Question

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons \(N \approx 4 \times 10^{27} \mathrm{~m}^{-3}\). Take \(\varepsilon_{0} \approx 10^{-11}\) and \(\mathrm{m} \approx 10^{-30}\), where these quantities are in proper SI units. (A) \(800 \mathrm{~nm}\) (B) \(600 \mathrm{~nm}\) (C) \(300 \mathrm{~nm}\) (D) \(200 \mathrm{~nm}\)

Short answer

The wavelength at which plasma reflection occurs is closest to (B) 600 nm.

Step-by-step solution

Write down the formula for plasma frequency

The plasma frequency, \(\omega_p\), can be determined using the formula \[\omega_p = \sqrt{\frac{Ne^2}{\varepsilon_0 m}}\], where \(N\) is the density of electrons, \(e\) is the charge of an electron (\(e = 1.6 \times 10^{-19} \text{C}\)), \(\varepsilon_0\) is the permittivity of free space, and \(m\) is the mass of an electron.

Calculate the plasma frequency

Substitute the given values into the plasma frequency formula: \[\omega_p = \sqrt{\frac{4 \times 10^{27} \times (1.6 \times 10^{-19})^2}{10^{-11} \times 10^{-30}}}\], and simplify to find \(\omega_p\).

Convert plasma frequency to angular frequency

Use the relation \(\lambda u = c\) to find the wavelength \(\lambda\), where \(c\) is the speed of light (\(c = 3 \times 10^8 \text{m/s}\)) and \(u\) is the frequency. First, find the frequency (\(u\)) using \(u = \frac{\omega_p}{2\pi}\), then calculate \(\lambda\) using \(\lambda = \frac{c}{u}\).

Compare the calculated wavelength with options

Compare the calculated wavelength value to the given options (A) 800 nm, (B) 600 nm, (C) 300 nm, and (D) 200 nm to determine which one is the most accurate estimate.

Key concepts

Plasma Frequency

When discussing plasmas in physics, understanding the plasma frequency is crucial. It’s a characteristic frequency at which the free electrons in a plasma or metal oscillate collectively when responding to electromagnetic disturbances. The formula for plasma frequency is given by \[\omega_p = \sqrt{\frac{Ne^2}{\varepsilon_0 m}}\]where

• \(N\) represents the electron density, or the number of free electrons per unit volume,
• \(e\) is the elementary charge of an electron,
• \(\varepsilon_0\) is the permittivity of free space, and
• \(m\) is the mass of an electron.

To interpret plasma frequency more tangibly, it can be thought of as the natural resonant frequency of the electron 'sea' in a conductor, acting against the restoring force provided by the positive ion background. This collective oscillation is key in determining the reflective properties of the metal at certain frequencies or wavelengths of light, hence its importance in the study of plasmas and electromagnetic properties of materials.

Electron Density

The term electron density, represented as \(N\), is used to indicate the concentration of free electrons in a given volume and is measured in units of electrons per cubic meter (\(\mathrm{m}^{-3}\)). In the context of plasma reflection wavelength, knowing the electron density is essential because higher electron densities typically result in higher plasma frequencies.

In metals, the free electrons are those that are not bound to any particular atom and can move more or less freely through the material, which makes them responsible for conducting electricity and heat. The electron density plays a pivotal role not only in electrical conductivity but also in optical properties such as reflectivity and absorption. Hence, for our exercise, the provided electron density of \(4 \times 10^{27} \mathrm{~m}^{-3}\) indicates a high concentration of free electrons, which in turn suggests a certain level of reflectivity at a specific wavelength.

Permittivity of Free Space

The permittivity of free space, denoted as \(\varepsilon_0\), is a fundamental constant that describes how easily an electric field can permeate through a vacuum. It can be perceived as the electric permittivity of a classical vacuum and is present in various equations in electromagnetism, including those related to the propagation of electromagnetic waves and the forces between charged particles.

The value of \(\varepsilon_0\) is approximately \(8.854 × 10^{-12} \mathrm{~F/m}\) (farads per meter), but in our problem, it is approximated as \(10^{-11} \mathrm{F/m}\) for simplicity. This constant is integral in the equation for the plasma frequency, which in turn determines the reflection characteristics of the metal in question. While \(\varepsilon_0\) is often considered in the context of a vacuum, its role in the plasma frequency formula bridges the gap between electromagnetism in free space and the behavior of charged particles within a material.