Minimum Gap of a Movable Plate Against Non-Linear Electrostatic Force

Problem Statement

On a smooth horizontal insulating floor, two thin metal plates A and B of area $S$S are placed facing each other vertically. Plate A is fixed to the floor. Plate B has mass $m$m and is attached to one end of a light insulating spring (spring constant $k$k) laid horizontally on the floor; the other end of the spring is fixed to a wall. Plate B can slide smoothly and without friction in the horizontal direction along the spring axis, always remaining parallel to plate A. When the spring is at its natural length, the distance between plates A and B is $d_0$d0​.

The permittivity of the medium between the plates is $\varepsilon$ε. Edge effects are negligible.

Plates A and B are connected in series with a switch and an ideal DC power supply of EMF $V$V. The electrical resistance and self-inductance of the circuit are negligibly small, and no electromagnetic radiation or Joule heat is generated due to plate motion.

As the initial condition, plate B is held at the natural length position (plate separation $d_0$d0​) with the switch closed for a sufficiently long time.

Then, with the switch still closed, the hand is gently released from plate B. Plate B begins to move toward plate A due to the electrostatic attractive force. Let $d_{min}$dmin​ be the plate separation when plate B is closest to plate A.

Find the value of $d_{min}$dmin​.

Constraints

• Initial plate separation: $d_0 = 1.7 \times 10^{-2} \text{ m}$d0​=1.7×10−2 m
• Spring constant: $k = 1.0 \times 10^1 \text{ N/m}$k=1.0×101 N/m
• Permittivity: $\varepsilon = 8.5 \times 10^{-12} \text{ F/m}$ε=8.5×10−12 F/m
• Plate area: $S = 2.0 \text{ m}^2$S=2.0 m2
• EMF of DC supply: $V = 4.0 \times 10^2 \text{ V}$V=4.0×102 V
• Mass of plate B: $m = 5.0 \times 10^{-1} \text{ kg}$m=5.0×10−1 kg

Input Format

Find $d_{min}$dmin​ in millimeters ($\text{mm}$mm) and give the numerical value as a positive integer.