Debye Length Calculator Formula

Debye Length Formula:

\[ \lambda_D = \sqrt{\frac{\varepsilon_0 \varepsilon_r k_B T}{e^2 n}} \]

Debye Length (λ_D):

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1. What is the Debye Length?

The Debye length (λ_D) is a fundamental parameter in plasma physics that represents the characteristic distance over which electric fields are screened in a plasma. It describes the scale at which mobile charge carriers (electrons and ions) respond to electric potential disturbances.

2. How Does the Calculator Work?

The calculator uses the Debye length formula:

\[ \lambda_D = \sqrt{\frac{\varepsilon_0 \varepsilon_r k_B T}{e^2 n}} \]

Where:

\( \varepsilon_0 \) — Vacuum permittivity (8.854 × 10⁻¹² F/m)
\( \varepsilon_r \) — Relative permittivity of the medium
\( k_B \) — Boltzmann constant (1.381 × 10⁻²³ J/K)
\( T \) — Temperature in Kelvin
\( e \) — Elementary charge (1.602 × 10⁻¹⁹ C)
\( n \) — Electron density (m⁻³)

Explanation: The Debye length increases with temperature and decreases with increasing charge carrier density, representing the competition between thermal energy and electrostatic interactions.

3. Importance of Debye Length Calculation

Details: The Debye length is crucial for understanding plasma behavior, electrostatic screening, and the validity of the plasma approximation. It determines whether a system can be treated as quasi-neutral and influences phenomena like plasma oscillations and sheath formation.

4. Using the Calculator

Tips: Enter temperature in Kelvin, electron density in particles per cubic meter, and relative permittivity (1 for vacuum, ~80 for water). All values must be positive with relative permittivity ≥ 1.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of Debye length?
A: It represents the scale over which electric fields are effectively screened in a plasma and determines the distance over which charged particles can influence each other.

Q2: How does temperature affect Debye length?
A: Debye length increases with temperature because higher thermal energy allows charges to move further against electrostatic attraction.

Q3: What are typical Debye length values?
A: In laboratory plasmas: micrometers to millimeters; in space plasmas: meters to kilometers; in semiconductors: nanometers.

Q4: What is the Debye sphere?
A: A sphere with radius equal to the Debye length, containing enough particles to maintain collective plasma behavior (typically requires many particles in the Debye sphere).

Q5: How is Debye length related to plasma frequency?
A: They are inversely related through the thermal velocity: \( \lambda_D \omega_p = v_{th} \), where \( \omega_p \) is plasma frequency and \( v_{th} \) is thermal velocity.