Definition: Ohm's Law

In an Ohmic material, the current density and electric field are related by $$\mathbf J = \sigma \mathbf E,$$ where $\sigma$ is the conductivity, equal to $1/\rho$, where $\rho$ is the resistivity. At the macroscopic level, for an entire circuit element, this equation will lead to the more familiar Ohm's Law, $$V = IR,$$ where $R$ is the \highlight{resistance}, a value related to the resistivity and the dimensions of the circuit element.

Theorem: Kirchoff's Laws

They are as follows:

1. Junction Rule: the sum of currents into any node is 0. This is simply a restatement of the conservation of charge.
2. Loop Rule: the sum of potential differences around a loop is 0. This is a statement of conservation of energy or the conservative nature of the electric field (this idea will be more clear once we get to electromagnetic induction and inductors).

Idea: Series \& Parallel

The equivalent resistance of two resistors $R_1$ and $R_2$ in series is $$\eq{R} = R_1 + R_2,$$ and in parallel it is $$\eq{R} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}.$$

Tip: Circuit Tricks

Now, although you can theoretically do any problem with these fundamental ideas, circuit problems can get quite tricky, so here are a couple tricks to remember.

1. If two points in a circuit have equal potential, you can freely attach and detach those two points.
2. If you have an infinite repeating resistor pattern, try recursion.
3. Sometimes you have to superpose different current injection arrangements, as we did in the infinite lattice problem.

Theorem: Thevenin \& Norton Equivalents

Any network composed of voltage sources, current sources, and resistors is equivalent to a single equivalent battery $\mathcal E_{\text{eq}}$ and a single equivalent resistor $R_{\text{eq}}$. $$V(I) = \mathcal E_{\text{eq}} + IR_{\text{eq}}.$$ Any network is also equivalent to a single current source $I_{\text{eq}}$ and a single equivalent resistor $R_{\text{eq}}$. $$I(V) = I_{eq} + V/R_{eq}.$$ Some methods for finding the equivalents:

• Use Kirchhoff's law to reduce the circuit to the desired equation
• Faster tricks: \begin{itemize}
• Consider an open circuit --- the voltage across the output nodes is $\mathcal E_{\text{eq}}$ or $I_{\text{eq}}R_{\text{eq}}$.
• Consider closing the circuit with a short --- the current through is $\mathcal E_{\text{eq}}/R_{\text{eq}}$ or $I_{\text{eq}}$.

\end{itemize}

Definition: Capacitors

The capacitance of a conductor is defined as the ratio between its charge and potential, $$C = \frac QV.$$ For an object composed of two conductors, we define $$C = \frac{Q}{\Delta V},$$where $Q$ is the magnitude of the charge on each and $\Delta V$ is the difference in potential.

Idea: Energy in a Capacitor

The energy stored in a capacitor is $$U = \frac 12 CV^2 = \frac 12 QV = \frac{Q^2}{2C}.$$

Idea: RC Circuits

If you connect a charged capacitor to a resistor in series, it will discharge exponentially, $$Q(t) = Q_0 e^{-t/\tau}, \tau = RC,$$ where $\tau$ is called the time constant. The voltage and other components decay similarly. With more complicated resistor and capacitor setups, you should use Kirchoff's laws to write down differential equations and solve.

Idea: Power

The energy dissipated per time across any circuit element is equal to $$P = IV.$$ This is because for a given charge $q$, the change in energy is $qV$, so the rate of change by taking the derivative is $IV$. For a resistor, this can be written in the following equivalent forms, $$P = IV = V^2/R = I^2R.$$