2.1.2 Conservation of linear momentum

The conservation of linear momentum is a fundamental principle in physics. It states that the total linear momentum of an isolated system remains constant. The momentum can be transferred between the different constituents of the system, provided that the momentum lost by one element is equal to the momentum gained by another. The principle can also be extended to open systems. In this case, the net change of momentum of the system equals the flux of momentum entering minus that exiting the system.

The principle is not only held by interacting particles but also by fields that carry momentum. Now, a balance equation for the conservation of the linear momentum can be written for every physical system. Under certain conditions, the conservation of linear momentum is useful to measure the forces involved in optical trapping. Starting from the Lorentz force and using the Maxwell's equations, one can write an expression for the conservation of linear momentum of an object that interacts with an electromagnetic field within a volume $ V$ (see Appendix A for the derivation),

$\displaystyle \frac{d}{dt}\left(\vec{P}_\mathrm{mech}+\vec{P}_\mathrm{field}\right)=\vec{\Phi}_\mathrm{field}$ (2.9)

where $ \vec{P}_\mathrm{mech}$ is the linear momentum of the object, $ \vec{P}_\mathrm{field}$ is the linear momentum of the electromagnetic field within the volume $ V$ and $ \vec{\Phi}_\mathrm{field}$ is the flux of linear momentum carried by an electromagnetic wave across the surface $ A$ that encloses the volume $ V$. The left hand side of the equation accounts for the overall change of linear momentum inside the volume $ V$ and the right hand side accounts for the net flux of linear momentum than enters the volume $ V$ across the surface $ A$. Figure 2.6a shows an schematic representation of the three terms of equation 2.9. The time variation of the linear momentum of the object is the force exerted on it

$\displaystyle \vec{F}=\frac{d\vec{P}_\mathrm{mech}}{dt}~.$ (2.10)

The total linear momentum of the electromagnetic field can be computed by summing the density momentum of the field $ \vec{g}$ within the volume $ V$ (see Appendix A)

$\displaystyle \vec{P}_\mathrm{field}=\int_V\vec{g}\cdot dV=\int_V\frac{\vec{S}}{c^2}dV$ (2.11)

where $ \vec{S}$ is the Poynting vector calculated from the electromagnetic field $ \vec{S}=\vec{E}\times\vec{H}$) and $ c$ is the speed of light. The total flux of linear momentum2.1 $ \vec{\Phi}_\mathrm{field}$ is calculated by summing the Maxwell stress tensor $ \overleftrightarrow{T}$ over the surface $ A$

$\displaystyle \vec{\Phi}_\mathrm{field}=\oint_A\overleftrightarrow{T}\cdot d\vec{a}~.$ (2.12)

The Maxwell's stress tensor accounts for the flux of electromagnetic linear momentum per unit of area2.2 and it is derived from the electromagnetic fields according to Eq. 2.8.

By using equations 2.10, 2.11 and 2.12 we can rewrite equation 2.9 as

$\displaystyle \vec{F}=\oint_A\overleftrightarrow{T}\cdot d\vec{a}-\frac{1}{c^2}\frac{d}{dt}\int_V\vec{S}dV$ (2.13)

which provides a recipe to calculate the electromagnetic force exerted on an object from the resulting electromagnetic fields of the interaction.

Assuming that the electromagnetic field is stationary within the volume $ V$ (i.e., all the temporal dependence is due to the complex phase $ \exp(i\omega t)$), the second term of the right side vanishes if a temporal average is performed. The resulting expression is equal to Eq. 2.7. Under these conditions, equation 2.9 can be rewritten as,

$\displaystyle \vec{F}=\vec{\Phi}_\mathrm{field}$ (2.14)

which is a simple expression for the conservation of light momentum (we do not write the temporal average $ \langle...\rangle$). The force exerted on a particle is equal to the stationary flux of linear momentum carried by the interacting electromagnetic field. In the next section it is described how to measure the flux of linear momentum.

Figure 2.6: Conservation of light momentum. (a) The surface $ A$ encloses the volume $ V$ where the radiation interacts with the material object. The net flux of momentum $ \vec{\Phi}_\mathrm{field}$ that enters/exits the volume is equal to the change in the linear momentum of the internal radiation ( $ \vec{P}_\mathrm{field}$) plus the change in the linear momentum (i.e., the force) of the objects ( $ \vec{P}_\mathrm{mech}$). (b) In a stationary state, the total flux of linear momentum of light (i.e., $ \vec{S}_\mathrm{in}-\vec{S}_\mathrm{out}$) that crosses the surface $ A$ is equal to the force applied to the object.

JM Huguet 2014-02-12