In Fig. 32-36, a capacitor with circular plates of radius R=18.0 cm is connected to a source of

In Fig. 32-36, a capacitor with circular plates of radius...

In Fig. $32-36,$ a capacitor with circular plates of radius $R=18.0 \mathrm{cm}$ is connected to a source of emf $8=8_{m}$ sin $\omega t,$ where $\mathscr{G}_{m}=220 \mathrm{V}$ and $\omega=130 \mathrm{rad} / \mathrm{s}$ . The maximum value of the displacement current is $i_{d}=7.60 \mu \mathrm{A}$ . Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current $i$ in the circuit? (b) What is the maximum value of $d \Phi_{E} / d t,$ where $\Phi_{E}$ is the electric flux through the region between the plates? (c) What is the separation $d$ between the plates? (d) Find the maximum value of the magnitude of $\vec{B}$ between the plates at a distance $r=11.0 \mathrm{cm}$ from the center.

In Fig. $32-36,$ a capacitor with circular plates of radius $R=18.0 \mathrm{cm}$ is connected to a source of emf $8=8_{m}$ sin $\omega t,$ where $\mathscr{G}_{m}=220 \mathrm{V}$
and $\omega=130 \mathrm{rad} / \mathrm{s}$ . The maximum value of the displacement current is $i_{d}=7.60 \mu \mathrm{A}$ . Neglect fringing of the electric field at the
edges of the plates. (a) What is the maximum value of the current $i$ in the circuit? (b) What is the maximum value of $d \Phi_{E} / d t,$ where $\Phi_{E}$
is the electric flux through the region between the plates? (c) What
is the separation $d$ between the plates? (d) Find the maximum
value of the magnitude of $\vec{B}$ between the plates at a distance
$r=11.0 \mathrm{cm}$ from the center.

Key Concepts

Alternating Current (AC) and Sinusoidal Waveforms

AC circuits, often driven by sinusoidally varying sources, exhibit time-dependent voltages and currents. Understanding the behavior of sinusoidal waveforms is vital when analyzing the temporal changes in electric fields and fluxes in a capacitor, which in turn affect the displacement current and the resulting magnetic fields.

Maxwell's Correction to Ampère's Law

Maxwell's correction to Ampère's Law incorporates the displacement current term, allowing the law to be applied to situations with time-varying electric fields. This extension ensures that a changing electric field produces a magnetic field in the same way that a conduction current does, maintaining consistency in electromagnetic theory.

Parallel Plate Capacitor Geometry

The geometry of a parallel plate capacitor, characterized by the area of the plates and the separation between them, determines its capacitance and the uniformity of the electric field between the plates. This geometric configuration is essential for calculating the electric field, potential difference, and, consequently, the displacement current in time-dependent scenarios.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for the changing electric field in regions where there is no conduction current, such as the gap between capacitor plates. It is defined as the rate of change of electric flux through a surface and plays a crucial role in ensuring the continuity of current in a circuit that includes capacitive elements.

Electric Flux and Its Time Derivative

Electric flux represents the amount of electric field passing through a given area and is integral to understanding the behavior of capacitors. The time derivative of the electric flux indicates how rapidly the electric field is changing over time, which is directly related to the displacement current in regions without free charge movement.

Transcript

00:01 In this problem, we have a capacitor with circular plates connected to an emf source.

00:06 And we have that the maximum value of the displacement current is 7 .6 micro -amperts.

00:14 And we neglect the fringing of the electric field at the edge of the plates.

00:19 We want to know what is the maximum value of the current in the circuit.

00:24 So at any instant, its placement current id in the gap between the plates is equal to the conductor conduction current i in the wires.

00:39 So we have id equals i.

00:43 So we have that the maximum value will be the maximum value of the current i max.

00:51 And this is 7 .6 microse.

00:55 For part b, we want to know what is the maximum value of the variation of the electric flux.

01:05 We can write the variation of the electric flux in terms of the induced current, displacement current.

01:13 So we have d5dt max is equal to imax divided by epsilon 0.

01:24 And we have the value here, so this is 7 .6, 10 to the minus 6, and the epsilon is 8 .85 times 10 to the minus 12.

01:35 And this gives us 8 .59 times 10 to the 5 volts, meters, divided by seconds.

01:47 Now for part c, you want to know what is the separation between the plates.

01:52 So we can find the separation between the plates because we know the variation of the potential...

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