Oscillating charge, larmor power dissipation

unscientific · Jan 4, 2014

Homework Statement

A z-polarized incident EM waves hits a small charge (-q) bounded to a massive charge (+q) with spring constant k0 and frequency ω.

(a) Write down the equation of motion

(b)Find time averaged total power dissipated.

(c) Find classical scattering cross section σ (thomson)

(d) Find value of E0 that makes magnetic forces non-negligible.

Diagram of a bound electron model: a light charge -q on a spring of natural length a0 and constant k above a heavy charge +q, with z(t) marked as its vertical displacement and an incident wave travelling along the x-hat direction, polarised along z-hat.

The Attempt at a Solution

Part (a) and (b)

Handwritten derivation: equation of motion m z̈ = -qE0 e^(i(ωt-kx)) - kz, solution z = Re(A e^(iωt)) with A = qE0/(m(ω²-ω0²)) and mω0² = k, and time-averaged Larmor power ⟨P⟩ = μ0 m² ω⁴ A²/(12πc).

Part(c)

Handwritten derivation of a scattering cross section: sigma equals the time-averaged power over the magnitude of the Poynting vector, reduced step by step to mu0 q squared over 6 pi c epsilon0, multiplied by (omega over 2 omega0) to the fourth power.

Part (d)
Because velocity of particle is slow, so magnetic force on it is low, giving it a small magnetic displacement compared to the electric displacement.

I related equated magnetic displacement to electric displacement to find a value of E₀.

Handwritten derivation: magnetic force F = qv x B, velocity v = iωA e^(iωt), and the magnitude |v| = ωA, followed by the magnetic displacement ratio and the resulting expression E0 = mcω0^2 / (qω).

unscientific · Jan 4, 2014

I'm not sure if part (d) is right.

unscientific · Jan 7, 2014

bumpp

unscientific · Jan 11, 2014

bumpp

TSny · Jan 11, 2014

As far as I can tell:

(a) OK
(b) The mass occurring in the fraction part of your answer should be replace by some other quantity. Otherwise, OK.
(c) Fixing (b) will affect the answer for (c).
(d) OK

Note, you should make a dimensional analysis of your results. This would help you see that your answers for (b) and (c) can't be correct.

unscientific · Jan 11, 2014

TSny said:

As far as I can tell:

(a) OK
(b) The mass occurring in the fraction part of your answer should be replace by some other quantity. Otherwise, OK.
(c) Fixing (b) will affect the answer for (c).
(d) OK

Note, you should make a dimensional analysis of your results. This would help you see that your answers for (b) and (c) can't be correct.

I mistakenly thought p = momentum, but it should be p = qx, so the m² should be replaced by charge, q².

TSny · Jan 11, 2014

Right.

Oscillating charge, larmor power dissipation

Homework Statement

The Attempt at a Solution

Discussion

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