a capacitor with circular plates of radius R=18.0 cm is connected to a source of emf ℰ=ℰm sinωt

A capacitor with circular plates of radius R=18.0 cm is...

a capacitor with circular plates of radius $R=18.0 \mathrm{~cm}$ is connected to a source of emf $\mathscr{E}=\mathscr{E}_{m} \sin \omega t,$ where $\mathscr{E}_{m}=220 \mathrm{~V}$ and $\omega=130 \mathrm{rad} / \mathrm{s} .$ The maximum value of the displacement current is $i_{d}=7.60 \mu$ 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.

a capacitor with circular plates of radius $R=18.0 \mathrm{~cm}$ is connected to a source of emf $\mathscr{E}=\mathscr{E}_{m} \sin \omega t,$
where $\mathscr{E}_{m}=220 \mathrm{~V}$ and $\omega=130 \mathrm{rad} / \mathrm{s} .$ The maximum value of the displacement current is $i_{d}=7.60 \mu$ 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

Sinusoidally Time-Varying Fields

Sinusoidally time-varying fields are characterized by quantities that change with time in a sinusoidal manner, such as emf given by a sine function. This concept is essential for analyzing AC circuits, where peak values of current, voltage, and displacement current are determined by the amplitude and the angular frequency of the sinusoidal variation. It also affects the phase relations and maximum values of dynamic quantities in the circuit.

Magnetic Field in Regions with Time-Varying Electric Fields

When the electric field between the plates of a capacitor varies with time, it gives rise to a magnetic field in the surrounding space, even in the absence of conduction current in that region. The magnetic field can be evaluated using the Ampère–Maxwell law, and its magnitude at a given distance from the center is related to the displacement current and the geometry of the region considered.

Electric Flux in Capacitor Gaps

Electric flux is defined as the product of the electric field and the area over which the field is uniform. In capacitors, particularly in the region between the plates where the electric field is considered uniform (neglecting edge effects), the flux quantifies the number of field lines passing through the plate area. The time derivative of this flux is directly related to the displacement current.

Capacitor Geometry and Capacitance

The behavior of a capacitor, including its capacitance, is fundamentally tied to its physical geometry—the area of the plates and the separation between them. In parallel-plate capacitors, the capacitance is directly proportional to the area of the plates and inversely proportional to the separation. This relationship is crucial for understanding how the stored charge and electric field evolve, especially in time-dependent situations.

Ampère–Maxwell Law

The Ampère–Maxwell law is an extension of Ampère’s circuital law that incorporates the contribution of displacement current. It relates the magnetic field in a region to both conduction currents and the rate of change of the electric flux, allowing one to calculate the magnetic field generated in regions where a time-varying electric field exists, such as between the plates of a capacitor.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for the changing electric field in a capacitor when there is no conduction current between the plates. It represents the rate of change of electric flux through a surface and plays a crucial role in modifying Ampère’s law to include time?varying fields, thereby ensuring continuity of current in circuits with capacitors.

Transcript

00:01 For this problem on the topic of magnetism, we have a capacitor with circular plates of radius 18 centimeters connected to a source of emf, where the emf amplitude is 220 volts and the angle of frequency 130 radiance per second.

00:13 The maximum value of the displacement current is 7 .6 microampers, and we want to find the maximum value of the current in the circuit, the maximum value of the rate of change of electric flux through the region between the plates, the separation d between the plates, as well as the maximum value of the magnitude of the magnetic field between the plates at a distance of 11 centimeters from the center.

00:37 Now for a, at any instant, the displacement current id in the gap between the plates equals the conduction current in the wires.

00:45 And so the maximum current in the circuit imax is equal to the maximum displacement current id max, which is as given 7 .6 microampiers.

01:02 Now for part b, we know that the displacement current id is equal to the constant epsilon not times the rate of change of electric flux, defy, dt, which means that the maximum displacement or rather the maximum electric flux, d -fi -d -t -max, is equal to the maximum displacement current divided by epsilon -0.

01:32 If we put our values in, this is 7 .6 times 10 to the minus 6 amperes divided by 8 .85 times 10 to the minus 12 ferrets per meter.

01:50 This gives the maximum rate of change of electric flux between the plates to be 8 .59 times 10 to the power of 5 volt meters per second.

02:09 For part c, we let the plate area be a and a plate separation bd.

02:16 The displacement current is id, which is epsilon knot, d5, d t, which is, epsilon not times d by d t and we can write the electric flux as the area a times the electric field e and so this can be written as epsilon not a times d by d t of v over d which we can write as epsilon not a over the plate separation d times d v d t now the potential difference of the capacitor is the same in magnitude as the emf of the generator.

03:06 So we know that v is equal to the emf amplitude, epsilon m times the sine of omega -t, which means that dv -d -t is equal to omega -epsalon -m cosine -omegat -t.

03:27 And so the displacement current id is equal to epsilon -0, a omega -ephsoulon m over d times the cosine of omega -t and the maximum displacement current therefore by letting the cosine term go to one is epsilon not a omega -epsoulon -m over d.

03:55 That means that the plate separation d is equal to epsilon not a omega -epsalon -m divided by the maximum displacement current id max...

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