Time average stored electric energy in a volume

Question

I am studying the book Microwave Engineering Third Edition by David M. Pozar. At the moment I am at chapter 1.6. At the beginning of this chapter Pozar states that

In the sinusoidal steady-state case, the time average stored electric energy in a volume \$V\$ is given by,

$$ W_e=\frac{1}{4}\,\operatorname{Re}\int\limits_{V}\bar{E}\cdot \bar{D}^{*} dv \tag{1.83},$$

which in the case of simple lossless isotropic, homogenous, linear media, were \$\varepsilon\$ is a real scalar constant, reduces to

$$ W_e=\frac{\varepsilon}{4}\int\limits_{V}\bar{E}\cdot \bar{E}^{*} dv \tag{1.84}$$

Similarly, the time-average magnetic energy stored in the volume \$V\$ is

$$ W_m=\frac{1}{4}\,\operatorname{Re}\int\limits_{V}\bar{H}\cdot \bar{B}^{*} dv \tag{1.85}$$

which becomes

$$ W_m=\frac{\mu}{4}\int\limits_{V}\bar{H}\cdot \bar{H}^{*} dv \tag{1.86}$$

for real, constant, scalar \$\mu\$.

Where the \${}^*\$ denotes complex conjugate. From basic physics, I know how the energy of a static continuous charge distribution is derived. However I am unable to figure out how I can derive equation 1.83 and 1.85. Please tell me how these equations can be derived.

Answer 1

The formulas for volume energy content are based on an assumption that you probably didn't want to see. They assume the field is a real thing which can store energy to a volume and that works also in empty space. The formulas are derived by moving charged particles or by charging a capacitor for E and by moving magnetic dipoles or by increasing inductor current for H. The work done is assumed to be stored to the field when the change has been so slow that radiation losses are neglible.

There can be some material which is polarized by the fields - E generates electric dipoles and H generates magnetic dipoles. Field quantities D and B cover them. The work used to generate such dipoles is a special case of moving charges to a new distribution and the energy which is not dissipated by friction is assumed to be stored to the field. The energy used to cause polarization is taken into the account by having D and B in the volume integrals. The integral is valid also when the polarization vs field strength is non-linear as long as we assume all work is stored to the field.

Maxwell's electromagnetism doesn't have any low level starting point which knows something fundamental of the space itself like how it can store energy. The electromagnetic energy content of a volume is a calculation result of different cases which need work to increase the field strength in the space and purely assume the work is stored to the field. The assumption has been a good one because the volume energy content formulas seem to be well in accordance with practical experiments. The experiments cannot measure the energy content of the space, but the experiments can measure how much work the field can do when we assume that the energy is taken from the field. An ultimate example of it is the Poynting vector for the power intensity of a radiowave.

You wrote you know how the electric field energy of a made-up charge concentration is calculated. As told above one could calculate the work needed to collect the charge from infinity. The work is done to win the resisting electric forces. Some work is needed also to create the magnetic field, but that's the other energy storing mechanism and only the electric part is the interesting one.

Applying some known vector field facts the actual charge density function nor the scalar potential are not needed. The whole energy density dW/dV can be calculated with field vectors. It's (E dot D)/2 where dot is the scalar product. It takes into the account the fact that non-isotropic materials exist.

Then there's also materials where changes of D has some latency when compared to the changes of E. For sinusoidal fields that means phase difference between E and D. The average of the product is calculated with the phasors of the vectors (three complex numbers per a vector) just like one calculates the average power with the phasors of AC voltage and current. The complex conjugate comes from it.

The real part is the average stored energy during one sinusoidal cycle, the imaginary part presents the energy flow forth and back to the source which makes the electric field fluctuate sinusoidally. It's analog with the reactive power in AC circuits. The denominator 4 is 2*2. The first 2 was already in the energy density formula. The other 2 comes from the phasors which have magnitudes = peak vector values.

I guess you can now find the corresponding explanation for magnetic energy.