevant to the gap distance.
Under the initial condition, we begin to deduce capacitance of the asymmetric capacitor [8][9][10] and then to estimate its lift force [11,12]. Deducing processes are as follows: (1) For S 2 ≫ S 1 , when high voltage is loaded on two plates, the electric field intensity on plate 1, E 1 � q/εS 1 , is larger than that on plate 2, E 2 � q/εS 2 , i.e., E 1 ≫ E 2 . It leads to the voltage drop ΔU � Δ d · E, which mainly centralizes around plate 1. So when calculating the capacitance d U , the field intensity near plate 1 should be taken into major consideration. at is to say, the capacitance calculation can be carried out by combining the following equations (2), (3), (4), (5), and (6): r ∝ S l (for thin wire and board plates), r ∝ � S √ (for sphere point and board plates).
Equations (5) or (6) can be also written as where r is the nominal dimensional size of plate 1, l is the length of plate 1, and k sh1 and k sh2 are the shape coefficient relevant to the plates' structure size.
(2) Because the distance d between two plates is far larger than the nominal size of plate 1 r, to simplify the calculation, we assume that surface charge of plate 1 is uniformly distributed, and voltage drop of thin wire plate or spherical capacitor plate is integrated for estimating the capacitance in magnitudes. e details are shown as follows. For a thin wire small plate capacitor, we can take where H 2 is the width of the board plate. Because referring to equation (3), we get Mainly considering the electric field variation beside the thin wire, we have Considering the effective fan-shaped part, we have Integrating both sides of equation (10), we obtain 2 Mathematical Problems in Engineering So we get the capacitance For a spherical small plate capacitor, we can take Combining equation (2), we get Referring to equation (3), we get Integrating both sides, we obtain For the distance d ≫ R 1 , we have So we get the capacitance (3) We can calculate the electric lift force of asymmetric capacitor loaded by high voltage with the capacitance C.
For thin wire small plate capacitor, using equation (14), is is the lift force formula about a normal lifter in thin wire asymmetric capacitor form under high voltage loaded.

Mathematical Problems in Engineering
Considering the condition S 2 ≫ S 1 , we have If simplifying calculation as a spherical plate, the surface area of plate 1 S 1 � 4πR 2 , we can get is is the concised formula that finally turned out, from which we can tell the maximum lift force produced by spherical asymmetric capacitor under high voltage loaded. e formulas are exerted on two applications to test their  validity. ey are, respectively, lift force estimation of a electricity lifter [13,14] and a high-voltage rising hair experiment [15,16]. On the initial conditions, using equation (21), we have

Lift Force
(25) at is to say, a lifter loaded with 30 kV voltage can produce a largest lift force of 11.7 gf.

Lift Force Estimation of High-Voltage Charged Conducting
Sphere. As we know, when a high voltage loads on human body, our hair may be lifted up by the static electricity [17]. But there is no precise data or concrete calculating method of the length of the lifted hair by the high voltage. By equation (24), the mentioned problem can be solved. We can use the formula to quantitatively calculate the hair length lifted by the static electric field. e details are shown as follows.

Target Problem.
When the voltage is loaded on the hair under the above initial conditions, what is the maximum length of the hair (l h � ?) that can be lifted up?

Solving Process.
In this case, the head and ground can be considered as the two plates of asymmetric, where the head may be regarded as a small plate and its area of sphere surface is S 1 and the distant ground as a large plate and its area of flat surface S 2 . However, S 2 ≫ S 1 , and the distance between the two plates d

Explanation and Conclusion
Based on some assumptions with simplified calculation, we derived lift force formula produced by an asymmetric capacitor in different conditions, with which the assess in certain survey and qualitative research can be undertaken in spite of unsatisfying precision. e method also provides a convenient way to calculate static electricity lift capacity produced by an asymmetric capacitor or lift force of lifters. It also contributes to the parameter optimization in designing [18] a larger load force of lifter formed by an asymmetric capacitor.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Mathematical Problems in Engineering 5