Problem 73 - University Physics with Modern Physics

Question

A rod of pure silicon (resistivity \(\rho=2300 \Omega \cdot \mathrm{m} )\) is carry-ing 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 \(\epsilon=\epsilon_{0}\) , 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 densitics become equal if \(\epsilon=\epsilon_{0}\) (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

Answer

Maximum conduction density: \( 1.957 \times 10^{-4} \, \mathrm{A/m}^2 \). Displacement density: \( 3.2 \times 10^{-11} \, \mathrm{A/m}^2 \). Equal density frequency: \( 7.91 \times 10^6 \, \mathrm{Hz} \); Phase difference: 90°.

Step 1: Understanding the given data

We are given these values: resistivity \( \rho = 2300 \, \Omega \cdot \mathrm{m} \), maximum electric field \( E_0 = 0.450 \, \mathrm{V}/\mathrm{m} \), frequency \( f = 120 \, \mathrm{Hz} \), and \( \omega = 2\pi f \). We need to find conduction current density and displacement current density under these conditions.

Step 2: Calculate maximum conduction current density

We use Ohm's law for a uniform field, \( \mathbf{J} = \sigma \mathbf{E} \), where \( \sigma = \frac{1}{\rho} \) is the conductivity. We find the conductivity as \( \sigma = \frac{1}{2300} \, \Omega^{-1} \cdot \mathrm{m}^{-1} \). Then, the maximum conduction current density is \( J_{\text{max}} = \sigma E_0 = \left( \frac{1}{2300} \right) \times 0.45 = 1.957 \times 10^{-4} \, \mathrm{A/m}^2 \).

Step 3: Calculate maximum displacement current density

The displacement current density is given by \( J_{d} = \epsilon \frac{dE}{dt} \). With \( E = 0.450 \sin(\omega t) \), we find \( \frac{dE}{dt} = 0.450 \omega \cos(\omega t) \). The maximum value of \( \cos(\omega t) \) is 1, so \( J_{d, \text{max}} = \epsilon_0 \times 0.450 \times \omega = \epsilon_0 \times 0.450 \times 2\pi \times 120 = 3.2 \times 10^{-11} \, \mathrm{A/m}^2 \).

Step 4: Calculate frequency for equal conduction and displacement densities

For conduction current density and displacement current density to be equal, \( \sigma E_0 = \epsilon_0 \omega E_0 \). This simplifies to \( \sigma = \epsilon_0 \omega \) \( \Rightarrow \) \( \omega = \frac{\sigma}{\epsilon_0} \). Solving for \( \omega \) gives \( \omega = \frac{1}{2300 \times 8.854 \times 10^{-12}} = 4.97 \times 10^7 \, \mathrm{rad/s} \). So \( f = \frac{\omega}{2\pi} \approx 7.91 \times 10^6 \, \mathrm{Hz} \).

Step 5: Phase between conduction and displacement currents at this frequency

Conduction current \( J = J_0 \sin(\omega t) \) and displacement current \( J_d = J_{d,0} \cos(\omega t) \). Thus, the conduction current lags the displacement current by 90 degrees (or \( \frac{\pi}{2} \) radians).

Key Concepts

Ohm's Law

Ohm's Law is a foundational principle in electromagnetism and electrical engineering. It relates the voltage, current, and resistance in a conductive material. The basic form of Ohm's Law is expressed as:

• \( V = IR \)

where

• \( V \) is the voltage across the conductor in volts,
• \( I \) is the current through the conductor in amperes, and
• \( R \) is the resistance of the conductor in ohms.

This equation demonstrates how the current flow through a conductor relates to the applied voltage and its inherent resistance. A high resistance material will restrict the current flow, much like a narrow pipe restricts water flow.
In electromagnetism, especially for uniform fields within materials, we often use a variant of Ohm's Law:

• \( \mathbf{J} = \sigma \mathbf{E} \)

where:

• \( \mathbf{J} \) represents the current density, describing how much electric current is flowing per unit area of the material,
• \( \sigma \) is the conductivity of the material (inverse of resistivity, \( \rho \)), and
• \( \mathbf{E} \) is the electric field applied across the material.

Here, the current density \( \mathbf{J} \) is directly proportional to the electric field \( \mathbf{E} \) and is contingent upon the material's conductivity. This relationship is crucial for understanding how materials conduct electricity and is especially pertinent when working with different materials like silicon in electronics.

Conduction Current Density

Conduction current density is a key concept in electromagnetism, describing how electric current flows through a conductor per unit area. It's particularly important in analyzing how efficiently materials like silicon conduct electrical currents. The conduction current density \( \mathbf{J} \) is expressed through the equation:

• \( \mathbf{J} = \sigma \mathbf{E} \)

where:

• \( \sigma \) represents the material's conductivity,
• \( \mathbf{E} \) is the electric field.

Conductivity \( \sigma \) is a measure of how easily a material allows current to pass through it. For any given electric field, a high conductivity implies a high current density.
In the context of the original exercise, the conductivity of silicon is calculated using its resistivity: \( \sigma = \frac{1}{\rho} \). Once we know \( \sigma \), we use that value with the electric field \( E_0 \) to find the maximum conduction current density. This is pivotal to understanding how currents behave in electrical components made from different materials and affects how we design circuits.

Displacement Current Density

Displacement current density is a fundamental concept introduced to extend the applicability of Ampere's law in electrodynamics. Unlike conduction current, displacement current does not involve actual charge carriers moving through the material. Instead, it is related to the time-varying electric field and is significant in scenarios where fields change over time, such as in capacitors and electromagnetic waves.
The displacement current density \( J_d \) is defined by the equation:

• \( J_d = \epsilon_0 \frac{dE}{dt} \)

where:

• \( \epsilon_0 \) is the permittivity of free space,
• \( \frac{dE}{dt} \) is the time derivative of the electric field.

In essence, displacement current fills in the gaps in Maxwell's equations, particularly in non-conductive regions where there's a changing electric field but no conduction current.
In the exercise, the electric field \( E \) is given as a sinusoidal function of time. By taking the derivative with respect to time, we calculate the rate of change of the electric field, which in turn helps us find the displacement current density. Understanding this concept is essential for analyzing circuits with alternating currents and electromagnetic radiation phenomena, ensuring comprehensive coverage of electrodynamics principles.