Resistance

Electric resistance, $R$ is the tendency of a material to resist the flow of an electric current, $I$. The SI unit of resistance is the Ohm ($\Omega$).

Ohm’s Law

The current flowing through a conductor is directly proportional to the voltage across its end, given that temperature is constant: $$ \begin{equation}\begin{aligned} V&=IR\\ \end{aligned}\end{equation} $$

Electric resistance is thus the ratio of the voltage applied to a conductor to the electric current that results from this voltage:

Resistance in series and parallel

Consider how voltage behaves in series: $$ \begin{equation}\begin{aligned} V_T&=V_1+V_2+V_3\\ \end{aligned}\end{equation} $$

Dividing throughout by the same current ($I_T=I_1=I_2=I_3$): $$ \begin{equation}\begin{aligned} \frac{V_T}{I_T}&=\frac{V_1}{I_1}+\frac{V_2}{I_2}+\frac{V_3}{I_3}\\ \therefore R_T&=R_1+R_2+R_3\\ \end{aligned}\end{equation} $$

For resistors in parallel, the current is additive: $$ \begin{equation}\begin{aligned} I_T&=I_1+I_2+I_3\\ \end{aligned}\end{equation} $$

The voltage is the same throughout so we can divide each term by the voltage ($V_T=V_1=V_2=V_3$): $$ \begin{equation}\begin{aligned} \frac{I_T}{V_T}&=\frac{I_1}{V_1}+\frac{I_2}{V_2}+\frac{I_3}{V_3}\\ \therefore \frac{1}{R_T}&=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\\ \end{aligned}\end{equation} $$

Power and resistance

Recall the formula for power in terms of current and voltage: $$ \begin{equation}\begin{aligned} P&=IV\\ \end{aligned}\end{equation} $$

Combining this with Ohm’s Law: $$ \begin{equation}\begin{aligned} P&=I(IR)\\ P&=I^2R\\ \end{aligned}\end{equation} $$

Also, $I=\frac{V}{R}$: $$ \begin{equation}\begin{aligned} P&=(\frac{V}{R})V\\ P&=\frac{V^2}{R}\\ \end{aligned}\end{equation} $$

Resistivity

Resistance is an extensive property (a property whose values changes based on the amount of matter present). This means that a wire’s resistance can vary according to its dimensions. Using resistance can be limiting because knowing the resistance of an object (e.g. a copper wire) will not help us to determine the resistance of another object (e.g. copper rod).

In other words, resistance does not translate across multiple items made of the same material. Resistivity, $\rho$ is an intensive property, meaning that its value does not depend on the amount of the material present. It can thus be applied to many objects as long as they are made of the same material.

The formula relating resistivity, $\rho$ to resistance, $R$ is:

$$ \begin{equation}\begin{aligned} R&=\frac{\rho L}{A}\\ \end{aligned}\end{equation} $$

Where $L$ is the length of the wire and $A$ is the cross-sectional area of the wire. Thus if we know the resistivity and have the length and cross-sectional area for the wire, we can easily calculate the resistance.