Chapter 10: Electromagnetic Oscillations and Waves

1. Undamped Free Oscillations

Initial State: The source $\mathcal{E}$ charges the capacitor $C$.

At $t = 0$, $i = 0$, $q = C\mathcal{E}$.

When the switch is toggled to position $a$, the capacitor begins to discharge. In time $dt$, the capacitor charge decreases by $dq$, thus $i = -\frac{dq}{dt}$.

$$ \frac{q}{C} - \left(-L\frac{di}{dt}\right) = 0 \implies \frac{q}{C} + L\frac{di}{dt} = 0 \implies i + LC \frac{d^2i}{dt^2} = 0 $$

Solving this differential equation yields: $i = i_m \sin\left(\frac{1}{\sqrt{LC}} t\right)$.

$$ \frac{di}{dt} = \frac{i_m}{\sqrt{LC}} \cos\left(\frac{1}{\sqrt{LC}} t\right) = \frac{q}{LC} $$

At $t=0$, $q = C\mathcal{E}$, so $\frac{i_m}{\sqrt{LC}} = \frac{C\mathcal{E}}{LC} = \frac{\mathcal{E}}{L} \implies i_m = \frac{\mathcal{E}}{\sqrt{L/C}}$.

$$ \therefore i = \frac{\mathcal{E}}{\sqrt{L/C}} \sin\left(\frac{1}{\sqrt{LC}} t\right) $$ $$ q = -\int i dt = -\frac{\mathcal{E}}{\sqrt{L/C}} \int \sin\left(\frac{1}{\sqrt{LC}} t\right) dt = -\mathcal{E} C \int \sin\left(\frac{1}{\sqrt{LC}} t\right) d\left(\frac{1}{\sqrt{LC}} t\right) = C\mathcal{E} \cos\left(\frac{1}{\sqrt{LC}} t\right) $$

Therefore, we have the following relations for voltage and energy:

$U_C = \frac{q}{C} = \mathcal{E} \cos\left(\frac{1}{\sqrt{LC}} t\right)$
$U_L = -U_C = -\mathcal{E} \cos\left(\frac{1}{\sqrt{LC}} t\right)$
$W_C = \frac{1}{2} \frac{q^2}{C} = \frac{(C\mathcal{E})^2}{2C} \cos^2\left(\frac{1}{\sqrt{LC}} t\right) = \frac{C\mathcal{E}^2}{4} \left[1 + \cos\left(\frac{2}{\sqrt{LC}} t\right)\right]$
$W_L = \frac{1}{2} L i^2 = \frac{L \mathcal{E}^2}{2 L/C} \sin^2\left(\frac{1}{\sqrt{LC}} t\right) = \frac{C\mathcal{E}^2}{4} \left[1 - \cos\left(\frac{2}{\sqrt{LC}} t\right)\right]$

2. Damped Free Oscillations

$$ \frac{q}{C} - \left(-L\frac{di}{dt}\right) + i R = 0 $$

Due to the resistance $R$, the energy of the oscillation gradually dissipates into heat.

3. Forced Oscillations

$$ U_m \sin(\omega t) - L \frac{di}{dt} = \frac{q}{C} + i R $$

Electrical Resonance: When the driving angular frequency $\omega$ matches the natural angular frequency $\omega_0$ of the circuit, the current amplitude reaches its maximum value, resulting in electrical resonance.

4. Electromagnetic Waves

1) Maxwell's Equations and Electromagnetic Waves

From Maxwell's equations in integral form:

$$ \oint \vec{E} \cdot d\vec{S} = \frac{\Sigma q}{\varepsilon_0} $$ $$ \oint \vec{B} \cdot d\vec{S} = 0 $$ $$ \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} = -\int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S} $$ $$ \oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right) = \mu_0 \int_S \left( \vec{j} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \right) \cdot d\vec{S} $$

We can derive the wave speed in a vacuum: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$.

The relationship between the electric field and magnetic field in an electromagnetic wave is: $\vec{B} = \frac{\hat{c} \times \vec{E}}{c}$, where $\hat{c}$ is the unit vector in the direction of propagation (Note: the text implies $B = E/c$).

Energy Density and Flux:

Energy Density: $\omega = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0} B^2 = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0} \left(\frac{E}{c}\right)^2 = \varepsilon_0 E^2$.
Energy Flux Density (Poynting Vector Magnitude): $S = \frac{\omega dS \cdot c dt}{dS \cdot dt} = c\omega = \frac{EB}{\mu_0}$. In vector form: $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$.
Intensity (Average Flux): $\bar{S} = I = \overline{c\omega} = c\varepsilon_0 \overline{E^2} = \frac{1}{2} c\varepsilon_0 E_m^2 \propto E_m^2$.

Radiation Pressure:

On a perfectly absorbing black body: $p_r = \frac{W/c}{\Delta t \cdot \Delta S} = \frac{W}{c \cdot \Delta t \cdot \Delta S} = \omega = \varepsilon_0 E^2$.
On a perfectly reflecting surface: $p_r = 2\varepsilon_0 E^2$.

2) Principles of Transmission and Reception

Open Circuit: An LC oscillator is coupled to an open circuit (antenna and ground) to radiate waves effectively.

Modulation (AM, FM, PM) & Amplification: The base signal modulates a high-frequency carrier wave for transmission.

Tuning: Adjusting the receiver's LC circuit to match the frequency of the incoming wave ($\omega_{\text{circuit}} = \omega_{\text{receive}} = 2\pi f = \frac{1}{\sqrt{LC}}$).
Demodulation (Detection): Extracting the original signal (sound, light, etc.) from the received high-frequency wave.