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:

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