Theoretical Question 1
Theoretical Question 1:
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
d t q
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
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 .
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 .