Question

The spectral density of the fluctuations \((2.3)\), \((2.4)\) is $$ S_{l}(\omega)=\frac{2}{\pi} \frac{k T}{R}, \quad S_{V}(\omega)=\frac{2}{\pi} k T R \quad \text { or } \quad S_{t}(f)=\frac{4 k T}{R}, \quad S_{Y}(f)=4 k T R $$

Short answer

The spectral densities \(S_{l}(\omega)\) and \(S_{V}(\omega)\) reflect fluctuations over angular frequency, inversely and proportionally related to resistance, respectively; while \(S_{t}(f)\) and \(S_{Y}(f)\) denote fluctuations over linear frequency with the same resistance dependency.

Step-by-step solution

Understand the context

Spectral density refers to the power per unit frequency (Hz) that quantifies the intensity distribution of a signal over various frequencies. The fluctuations denoted as \(S_l(\omega)\) and \(S_V(\omega)\) are expressed in frequency domain as a function of angular frequency \(\omega\), whereas \(S_t(f)\) and \(S_Y(f)\) are expressed as a function of the frequency \(f\). \(k\) is the Boltzmann's constant, \(T\) is the absolute temperature, and \(R\) is the resistance.

Examine the formulas

The formulas given represent two pairs of spectral densities each for different parameters. The first pair \(S_l(\omega)\) and \(S_V(\omega)\) is related to angular frequency \(\omega\) and the second pair \(S_t(f)\) and \(S_Y(f)\) to linear frequency \(f\). The presence of \(T\) and \(R\) in all equations indicates their dependence on temperature and resistance respectively.

Analyze the relationships

By comparing the two pairs of equations, we notice that the spectral density of fluctuations relating to voltage and linear frequency \(S_Y(f)\) is proportional to \(R\), and the one relating to current and linear frequency \(S_t(f)\) is inversely proportional to \(R\). Taking into account that angular frequency \(\omega\) relates to linear frequency \(f\) through the relation \(\omega = 2\pi f\), the equations can be interconverted if necessary.

Key concepts

Fluctuations

Fluctuations in physical systems refer to the random variations that occur over time in the properties of a system. In the context of spectral density, these fluctuations are important as they determine the noise power present within different frequency components of a signal.
For instance, the fluctuation of current or voltage in an electrical circuit due to temperature can be described by spectral density functions such as \(S_{l}(\omega)\) and \(S_{V}(\omega)\), where the variations are not constant but rather distributed across various frequencies.

Boltzmann's Constant

Boltzmann's constant \(k\) is a fundamental physical constant that bridges the gap between macroscopic and microscopic physics. It relates the average kinetic energy of particles in a gas with the temperature of the gas, embodying the relationship between energy and temperature at the particle level.
In our spectral density equations, \(k\) appears to directly relate temperature to the fluctuations, cementing the idea that temperature is a key factor in determining the level of noise due to thermal fluctuations in a resistor.

Angular Frequency

Angular frequency \(\omega\) describes the rate of change of an angular position over time and is used to quantify oscillations in fields like physics and engineering. It is related to the linear frequency \(f\) by the equation \(\omega = 2\pi f\), where \(f\) is typically measured in hertz (Hz).
When discussing spectral density, angular frequency is used to analyze how the fluctuation spreads over different frequencies in a waveform, allowing for a deeper understanding of the distribution of noise power in a signal.

Linear Frequency

Linear frequency, often denoted as \(f\), represents the number of cycles or oscillations that occur per unit time and is expressed in hertz (Hz). Spectral density functions like \(S_{t}(f)\) and \(S_{Y}(f)\) use linear frequency to demonstrate how the power of the noise varies with different regular frequency components.
The understanding of linear frequency is pivotal when dealing with electronic signals, as it helps us gauge how fast the variables in a system change with time.

Temperature Dependence

In the realm of physics, particularly thermodynamics and statistical mechanics, temperature dependence is a crucial concept that indicates how various physical properties of a system change as the temperature varies.
With the increase in temperature, the thermal vibrations of particles increase as well, leading to greater electrical noise or fluctuations within a system. In the spectral density formulas, the direct proportionality to temperature \(T\) suggests that higher temperatures will lead to greater intensity in the fluctuations of electrical parameters like voltage and current.

Electrical Resistance

Electrical resistance, symbolized by \(R\), is a measure of the opposition to the flow of electric current through a conductor. It arises due to collisions between the current-carrying charged particles and the fixed atoms within the conductive material.
The spectral density of fluctuations in the context of resistance reveals the interaction between thermal noise and electrical resistance. For instance, in the expressions \(S_{l}(\omega)\) and \(S_{t}(f)\), resistance plays a pivotal role, inversely affecting current noise while being directly proportional to the voltage noise, underscoring the intricate balance in electrical phenomena.