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The time-averaged power flow (according to the instantaneous Poynting vector averaged over a full cycle, for instance) is then given by the real part of S m. The imaginary part is usually ignored, however, it signifies "reactive power" such as the interference due to a standing wave or the near field of an antenna.
where = [] is the time averaged Poynting vector. If W a > 0 {\displaystyle W_{\text{a}}>0} energy is absorbed within the sphere, otherwise energy is being created within the sphere. We will not consider this case here.
All but the last term of can be written as the tensor divergence of the Maxwell stress tensor, giving: = +, As in the Poynting's theorem, the second term on the right side of the above equation can be interpreted as the time derivative of the EM field's momentum density, while the first term is the time derivative of the momentum density for ...
The result is a gradual spiral of dust grains into the Sun. Over long periods of time, this effect cleans out much of the dust in the Solar System. While rather small in comparison to other forces, the radiation pressure force is inexorable. Over long periods of time, the net effect of the force is substantial.
The Poynting vector for a wave is a vector whose component in any direction is the irradiance (power per unit area) of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is 1 / 2 Re{E × H ∗}, where E and H are due only to the
where: is the rate of change of the energy density in the volume. ∇•S is the energy flow out of the volume, given by the divergence of the Poynting vector S. J•E is the rate at which the fields do work on charges in the volume (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product).
The purpose of the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution y ( t ) {\displaystyle y(t)} approximates x ( t ) {\displaystyle x(t)} for times t = O ( 1 / ε ) . {\displaystyle t={\mathcal {O}}(1/\varepsilon ).}
The radius vector, , is the distance from the charged particle's position at the retarded time to the point of observation of the electromagnetic fields at the present time, is the charge's velocity divided by , ˙ is the charge's acceleration divided by , and = /.