Maxwell's Equations and Displacement Current

Time-varying Fields and Maxwell's Equations

• Maxwell's Equations comprise a set of equations explaining all classical electromagnetism (EM) phenomena.
• Displacement current is Maxwell's prediction of a magnetic field being produced by a time-varying electric field.
• The displacement current density is expressed as $\vec{J}_d = \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$.
• Generalized Ampere's Law, incorporating displacement current, is also known as Ampere-Maxwell's Law: $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$.
• The integral form of Ampere-Maxwell's Law is $\oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E = \int \vec{E} \cdot d\vec{A}$.

Application of Ampere's Law

• While charging a cylindrical capacitor, Ampere's Law can be applied to find the magnetic field between the plates.
• Along an Amperian loop, $\oint \vec{B} \cdot d\vec{l} = B(2\pi r) = \mu_0 I_{enc}$.
• For $r > R$, the magnetic field is $B = \frac{\mu_0 I}{2\pi r}$.

Free Space

• In free space, Maxwell's Equations are:
  • $\nabla \cdot \vec{E} = 0$.
  • $\nabla \cdot \vec{B} = 0$.
  • $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$.
  • $\nabla \times \vec{B} = \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$.
• These equations predict the existence of electromagnetic (EM) waves.
• EM waves propagate in free space at a speed of $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 3.0 \times 10^8 \text{ m/s}$.
• Light is an electromagnetic wave.

Properties of Plane Electromagnetic Waves

• Electric and magnetic fields are perpendicular.
• Wave propagation occurs in the direction given by $\vec{E} \times \vec{B}$.
• Fields have the relationship $E = cB$.
• Polarization of an EM wave is defined by the direction of the $\vec{E}$ field.

Energy and Momentum of EM Waves

• Energy flow per unit area is described by the Poynting vector $\vec{S}$, given by $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$.
• The direction of $\vec{S}$ indicates the direction of the wave propagation, and the magnitude is the power per unit area.
• Intensity is the time-averaged value of the Poynting vector: $I = \langle S \rangle = \frac{E_0 B_0}{2\mu_0} = \frac{E_0^2}{2\mu_0 c} = \frac{c B_0^2}{2\mu_0}$.
• EM waves exert radiation pressure on surfaces.
• If completely absorbed, the radiation pressure is $P = \frac{I}{c}$.
• If completely reflected, the radiation pressure is $P = \frac{2I}{c}$.

Channel Capacity

Channel Capacity Definition

• Channel capacity of a communication channel represents the maximum rate of information that can be reliably transmitted.
• Reliable communication means the probability of error can be made arbitrarily small.

Discrete Memoryless Channel (DMC) Definition:

• Input alphabet: $\mathcal{X}$
• Output alphabet: $\mathcal{Y}$
• Transition probabilities: $p(y|x)$, $x \in \mathcal{X}$, $y \in \mathcal{Y}$
• Channel Capacity: $C = \max_{p(x)} I(X;Y)$ in bits per channel use.
• $I(X;Y)$ is the mutual information between the input $X$ and the output $Y$, with the maximum taken over all possible input distributions $p(x)$.

Examples of Channel Capacity Calculations

• Noiseless Binary Channel:

  • Transmission: $X \rightarrow Y$, with $Y = X$
  • Channel Matrix: $$ P = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} $$
  • Channel Capacity: $C = 1$ bit
    • Achieved by using the channel once per bit.

• Noisy Channel with Non-Overlapping Outputs:

  • Transmission: Each input leads to a unique output.
  • Channel Capacity: $C = \log |\mathcal{X}|$
    • $|\mathcal{X}|$ is the size of the input alphabet.

• Noisy Typewriter:

  • Input: English text (26 letters)
  • Output: if $X = a$ then Y = $a$ or Y = $b$ with equal probability
  • Channel Capacity: $C = \log 13$

• Binary Symmetric Channel (BSC):

  • Transition Probabilities: $$ P(Y=0|X=0) = P(Y=1|X=1) = 1-p $$ $$ P(Y=0|X=1) = P(Y=1|X=0) = p $$ where $0 \le p \le 1$.
  • Channel Capacity: $C = 1 - H(p)$ bits
    • $H(p) = -p \log p - (1-p) \log (1-p)$ is the binary entropy function.

• Binary Erasure Channel (BEC):

  • Input Alphabet: $\mathcal{X} = {0, 1}$
  • Output Alphabet: $\mathcal{Y} = {0, 1, e}$ ('e' denotes erasure)
  • Transition Probabilities: $P(Y=0|X=0) = 1 - \alpha$, $P(Y=e|X=0) = \alpha$ $P(Y=1|X=1) = 1 - \alpha$, $P(Y=e|X=1) = \alpha$
  • Channel Capacity: $C = 1 - \alpha$ bits