Chapter 30, Problem 33P

(II) In some experiments, short distances are measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. (a) If charge is found to oscillate in this circuit at frequency f = ω/2π when the capacitor plates are separated by distance x, show that x = 4π²A∈₀f²L. (b) When the plate separation is changed by Δx, the circuit’s oscillation frequency will change by Δf. Show that Δx/x ≈ 2(Δf/f). (c) If f is on the order of 1 MHz and can be measured to a precision of Δf = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitor’s edges can be neglected.