Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The time-average electric and magnetic energies follow from the energy densities $\frac{ε}{4} ∣ E ∣^{2}$ and $\frac{μ}{4} ∣ H ∣^{2}$ of time-harmonic fields (Section 6.1). Every integral of a squared sine or cosine over a whole number of half periods is half the length, so

W_{e} = \frac{ε}{4} \int_{V} ∣ E_{y} ∣^{2} d v = \frac{ε}{4} ∣ E_{0} ∣^{2} \frac{a}{2} b \frac{d}{2} = \frac{ε ∣ E _{0} ∣ ^{2} ab d}{16} . (7.8.15)

For the magnetic energy, each component contributes the same factor $ab d /4$ , and Eq. (7.8.13) closes the bracket:

W_{m} = \frac{μ}{4} \int_{V} (∣ H_{x} ∣^{2} + ∣ H_{z} ∣^{2}) d v = \frac{μ}{4} \frac{∣ E _{0} ∣ ^{2}}{η ^{2}} \frac{ab d}{4} [(\frac{λ}{2 d})^{2} + (\frac{λ}{2 a})^{2}] = \frac{ε ∣ E _{0} ∣ ^{2} ab d}{16} = W_{e}, (7.8.16)

since $μ / η^{2} = ε$ .

Stored energy of the TE₁₀₁ mode

At resonance the time-average electric and magnetic energies are equal, and the total stored energy is

W = W_{e} + W_{m} = \frac{ε ∣ E _{0} ∣ ^{2} ab d}{8} [J] . (7.8.17)

In plain words

The energy goes back and forth between $E$ and $H$ twice per period, and its total stays constant. Equal average electric and magnetic energies are the very definition of resonance, for a cavity as for an $L C$ circuit.

In the time domain Eq. (7.8.14) gives the instantaneous energies $w_{e} (t) = W cos^{2} ω t$ and $w_{m} (t) = W sin^{2} ω t$ , whose sum is $W$ at every instant (Figure 7.8.6). Away from resonance $W_{e} \neq = W_{m}$ , and the difference must be supplied by the source in every cycle as reactive power, just as for a detuned $L C$ circuit.

The energy of the TE_{101} mode moves between the electric field (w_e=W\cos^2\omega t) and the magnetic field (w_m=W\sin^2\omega t) twice per period T. The sum stays W. — **Figure 7.8.6.** The energy of the TE $_{101}$ mode moves between the electric field ( $w_{e} = W cos^{2} ω t$ ) and the magnetic field ( $w_{m} = W sin^{2} ω t$ ) twice per period $T$ . The sum stays $W$ .

7.8.7 Stored energy