Driven electron cloud (Lorentz model) under an alternating electric field If m is the electron mass in an atom's bound electron cloud, write the standard scalar equation of motion for the electron displacement x(t) when the dielectric is excited by a sinusoidal electric field E(t) = E0 cos(ω t). Include a linear restoring term and damping representative of collisions (classical Lorentz oscillator model).

Introduction / Context:
The classical Lorentz oscillator model explains frequency-dependent dielectric behavior by modeling bound electrons as damped, driven harmonic oscillators. This provides physical intuition for dispersion, dielectric loss, and resonance phenomena observed in real materials across optical, RF, and microwave regimes.

Given Data / Assumptions:

• Electron of mass m and charge −e bound to an equilibrium position by a linear restoring force.
• Sinusoidal electric field E(t) = E0 cos(ω t) applied.
• Linear viscous damping proportional to velocity with coefficient m γ.
• Small displacements so linearization holds.

Concept / Approach:

Newton's second law balances inertia, damping, restoring force, and electric driving force. For a bound electron: inertia m d^2x/dt^2, damping m γ dx/dt, restoring m ω0^2 x. The electric force on charge −e is F = −e E(t), giving the standard differential equation for x(t). The solution yields complex susceptibility with resonance near ω0 and loss represented by γ.

Step-by-Step Solution:

Verification / Alternative check:

Solving in steady state with x(t) = Re{X e^{jωt}} yields complex amplitude X ∝ (−e/m) / (ω0^2 − ω^2 + j γ ω). The resulting polarization P = N (−e) x establishes ε′(ω) and ε′′(ω), matching dispersion curves.

Why Other Options Are Wrong:

Option B omits restoring and damping; C has dimensional inconsistencies; D ignores driving and restoring terms; E is missing the damping and correct frequency dependence.

Common Pitfalls:

Forgetting the negative sign of electron charge, or omitting damping which is necessary to model loss (finite tan δ).

Final Answer:

m d^2x/dt^2 + m γ dx/dt + m ω0^2 x = − e E0 cos(ω t)