Resistance

As electric charges pass through a conductor, it constitutes an electric current. This current experiences an opposition called resistance.

There are different factors that affect resistance. Resistance depends on the length, cross-sectional area, temperature and the structure of the material.


1. Length

The resistance is directly proportional to the length of the conductor, \(R \propto L\). Thus, if we have two conductors with different length, the longer one will have a greater resistance than the shorter conductor.

\(\frac {R_1}{R_2}=\frac {L_1}{L_2}\).

Example:

An 85-m conductor has a resistance of \(0.2\;\Omega\). What is the resistance of the same wire with a length of 150 m?

Given:

\(R_1=0.2\;\Omega\\ L_1=85\;m\\ L_2=150\;m\) 

Solution:

From the given formula \(\frac {R_1}{R_2}=\frac {L_1}{L_2}\), we derive the formula for \(R_2\) which is

\(R_2=R_1 \begin{pmatrix} \frac{L_2}{L_1} \end{pmatrix}\\ \quad\;=\frac{(0.2\;\Omega)(150 \;m)}{85\;m}\\ \quad\;=\frac{30\;\Omega m}{85\;m}\\ \quad\;=0.4\;\Omega \)

Since material 2 is longer than material 1, it has a greater resistance of \(0.4\;\Omega\).


2. Cross-sectional Area

The resistance of a conductor varies inversely as its cross-sectional area, \(R \propto A\). Therefore, the greater the area, the lesser the resistance. This relationship may be expressed as

\(\frac{R_1}{R_2}=\frac{A_2}{A_1}\)

Example:

The copper wire with cross-sectional area, \(A=4\;m^2 \) has a resistance of \(1.0\; \Omega\). What will be the resistance if the area of a copper wire is increased to \(16\;m^2\)?

Given:

\(R_1=1\;\Omega\\ A_1=4\;m^2\\ A_2=16\;m^2\)

Solution:

From rhe formula, \(\frac{R_1}{R_2}=\frac{A_2}{A_1}\), we have

\(R_2=R_1 \begin{pmatrix} \frac{A_1}{A_2} \end{pmatrix}\\ \quad\;=1\;\Omega \begin{pmatrix} \frac{4\;m^2}{16\;m^2} \end{pmatrix}\\ \quad\;=0.25\;\Omega\)

The wire with greater area has lesser resistance than the wire with smaller area.


3. Temperature

Resistance also depend on the temperature of a material. Most materials have increasing resistance with increasing temperature. But, there are some materials, like glass and carbon have decreasing resistance when the temperature increases.


4. Material

The structure of a material also affects its resistance. Good conductors of electricity have very low resistances while poor conductors have higher resistances.


Resistivity is the resistance per unit of a material. It is directly proportional to the temperature of the material. It can be stated mathematically as \(R\propto \rho\).

The equation below is the combination of all the factors that affect the resistance of a material.

\(R=\rho \frac LA\)

where,

\(R=\text{resistance}\; (\Omega)\\ \rho=\text{resistivity}\;(\Omega\cdot m)\\ L=\text{length}\;(m)\\ A=\text{area}\;(m^2)\)

Example:

A wire 0.85-m long has an area of \(2.8\times10^{-3}\;m^2\). Calculate the resistance of the material if its resistivity is \(0.7\;\Omega\cdot m\).

Given:

\(L=0.85\;m\\ A=2.8 \times10^{-3}\;m^2\\ \rho=0.7\; \Omega \cdot m\)

Solution:

\(R=\rho \frac LA\\ \;\;\,=0.7\;\Omega\cdot m \begin{pmatrix} \frac{0.85\;m}{2.8\times 10^{-3}m^2} \end{pmatrix}\\ \;\;\,=212.5\;\Omega \)