(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.
It is desired to tune the natural frequency of an LC circuit to match the 91.3-MHz broadcastsignal of an FM radio station. If a capacitor of 2.45 * 10-12 F is to be used in the circuit,what inductance is required?
A transmission line whosecharacteristic impedance is 500,find that the speed of the wavetransmitted by this line is equal tothe speed of light in a vacuum, findboth the inductance per unit lengthand the amplitude per unit length ofthis line
When you connect two loudspeakers to the output of a stereoamplifier, should you be sure the lead-in wires are equal inlength to avoid a time lag between speakers? Explain.
Chapter 30 Solutions
Physics for Scientists and Engineers, Vol 1 (Chapters 1-20)
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