tceic.com

# 物理竞赛试题1 (20)

Theoretical Question 1

Page 1/2

Theoretical Question 1:

"Ping-Pong" Resistor
A capacitor consists of two circular parallel plates both with rad

ius R separated by distance d, where d << R , as shown in Fig. 1.1(a). The top plate is connected to a constant voltage source at a potential V while the bottom plate is grounded. Then a thin and small disk of mass m with radius r ( << R, d ) and thickness t ( << r ) is placed on the center of the bottom plate, as shown in Fig. 1.1(b). Let us assume that the space between the plates is in vacuum with the dielectric constant ε 0 ; the plates and the disk are made of perfect conductors; and all the electrostatic edge effects may be neglected. The inductance of the whole circuit and the relativistic effects can be safely disregarded. The image charge effect can also be neglected.
side view R d V

+V

d t q

r

mg

(a)

(b)

Figure 1.1 Schematic drawings of (a) a parallel plate capacitor connected to a constant voltage source and (b) a side view of the parallel plates with a small disk inserted inside the capacitor. (See text for details.) (a) [1.2 points] Calculate the electrostatic force Fp between the plates separated by d before inserting the disk in-between as shown in Fig. 1.1(a). (b) [0.8 points] When the disk is placed on the bottom plate, a charge q on the disk of Fig. 1.1(b) is related to the voltage V by q = χV . Find χ in terms of r , d , and ε 0 . (c) [0.5 points] The parallel plates lie perpendicular to a uniform gravitational field g . To lift up the disk at rest initially, we need to increase the applied voltage beyond a

Theoretical Question 1

Page 2/2

threshold voltage Vth . Obtain Vth in terms of m , g , d , and χ . (d) [2.3 points] When V > Vth , the disk makes an up-and-down motion between the plates. (Assume that the disk moves only vertically without any wobbling.) The collisions between the disk and the plates are inelastic with the restitution coefficient η ≡ ( v after / v before ) , where v before and v after are the speeds of the disk just before and after the collision respectively. The plates are stationarily fixed in position. The speed of the disk just after the collision at the bottom plate approaches a "steady-state speed" vs , which depends on V as follows:
v s = αV 2 + β .

(1.1)

Obtain the coefficients α and β in terms of m , g , χ , d , and η . Assume that the whole surface of the disk touches the plate evenly and simultaneously so that the complete charge exchange happens instantaneously at every collision. (e) [2.2 points] After reaching its steady state, the time-averaged current I through the capacitor plates can be approximated by I = γV 2 when qV >> mgd . Express the coefficient γ in terms of m , χ , d , and η . (f) [3 points] When the applied voltage V is decreased (extremely slowly), there exists a critical voltage Vc below which the charge will cease to flow. Find Vc and the corresponding current I c in terms of m , g , χ , d , and η . By comparing Vc with the lift-up threshold Vth discussed in (c), make a rough sketch of the I V characteristics when V is increased and decreased in the range from V = 0 to 3 Vth .

### 2007第20届北京市高一物理竞赛预赛试题(印刷版)

20 届北京市高一物理竞赛预赛试题(北京二中杯)(全卷满分 100 分) 三题分号数一二 20 21 22 23 总分 2007 年 5 月 阅卷人 一、选择题(共 33 分,每...

### 2007年上海市第二十一届初中物理竞赛初赛试题及答案_免...

2007年上海市第二十一届初中物理竞赛初赛试题及答案2007年上海市第二十一届初中物理竞赛初赛试题及答案隐藏>> 上海市第二十一届初中物理竞赛(大同中学杯) 初赛试卷...

### 2012年第二十二届全国初中应用物理知识竞赛试题及答案(...

2012 年第二十二届全国初中应用物理竞赛试题一、选择题: 1. 早晨, 小明骑着自行车沿平直的公路驶向学校, 强劲的北风迎面吹来, 此时地面对车前轮的摩擦力为 f1...