Resistors

Resistors are any module in a circuit that hinders the passing of current.

There are 2 measures for resistors: resistivity and resistance.

Resistivity measures how poorly a material conducts electricity. It is defined as follows

$$\rho =\frac{\vec{E}}{\vec{J}}={\rho }_{293k}(1+\alpha (T-T_{293k}))=\frac{\Delta V}{dJ}=\frac{\Delta VA}{dI}\propto \frac{1}{\sigma }$$

On the other hand, resistance measures how poorly a wire/resistor conducts electricty.

$$R=\frac{\Delta V}{I}=\frac{L\rho }{A}$$

The formula above is called Ohm’s Law.

Series §

Since we know that the current doesn’t change from each resistor but voltage adds up, we can solve to find the total resistance of resistors in series.

$$\begin{array}{rl}V& =IR\\ V_{tot}& =V_1+V_2+V_3\\ I{R}_{tot}& =I{R}_1+I{R}_2+I{R}_3\\ R_{tot}& =R_1+R_2+R_3\end{array}$$

Parallel §

Since we know that the current splits and adds up but the voltage stays the same, we can solve for the total resistance of resistors in parallel.

$$\begin{array}{rl}I& =\frac{V}{R}\\ I_{tot}& =I_1+I_2+I_3\\ \frac{V}{R_{tot}}& =\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}\end{array}$$