**Problem Statement:**Lightning strikes a dielectric sphere of radius 20 mm for which \( \varepsilon_r = 2.5 \), \( \sigma = 5 \times 10^{-6} \) S/m and deposits uniformly a charge of 1 C. Determine the initial volume charge density and the volume charge density 2 μs later.---**Explanation:**- **Dielectric Sphere Characteristics:** - Radius: 20 mm - Relative permittivity (\( \varepsilon_r \)): 2.5 - Conductivity (\( \sigma \)): \( 5 \times 10^{-6} \) S/m- **Charge Characteristics:** - Total charge deposited: 1 Coulomb - Initial volume charge density refers to the charge distributed within a given volume initially. - Volume charge density 2 microseconds later examines how the distribution changes with time due to the sphere's conductivity.To solve the problem:1. Calculate the initial volume charge density using the formula: \[ \rho_0 = \frac{Q}{\frac{4}{3} \pi r^3} \] where \( Q \) is the total charge and \( r \) is the radius of the sphere. 2. Find how the volume charge density changes over time taking into account the conductivity \( \sigma \), which affects how charge redistributes over time. This question might involve exponential decay governed by material properties.No diagrams or graphs are present in the image.

Given: In this question, the given details are, Lightning strikes a dielectric sphere of radius 20 mm for which εr=2.5 , σ=5×10-6S/m and deposits uniformly a charge of 1 C. To find the value of the initial volume charge density and the volume charge density 2 μs later.Solution: We want to find the value of the initial volume charge density and the volume charge density 2 μs later. Now, to find the relaxation time by using the formula, Tr=εσTr=εrεoσ Substitute the values in above equation, Tr=(2.5)10-936π5×10-6Tr=4.421×10-6Tr=4.421 μsHere, to find the initial volume charge density , by using the formula, ρvo=QVolume of the sphereρvo=Q4πr33ρvo=3Q4πr3 substitute the values in above equation, and we get, ρvo=(3)(1)4π(20×10-3)3ρvo=29.84×103ρvo=29.84 kC/m3 The volume charge density is, ρv=ρvoe-tTrNow, putting all the known values in above equation, ρv=29.84×103e-t4.421×10-6ρv=29.84e-2.262×105t×103ρv=29.84e-2.262×105t kC/m3 We want to find the volume charge density 2 μs later, Therefore, the value of t is 2×10-6s, ρv=29.84e-2.262×105(2×10-6) ×103ρv=18.98×103ρv=18.98 kC/m3Final answer: The initial volume charge density is ρvo=29.84 kC/m3 And the volume charge density 2 μs later is ρv=18.98 kC/m3

Engineering

Electrical Engineering

Lightning strikes a dielectric sphere of radius 20 mm for which &, = 2.5, σ = 5 x 10-6S/m and deposits uniformly a charge of 1 C. Determine the initial volume charge density and the volume charge density 2 us later.

Lightning strikes a dielectric sphere of radius 20 mm for which &, = 2.5, σ = 5 x 10-6S/m and deposits uniformly a charge of 1 C. Determine the initial volume charge density and the volume charge density 2 us later.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Concept explainers

Electric field

A physical field present around an electrically charged particle that exerts a force on all other charged particles is called an electric field. The charged particles present in the field either repel or attract. From electric charges or time-varying magnetic fields, the electric field originates. One of the manifestations of electromagnetic force which is one of the four fundamental forces present in nature are the electric fields and magnetic fields.

Question

**Problem Statement:**

Lightning strikes a dielectric sphere of radius 20 mm for which \( \varepsilon_r = 2.5 \), \( \sigma = 5 \times 10^{-6} \) S/m and deposits uniformly a charge of 1 C. Determine the initial volume charge density and the volume charge density 2 μs later.

---

**Explanation:**

- **Dielectric Sphere Characteristics:**
- Radius: 20 mm
- Relative permittivity (\( \varepsilon_r \)): 2.5
- Conductivity (\( \sigma \)): \( 5 \times 10^{-6} \) S/m

- **Charge Characteristics:**
- Total charge deposited: 1 Coulomb
- Initial volume charge density refers to the charge distributed within a given volume initially.
- Volume charge density 2 microseconds later examines how the distribution changes with time due to the sphere's conductivity.

To solve the problem:
1. Calculate the initial volume charge density using the formula:
\[
\rho_0 = \frac{Q}{\frac{4}{3} \pi r^3}
\]
where \( Q \) is the total charge and \( r \) is the radius of the sphere.

2. Find how the volume charge density changes over time taking into account the conductivity \( \sigma \), which affects how charge redistributes over time. This question might involve exponential decay governed by material properties.

No diagrams or graphs are present in the image.

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