Idea: Non-linear Circuit Elements

Resistors have linear $I-V$ relationships, but many elements can have weirder ones. Examples of some are:

• Diodes: elements that only allow current in one direction after a given forward bias voltage.
• Zener Diodes: two way diodes with a forward bias and a reverse bias.

If a more exotic element is in a problem, they will define it for you. The best strategy for dealing with these non-linear circuit elements is to carefully go through the cases.

Theorem: Ampere-Maxwell's Law

In the case that there is a changing electric field, Ampere’s Law has a correction: $$\oint \mathbf B \cdot d \mathbf s = \mu_0 I_{\text{enc}} + \mu_0\epsilon_0 \frac{d\Phi_E}{dt}.$$

Theorem: Maxwell's Equations

The Ampere-Maxwell Law completes the set of Maxwell's equations: \begin{align*} \nabla \cdot \mathbf E &= \frac \rho{\epsilon_0}
\nabla \cdot \mathbf B &= 0
\nabla \times \mathbf E &= -\frac{\partial \mathbf B}{\partial t}
\nabla \times \mathbf B &= \mu_0 \mathbf J + \mu_0\epsilon_0 \frac{\partial \mathbf E}{\partial t} \end{align*} These equations tell us the existence of \highlight{electromagnetic waves} which move at speed $c =1/\sqrt{\mu_0\epsilon_0}$ as you'll derive in one of the problems below.

Definition: Poynting Vector

Electromagnetic waves carry energy, and one way to quantify that is the \highlight{Poynting vector}, which is defined as $$\mathbf S = \frac{\mathbf E \times \mathbf B}{\mu_0},$$ and it gives the flux density of the rate of energy transfer in an electromagnetic field (not necessarily waves).

Theorem: Radiation Pressure

The poynting vector also gives the momentum density in the electromagnetic field as $$p = \frac{\mathbf S}{c^2}.$$ Using this, as we derived in class, the radiation pressure due to EM fields is $$P_{\text{rad}} = \frac{\langle S\rangle}{c}.$$

Theorem: Larmor Formula

A charge $q$ accelerating at acceleration $a$ radiates energy through EM waves at a rate $$P = \frac{q^2a^2}{6\pi \epsilon_0 c^3}.$$ This is known as the Larmor Formula.

Idea

In many cases where classical mechanics quantities like angular momentum or momentum are not conserved, another quantity can be conserved. It is a general physics principle where symmetries lead to conserved quantities. In the presence of a magnetic field, the canonical angular momentum can be conserved, $$\mathbf L_{\text{can}} = \mathbf r \times (\mathbf p + q\mathbf A),$$ but the form may look different from problem to problem.