A parallel-plate capacitor with circular plates of radius R is being charged as in Figure (a). Derive an expression for the current density of the displacement current J, forrs R. State your answers in terms of ɛ0, E, and t. • 2 •2 E X x X x B B. X X X X X x x x X x XX B X X X R X XX B. (a) (b) x X X x/X +
A parallel-plate capacitor with circular plates of radius R is being charged as in Figure (a). Derive an expression for the current density of the displacement current J, forrs R. State your answers in terms of ɛ0, E, and t. • 2 •2 E X x X x B B. X X X X X x x x X x XX B X X X R X XX B. (a) (b) x X X x/X +
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Chapter1: Units, Trigonometry. And Vectors
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![### Chapter 32, Problem 014
**Status:** Incorrect.
**Problem Statement:**
A parallel-plate capacitor with circular plates of radius \( R \) is being charged as in Figure (a). Derive an expression for the current density of the displacement current \( J_d \) for \( r \leq R \). State your answers in terms of \( \varepsilon_0 \), \( E \), and \( t \).
**Explanation and Diagrams:**
**Figure (a)** shows a parallel-plate capacitor with circular plates. The electric field (\(\mathbf{E}\)) is directed between the two plates, and the current (\(i\)) is flowing towards the plates as shown. The plates are separated by a distance, and the electric field lines are perpendicular to the surface of the plates.
**Figure (b)** shows a top view of the capacitor where the region between the plates is depicted. The circular area has a radius \(R\), and magnetic fields (\(\mathbf{B}\)) are present. The magnetic field lines (\(\mathbf{B}\)) are shown to circulate around the path within the capacitor. The direction of these fields is indicated by the right-hand rule, with the cross (\(\times\)) symbols indicating the direction of the magnetic field into the page and dots (\(\cdot\)) indicating the magnetic field out of the page.
**Mathematical Notation:**
The relationship for the displacement current density (\(J_d\)) is given by:
\[ J_d = \varepsilon_0 \left( \frac{dE}{dt} \right) \]
Where:
- \( \varepsilon_0 \) is the permittivity of free space.
- \( E \) is the electric field.
- \( t \) is time.
- \( \frac{dE}{dt} \) is the rate of change of the electric field with respect to time.
This expression is encapsulated within a red-bordered box in the problem statement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3861071f-90cb-4097-84b9-a523b25fee85%2Fbd04b5a6-6d85-4ade-b287-ee3065cd15b6%2F2c7kmv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Chapter 32, Problem 014
**Status:** Incorrect.
**Problem Statement:**
A parallel-plate capacitor with circular plates of radius \( R \) is being charged as in Figure (a). Derive an expression for the current density of the displacement current \( J_d \) for \( r \leq R \). State your answers in terms of \( \varepsilon_0 \), \( E \), and \( t \).
**Explanation and Diagrams:**
**Figure (a)** shows a parallel-plate capacitor with circular plates. The electric field (\(\mathbf{E}\)) is directed between the two plates, and the current (\(i\)) is flowing towards the plates as shown. The plates are separated by a distance, and the electric field lines are perpendicular to the surface of the plates.
**Figure (b)** shows a top view of the capacitor where the region between the plates is depicted. The circular area has a radius \(R\), and magnetic fields (\(\mathbf{B}\)) are present. The magnetic field lines (\(\mathbf{B}\)) are shown to circulate around the path within the capacitor. The direction of these fields is indicated by the right-hand rule, with the cross (\(\times\)) symbols indicating the direction of the magnetic field into the page and dots (\(\cdot\)) indicating the magnetic field out of the page.
**Mathematical Notation:**
The relationship for the displacement current density (\(J_d\)) is given by:
\[ J_d = \varepsilon_0 \left( \frac{dE}{dt} \right) \]
Where:
- \( \varepsilon_0 \) is the permittivity of free space.
- \( E \) is the electric field.
- \( t \) is time.
- \( \frac{dE}{dt} \) is the rate of change of the electric field with respect to time.
This expression is encapsulated within a red-bordered box in the problem statement.
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