Resistance Definition: It is the opposition to current and is symbolized by R. Its unit is the ohm, symbolized by the Greek letter Ω.
A voltage applied to a conductor causes a net drift of free electrons along the length of the conductor. Repulsion from electrons entering the conductor from the circuit’s energy source accelerates free electrons along the conductor. Thus, energy from the source is transferred to the free electrons as kinetic energy.
As the moving electrons collide with atoms in the conductor, some kinetic energy transfers from the electrons to the atoms. The transferred energy appears as heat since it increases the vibration of the atoms within the lattice structure of the conductor.
The collisions between the free electrons and the atoms reduce the speed at which the electrons drift in response to the applied voltage. The greater the rate of collisions in a material, the greater its resistance.
Current flowing through a resistance always produces heat. Devices such as stove elements and incandescent lamps apply this heat.
In many circuits, however, the heat produced is an unavoidable loss of energy from the system. Some applications require ventilation or other cooling to prevent the waste heat from damaging the circuit.
A resistor is a component made to have a specific resistance. If we connect a resistor between a lamp and a voltage source, as shown in Figure 1, the total resistance of the circuit increases and the current is correspondingly reduced.
The rate at which the source transfers energy to the circuit is also reduced, and that energy is divided between the lamp and the resistor. Thus, adding the resistor to the circuit dims the lamp.
Figure 1 Using a resistor to limit current
Factors Affecting Electrical Resistance
Suppose we have two pieces of wire that are identical except that one is twice the length of the other.
For free electrons traveling the full length of these wires, the average interval between collisions with atoms in the wire is the same. So, electrons passing through the longer wire have twice as many collisions as electrons passing through the shorter wire.
Consequently, the opposition of the longer wire to electric current is twice as great as that of the shorter wire. We can generalize this comparison:
The resistance of a conductor is directly proportional to its length.
\[R\alpha l\]
Next, we compare two pieces of wire that are identical except that one has twice the cross-sectional area of the other.
The thicker wire has the same cross-sectional area as two pieces of the thinner wire joined together at both ends (connected in parallel).
Each of the two smaller diameter wires will pass the same current when connected to a given voltage source. Therefore the thicker wire passes twice as much current as the thinner wire for a given applied voltage.
Since R = V/I, the thicker wire has half the resistance of the thinner wire. Again, we can generalize the comparison:
The resistance of a conductor is inversely proportional to its cross-sectional area.
\[R\alpha \frac{1}{A}\]
As we know that some materials possess more free electrons per unit volume than others. The intervals between collisions of a free electron with atoms in a material and the energy transferred by the collisions also depend on the molecular structure of the material.
For example, a silver wire has a lower resistance than a copper wire with the same dimensions, and the copper wire has a lower resistance than an aluminum wire with the same dimensions.
The resistance of a conductor is dependent on the composition of the conductor.
Resistivity
The resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. Therefore, we can calculate the resistance for any dimensions of a conductor if we know the resistance of a length of the material with a uniform cross-sectional area, assuming no change in temperature.
\[\begin{matrix} \frac{{{R}_{2}}}{{{R}_{1}}}=\frac{{{l}_{2}}}{{{l}_{1}}}\times \frac{{{A}_{1}}}{{{A}_{2}}} & {} & \left( 1 \right) \\\end{matrix}\]
Where R is the resistance of the conductor, l is the length, and A is the cross-sectional area.
Electrical Resistance Calculation Example 1
A conductor 1.0 m long with a cross-sectional area of 1.0 mm2 has a resistance of 0.017 V. Find the resistance of 50 m of wire of the same material with a cross-sectional area of 0.25 mm2.
Solution
\[{{R}_{2}}=0.017\Omega \times \frac{50m}{1m}\times \frac{1m{{m}^{2}}}{0.25m{{m}^{2}}}=3.4\Omega \]
Resistivity is a convenient quantity for calculating the resistance of a given conductor.
Resistivity Definition: The resistivity of a material is the resistance of a unit length of the material with the unit cross-sectional area. The letter symbol for resistivity is the Greek letter ρ (rho).
In SI, the resistivity of a material is the resistance between opposite faces of a cube of the material measuring 1 m along each side. We can apply dimensional analysis to determine the units for resistivity.
\[\begin{align} & \rho =resistance\text{ }of\text{ }unit\text{ }area\text{ }per\text{ }unit\text{ }length \\ & =\frac{ohms\times mete{{r}^{2}}}{meter}=ohm-meter \\\end{align}\]
Unit: The ohm meter is the SI unit of resistivity. The unit symbol for ohm meter is Ωm.
Since temperature affects resistance, values of resistivity are given for a specified temperature, usually 20°C. Table 1 lists the resistivities of the more common metallic conductor materials at 20°C.
| Material | Resistivity (nΩm) |
| Silver | 16.4 |
| Copper | 17.2 |
| Aluminum | 28.3 |
| Tungsten | 55 |
| Nickel | 78 |
| Iron | 120 |
| Constantan | About 490 |
| Nichrome II | About 1100 |
Table 1 Resistivity of some common conductors at 20°C
Electrical Resistance Formula: If we set l1 =1 m and A1 =1 m2 in Equation 1, R1 becomes ρ. Rearranging the equation gives a formula for the resistance of any conductor:
\[\begin{matrix} R=\rho \frac{l}{A} & {} & \left( 2 \right) \\\end{matrix}\]
Where R is the resistance of the conductor in ohms, l is the length of the conductor in meters, A is the cross-sectional area in square meters, and ρ is the resistivity of the conductor material in ohm-meters.
Electrical Resistance Calculation Example 2
Find the resistance at 20°C of 200 m of an aluminum conductor with a cross-sectional area of 4.0 mm2.
Solution
Using the value of ρ for aluminum from Table 1,
\[R=\rho \frac{l}{A}=2.83\times {{10}^{-8}}\Omega m\times \frac{200m}{4\times {{10}^{-6}}{{m}^{2}}}=1.4\Omega \]
In this solution, converting square millimeters to square meters allows us the cancel out all the meter units.
Most conductors have a circular cross-section. If we know the diameter of the conductor, we can calculate the cross-sectional area.
Electrical Resistance Calculation Example 3
Find the resistance at normal room temperature of 60 m of copper wire having a diameter of 0.64 mm.
Solution
$\begin{align} & A=\pi {{r}^{2}}=\frac{\pi }{4}{{d}^{2}}=\frac{\pi }{4}\times {{0.64}^{2}}m{{m}^{2}}=3.217\times {{10}^{-7}}{{m}^{2}} \\ & R=\rho \frac{l}{A}=1.72\times {{10}^{-8}}\Omega m\times \frac{60m}{3.217\times {{10}^{-7}}{{m}^{2}}}=3.2\Omega \\\end{align}$
