(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 /2 π when the capacitor plates are separated by distance x , show that x = 4 π 2 A ∈ 0 f 2 L . ( b ) When the plate separation is changed by Δ x , the circuit’s oscillation frequency will change by Δ f . Show that Δ x / x ≈ 2(Δ f / f ). ( c ) If f is on the order of 1 MHz and can be measured to a precision of Δ f = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitor’s edges can be neglected.
(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 /2 π when the capacitor plates are separated by distance x , show that x = 4 π 2 A ∈ 0 f 2 L . ( b ) When the plate separation is changed by Δ x , the circuit’s oscillation frequency will change by Δ f . Show that Δ x / x ≈ 2(Δ f / f ). ( c ) If f is on the order of 1 MHz and can be measured to a precision of Δ f = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitor’s edges can be neglected.
(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.
ii) A 800 2 resistor, a 0.05265 H inductor, and 301 µF capacitor are connected in series
with an AC source. If the capacitive reactance is cqual to the inductive reactance of
the circuit, estimate:
(a) the frequency of the source and
|(b) the impedance of the circuit.
(I) What is the reactance of a 6.20μ F capacitor at a frequency of (a) 60.0 Hz, (b) 1.00 MHz?
ii) A 75 Q resistor, a 3.6 H inductor, and 20 µF capacitor are connected in series with an
A.C. source of effective voltage 320 V, 90 Hz. estimate:
(a) impedance,
(b) observed current and
(c) power dissipated in the circuit.
Chapter 30 Solutions
Physics for Scientists and Engineers with Modern Physics
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.