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The potential distribution in a dielectric material (ɛ and P at point (-2, 5, 3). 8ɛ,) is V = 4x yz'V. Find V, E, ||

The potential distribution in a dielectric material (ɛ and P at point (-2, 5, 3). 8ɛ,) is V = 4x yz'V. Find V, E, ||

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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The potential distribution in a dielectric material \((\varepsilon = 8\varepsilon_0)\) is given by \(V = 4x^2yz^3\) V. Find \(V\), \(E\), and \(P\) at the point \((-2, 5, 3)\).
Transcribed Image Text:The potential distribution in a dielectric material \((\varepsilon = 8\varepsilon_0)\) is given by \(V = 4x^2yz^3\) V. Find \(V\), \(E\), and \(P\) at the point \((-2, 5, 3)\).
**Electromagnetic Fields in Dielectric Materials** In a dielectric material where the permittivity \(\varepsilon = 5\varepsilon_0\), the potential field is given by: \[ V = 10x^2yz - 5z^2 \, V. \] Determine the following: (a) **Electric Field** (\(\mathbf{E}\)): The electric field \(\mathbf{E}\) is related to the potential \(V\) by \(\mathbf{E} = -\nabla V\). (b) **Electric Displacement Field** (\(\mathbf{D}\)): This is defined as \(\mathbf{D} = \varepsilon \mathbf{E}\). (c) **Polarization** (\(\mathbf{P}\)): The polarization \(\mathbf{P}\) in a dielectric is given by \(\mathbf{P} = \mathbf{D} - \varepsilon_0 \mathbf{E}\). (d) **Volume Charge Density** (\(\rho_v\)): It can be determined from the relation \(\nabla \cdot \mathbf{D} = \rho_v\). This exercise involves applying the concepts of electromagnetic theory to calculate various electric properties within a dielectric medium. These calculations are crucial for understanding the behavior of materials in electric fields.
Transcribed Image Text:**Electromagnetic Fields in Dielectric Materials** In a dielectric material where the permittivity \(\varepsilon = 5\varepsilon_0\), the potential field is given by: \[ V = 10x^2yz - 5z^2 \, V. \] Determine the following: (a) **Electric Field** (\(\mathbf{E}\)): The electric field \(\mathbf{E}\) is related to the potential \(V\) by \(\mathbf{E} = -\nabla V\). (b) **Electric Displacement Field** (\(\mathbf{D}\)): This is defined as \(\mathbf{D} = \varepsilon \mathbf{E}\). (c) **Polarization** (\(\mathbf{P}\)): The polarization \(\mathbf{P}\) in a dielectric is given by \(\mathbf{P} = \mathbf{D} - \varepsilon_0 \mathbf{E}\). (d) **Volume Charge Density** (\(\rho_v\)): It can be determined from the relation \(\nabla \cdot \mathbf{D} = \rho_v\). This exercise involves applying the concepts of electromagnetic theory to calculate various electric properties within a dielectric medium. These calculations are crucial for understanding the behavior of materials in electric fields.
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