Question

Intensity of electromagnetic wave equation

Answer 1

The answer is : Electromagnetic waves carry energy, and as they propagate through space they can transfer energy to objects placed in their path. The rate of flow of energy in an electromagnetic wave is described by a vector S, called the Poynting vector, which is defined by the expression The magnitude of the Poynting vector represents the rate at which energy flows through a unit surface area perpendicular to the direction of wave propagation. Thus, the magnitude of the Poynting vector represents power per unit area. The direction of the vector is along the direction of wave propagation (Fig. 34.7). The SI units of the Poynting vector are J/s.m2 = W/m2. As an example, let us evaluate the magnitude of S for a plane electromagnetic wave where |Ex B| = EB. In this case, Because B = E /c, we can also express this as These equations for S apply at any instant of time and represent the instantaneous rate at which energy is passing through a unit area. 
What is of greater interest for a sinusoidal plane electromagnetic wave is the time average of S over one or more cycles, which is called the wave intensity I. When this average is taken, we obtain an expression involving the time average of cos2 (kx - ωt), which equals ½. Hence, the average value of S (in other words, the intensity of the wave) is Recall that the energy per unit volume, which is the instantaneous energy density uEassociated with an electric field, is given by Equation 26.13, and that the instantaneous energy density uBassociated with a magnetic field is given by Equation 32.14: Because E and B vary with time for an electromagnetic wave, the energy densities also vary with time. When we use the relationships B = E/c and c = 1 / √(μ0ε0), Equation 32.14 becomes Comparing this result with the expression for uE , we see that That is, for an electromagnetic wave, the instantaneous energy density associated with the magnetic field equals the instantaneous energy density associated with the electric field. Hence, in a given volume the energy is equally shared by the two fields. 
The total instantaneous energy density u is equal to the sum of the energy densities associated with the electric and magnetic fields: When this total instantaneous energy density is averaged over one or more cycles of an electromagnetic wave, we again obtain a factor of ½. Hence, for any electromagnetic wave, the total average energy per unit volume is Comparing this result with Equation 34.20 for the average value of S, we see that In other words, the intensity of an electromagnetic wave equals the average energy density multiplied by the speed of light.