**Problem: Electric Field and Displacement Current**A uniform electric field perpendicular to a spherical surface of radius 5.00 cm is increasing at a rate of 4.00 V/m·µs. Find the displacement current through the spherical surface.In this problem, we need to calculate the displacement current, which arises due to the changing electric field across the surface. The increasing electric field suggests that we can use the concept of displacement current density to find the total displacement current. **Key Concepts:**- **Electric Field (E):** A representation of the force experienced by a charge in a specific area, measured in volts per meter (V/m). - **Displacement Current (Id):** A term used in Maxwell’s equations representing the rate of change of electric displacement field across a region.**Formula to Use:**\[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \]Where:- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).- \( \frac{d\Phi_E}{dt} \) is the rate of change of electric flux.**Electric Flux (\( \Phi_E \)):**\[ \Phi_E = E \cdot A \]Where:- \( E \) is the uniform electric field.- \( A \) is the area of the spherical surface.**Steps to Solve:**1. Calculate the area of the spherical surface: \[ A = 4\pi r^2 \]2. Calculate the rate of change of electric flux: \[ \frac{d\Phi_E}{dt} = \frac{d}{dt}(E \cdot A) \]3. Plug the values into the displacement current formula to find \( I_d \).This approach will help you determine the displacement current through the spherical surface effectively.

From the question, it is given that- Radius of spherical surface: r = 5.00 cm = 5.00 x 10-2 m Rate…

Answered: A uniform electric field perpendicular…

A uniform electric field perpendicular to a spherical surface of radius 5.00 cm is increasing at a rate of 4.00 V/m•µs. Find the displacement current through the spherical surface.

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

Question

**Problem: Electric Field and Displacement Current**

A uniform electric field perpendicular to a spherical surface of radius 5.00 cm is increasing at a rate of 4.00 V/m·µs. Find the displacement current through the spherical surface.

In this problem, we need to calculate the displacement current, which arises due to the changing electric field across the surface. The increasing electric field suggests that we can use the concept of displacement current density to find the total displacement current.

**Key Concepts:**

- **Electric Field (E):** A representation of the force experienced by a charge in a specific area, measured in volts per meter (V/m).

- **Displacement Current (Id):** A term used in Maxwell’s equations representing the rate of change of electric displacement field across a region.

**Formula to Use:**
\[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \]

Where:
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).
- \( \frac{d\Phi_E}{dt} \) is the rate of change of electric flux.

**Electric Flux (\( \Phi_E \)):**
\[ \Phi_E = E \cdot A \]

Where:
- \( E \) is the uniform electric field.
- \( A \) is the area of the spherical surface.

**Steps to Solve:**

1. Calculate the area of the spherical surface:
\[ A = 4\pi r^2 \]

2. Calculate the rate of change of electric flux:
\[ \frac{d\Phi_E}{dt} = \frac{d}{dt}(E \cdot A) \]

3. Plug the values into the displacement current formula to find \( I_d \).

This approach will help you determine the displacement current through the spherical surface effectively.

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