91Ó°ÊÓ

CALC A rod of pure silicon (resistivity \(\rho=2300 \Omega \cdot \mathrm{m} )\) is carrying a current. The electric field varies sinusoidally with time according to \(E=E_{0}\) sin \(\omega t,\) where \(E_{0}=0.450 \mathrm{V} / \mathrm{m}, \omega=2 \pi f\) and the frequency \(f=120 \mathrm{Hz}\) (a) Find the magnitude of the maximum conduction current density in the wire. (b) Assuming \(\mathcal{E}=\mathcal{E}_{0},\) find the maximum displacement current density in the wire, and compare with the result of part (a). (c) At what frequency \(f\) would the maximum conduction and displacement densities become equal if \(\mathcal{E}=\mathcal{E}_{0}\) (which is not actually the case)? (d) At the frequency determined in part (c), what is the relative phase of the conduction and displacement currents?

Step by step solution

01

Understand the Problem

We have a rod of pure silicon with a given resistivity, and the electric field within the rod varies sinusoidally with time. We need to find the maximum conduction current density and the maximum displacement current density, compare them, and find the frequency at which they are equal.

02

Find Maximum Conduction Current Density

Using Ohm's Law, the conduction current density \( J_c \) in a conductor is given by \( J_c = \frac{E}{\rho} \). The maximum conduction current density occurs when the electric field is at its maximum, \( E = E_0 \). Thus, the maximum \( J_{c,\text{max}} = \frac{E_0}{\rho} = \frac{0.450 \text{ V/m}}{2300 \Omega \cdot \text{m}} = 1.96 \times 10^{-4} \text{ A/m}^2 \).

03

Find Maximum Displacement Current Density

The displacement current density \( J_d \) is given by \( J_d = \varepsilon_0 \frac{dE}{dt} \), where \( \varepsilon_0 \) is the permittivity of free space. Since \( E = E_0 \sin{\omega t} \), we have \( \frac{dE}{dt} = E_0 \omega \cos{\omega t} \). The maximum \( J_d \) is \( \varepsilon_0 E_0 \omega = \varepsilon_0 E_0 2 \pi f \). Substituting the given values, we get \( J_{d,\text{max}} = 8.85 \times 10^{-12} \text{ F/m} \cdot 0.450 \text{ V/m} \cdot 2\pi \cdot 120 \text{ Hz} = 3.00 \times 10^{-10} \text{ A/m}^2 \).

04

Compare Conduction and Displacement Current Densities

From the results, the maximum conduction current density \( J_{c,\text{max}} = 1.96 \times 10^{-4} \text{ A/m}^2 \) is much larger than the maximum displacement current density \( J_{d,\text{max}} = 3.00 \times 10^{-10} \text{ A/m}^2 \).

05

Determine Frequency for Equal Current Densities

We solve for the frequency \( f \) where \( J_{c,\text{max}} = J_{d,\text{max}} \). This gives \( \frac{E_0}{\rho} = \varepsilon_0 E_0 2 \pi f \). Simplifying this, we find \( f = \frac{1}{2 \pi \varepsilon_0 \rho} = \frac{1}{2 \pi \times 8.85 \times 10^{-12} \times 2300} \approx 7.75 \times 10^9 \text{ Hz} \).

06

Find Phase Difference at Equal Current Densities

When the conduction and displacement current densities are equal, they oscillate with the same frequency and hence have a relative phase of \( \frac{\pi}{2} \) because the displacement current is proportional to the cosine of the phase of the electric field which is sine.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conduction Current

Conduction current is the flow of electric charge through a material due to an applied electric field. In the case of conductive materials like metals or silicon, electrons move through the lattice structure creating a current. This flow is described mathematically by Ohm's Law.
The formula, derived from Ohm's Law, to calculate conduction current density \( J_c \) is:

• \( J_c = \frac{E}{\rho} \)

where \( E \) is the electric field and \( \rho \) is the resistivity of the material. The conduction current is maximum when the electric field reaches its peak value. This type of current is essential in electronics, allowing circuit components to function by transmitting power efficiently through conductive paths.

Displacement Current

Displacement current density \( J_d \) arises in situations where the electric field varies with time. For example, in the sinusoidal electric fields discussed here, the displacement current fills the gap that conduction current cannot cover, especially in regions with changing electric fields like capacitors.
Its formula is:

• \( J_d = \varepsilon_0 \frac{dE}{dt} \)

where \( \varepsilon_0 \) is the permittivity of free space and \( \frac{dE}{dt} \) is the rate of change of the electric field. Displacement current plays a crucial role in AC circuits and electromagnetic theories by helping to maintain continuity in electric fields across different media.

• The maximum value of \( J_d \) is particularly significant when comparing it to conduction current density, showcasing the prominence of displacement currents in specific scenarios.

Sinusoidal Electric Fields

When an electric field varies sinusoidally, it means the electric field strength oscillates following a sine wave pattern over time. This kind of behavior is characterized by the equation \( E = E_0 \sin{\omega t} \), where \( E_0 \) is the maximum field strength, and \( \omega \) (omega) is the angular frequency.
Sinusoidal fields are common in alternating current (AC) systems and are vital for transmitting power in an efficient, cyclical manner. Key aspects when dealing with these fields include:

• Frequency \( f \), where \( \omega = 2\pi f \), indicating how fast the fields oscillate.
• A consistent, repeated cycle that helps to generate energy waves utilized in communication and power technologies.
• Operational at frequencies that range from household power supply Hz to microwave Hz, thus impacting both domestic and industrial applications.

Ohm's Law

Ohm’s Law is a fundamental principle that defines the relationship between voltage (V), current (I), and resistance (R) in electrical circuits. It is essential for calculating how current flows in a conductor under a given electric potential difference. The famous formula is:

• \( V = IR \)

For current density in materials like silicon, it translates to \( J_c = \frac{E}{\rho} \).
Here’s why Ohm’s Law remains vital:

• It provides a clear understanding of how electrical components behave under different electrical pressures.
• Utilizes the concept of electrical resistivity, a measure of how much a material opposes the flow of current, influencing the magnitude of conduction current density.
• Helps predict and design circuit behavior in practical applications across multiple electrical and electronic disciplines.

Permittivity of Free Space

Permittivity of free space, often denoted as \( \varepsilon_0 \), is a physical constant that characterizes the ability of vacuum to permit electric field lines. It defines the interaction between electric fields and insulating materials in a vacuum, establishing the strength of electric fields that can exist in free space.

• \( \varepsilon_0 \approx 8.85 \times 10^{-12} \; \text{F/m} \), representing how "permissive" the space is for electric field lines.
• Used in calculating displacement current \( J_d = \varepsilon_0 \frac{dE}{dt} \), among various other electromagnetic phenomena.
• Essential for understanding capacitors and the propagation of electromagnetic waves, where free-space acts almost like a "bare-bones" platform allowing for interaction without additional external interferences.

Understanding permittivity is essential for engineers and physicists, as it sets foundational limits and capabilities for electric field manipulations in various technologies.