Question

Derive a relation between intensity and energy density of an emw

Answer 1

While we have not emphasized it so far, electric and magnetic fields both contain energy. The total amount of energy depends on the values of the fields everywhere, so it is more convenient to define the energy density of the fields. This is the amount of energy per unit volume contained within the fields.  Unlike total energy, energy density can be defined easily for specific locations.  To obtain it for one position, we only need to know the value of the electric and magnetic fields at that position. This is similar to our motivation for introducing energy densities when we discussed fluids in Physics 7B. The energy density of the electric and magnetic fields are

uE=12μ0E2c2(1)

uB=12μ0B2(2)

where  μ0  is the magnetic permeability of free space,  4π×10−7 N/A2 . These results are a little bit tricky to derive from what we already know, so we do not attempt a derivation here. We will just accept this energy density as given.

The definitions for energy density apply both to electromagnetic waves and to static electric and magnetic fields.  For electromagnetic waves, both the electric field and the magnetic field contribute to the energy density.  The total energy density is the sum of these contributions.

utotal=uE+uB(3)

We know that when the electric and magnetic fields hit zero then the energy density in the fields also goes to zero. The energy density is greatest when the magnetic and electric fields experience their peaks:

umax, EM=uE max+uB max(4)

umax, EM=12μ0E20c2+12μ0B02(5)

where  B0  and  E0  are the maximum displacement of the magnetic and electric fields, respectively.  Recall from the previous section that these two quantities are related by the equation  E0=cB0 .  Rewriting the equations, we find that the maximum energy density of an electromagnetic wave is

umax EM=1μ0B02(6)

Energy density at a point along an electromagnetic wave oscillates between 0 and  umax EM . It's been shown (both mathematically and experimentally) that the average energy density is half of the maximum value

uEM avg=12μ0B02(7)