31. Alternating Current

Will a frequency f = 60 Hz or ω = 75 s⁻¹ produce a larger max current in an inductor connected to an AC source?

In some experiments, very tiny distances or spaces ( ≈ nm ) can be 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 = 4π² Aε₀f²L.

In some experiments, very tiny distances or spaces ( ≈ nm ) can be measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. (b) When the plate separation is changed by ∆x, the circuit’s oscillation frequency will change by ∆f. Show that ∆x/x ≈ 2(∆f/f).

In some experiments, very tiny distances or spaces ( ≈ nm ) can be measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. (c) If f is on the order of 1 MHz and can be measured to a precision of ∆f = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitor’s edges can be neglected.

(II) A 75-W incandescent lightbulb is designed to operate with an applied ac voltage of 120 V rms. The bulb is placed in series with an inductor L, and this series combination is then connected to a 60.0-Hz 240-V rms voltage source. For the bulb to operate properly, determine the required value for L. Assume the bulb has resistance R and negligible inductance.

At t = 0, the current through a 60.0-mH inductor is 50.0 mA and is increasing at the rate of 78.0 mA/s. What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 8.0 from the initial value?

(III) Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. (a) In the region between the plates, show that the magnitude of the induced magnetic field is given by B = B₀(r) cos (2𝝅ft), where B₀(r) is a function of the radial distance r from the capacitor’s central axis.

(III) Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. (b) Determine the expression for the amplitude B₀(r) of this time-dependent (sinusoidal) field when r ≤ r₀ and when r > r₀.

(III) Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. (c) Plot B₀(r) in tesla for the range 0 ≤ r ≤ 10 cm.