Laws of Resistance

Unit of Resistivity

Temperature Coefficient of Resistance

Resistance Variation with Temperature

## Definition of Resistance

**Electrical resistance may be defined as the basic property of any substance due to which it opposes the flow of electric current through it.**

While a voltage is applied across any substance, electric current starts flowing through it. But if we observe carefully, the electric current flows through the all substances are not equal even when the same voltage is applied across each of the substances. This is because current carrying capacity of all substances is not equal. The electric current depends upon the number of electrons' crosses the cross-section per unit time. Again this number of electrons crossing the cross-section is dependable on the free electrons available in the substances. If free electrons are plenty in a substance, the amount of current is more for same applied voltage across the substances. The electric current through a substance not only depends upon the number of free electrons in it, but also depends upon the length of path an electron has to travel to reach from lower potential end to higher potential end of the substance. In addition to that, every electron has to collide randomly with other atoms and electrons in numbers of times during its traveling.

So, every substance has a property to resist current through it and this property is known as electric resistance.

If one volt across a conductor produces one ampere of current through it, then the resistance of the conductor is said to be one ohm (Ω).

## Laws of Resistance

There are mainly two **laws of resistance** from which the resistivity or specific resistance of any substance can easily be determined. One law is related to cross-sectional area of the conductor and other law is related with its length.

As stated earlier, the current through any conductor depends upon number of electrons passes through a cross-section per unit time. So if cross section of any conductor is large then more electrons can cross it that means more electric current can flow through the conductor. For fixed voltage, more current means less electrical resistance. So it can be concluded like that resistance of any conductor is inversely proportional to its cross-sectional area.

If the length of the conductor is increased, the path traveled by the electrons is also increased. If electrons travel long, they collide more and consequently the number of electron passing through the conductor becomes less; hence current through the conductor is reduced. In other word, resistance of the conductor increases with increase in length of the conductor.

The **laws of resistance** state that,

Electrical resistance R of a conductor or wire is

- directly proportional to its length, l i.e. R ∝ l,
- inversely proportional to its area of cross-section, a i.e.

Combining these two laws we get,

Where ρ (rho) is the proportionality constant and known as

**resistivity**or

**specific resistance**of the material of the conductor or wire. Now if we put, l = 1 and a = 1 in the equation,

We get, R = ρ. That means resistance of a material of unit length having unit cross - sectional area is equal to its

**resistivity**or

**specific resistance**.

**Resistivity of a material**can be alliteratively defined as the

**electrical resistance**between opposite faces of a unit cube of that material. Hence we have seen that laws of resistance are very simple.

### Unit of Resistivity

The **unit of resistivity** can be easily determined form its equation

The

**unit of resistivity**is Ω-m in MKS system and Ω-cm in CGS system and

1 Ω-m = 100 Ω-cm.

### Temperature Coefficient of Resistance and Inferred Zero Resistance Temperature

Materials | Resistivity in μ Ω-cm at 20^{o}C |
Temperature Coefficient of Resistance in Ω per ^{o}C at 20^{o}C |
Inferred Zerro Resistance Temperature in ^{o}C |
---|---|---|---|

Aluminium | 2.82 | 0.0039 | − 236 |

Brass | 6 to 8 | 0.0020 | − 480 |

Carbon | 3k to 7k | 0.00005 | |

Constantan | 49 | 0.000008 | − 125000 |

Copper | 1.72 | 0.00393 | − 234.5 |

Gold | 2.44 | 0.0034 | − 274 |

Iron | 12.0 | 0.005 | − 180 |

Lead | 22.0 | 0.0039 | − 236 |

Manganin | 42 to 74 | 0.00003 | − 236 |

Mercury | 96 | 0.00089 | − 1100 |

Nickel | 7.8 | 0.006 | − 147 |

Silver | 1.6 | 0.0038 | − 310 |

Tungsten | 5.51 | 0.0045 | − 200 |

Zinc | 6.3 | 0.004 | − 230 |

## Resistance Variation with Temperature

There are some materials mainly metals, such as silver, copper, aluminum, which have plenty of free electrons. Hence this type of materials can conduct current easily that means they are least resistive. But the resistivity of these materials is highly dependable upon their temperature. Generally metals offer more electrical resistance if temperature is increased. On the other hand the resistance offered by a non - metallic substance normally decreases with increase of temperature.

If we take a piece of pure metal and make its temperature 0° by means of ice and then increase its temperature from gradually from 0°C to to 100°C by heating it. During increasing of temperature if we take its resistance at a regular interval, we will find that electrical resistance of the metal piece is gradually increased with increase in temperature. If we plot the **resistance variation with temperature** i.e. resistance Vs temperature graph, we will get a straight line as shown in the figure below. If this straight line is extended behind the resistance axis, it will cut the temperature axis at some temperature, − t_{0}°C. From the graph it is clear that, at this temperature the electrical resistance of the metal becomes zero. This temperature is referred as *inferred zero resistance temperature*. Although zero resistance of any substance cannot be possible practically. Actually rate of **resistance variation with temperature** is not constant throughout all range of temperature. Actual graph is also shown in the figure below.

Let's R_{1} and R_{2} are the measured resistances at temperature t_{1}°C and t_{2}°C respectively. Then we can write the equation below,

From the above equation we can calculate resistance of any material at different temperature. Suppose we have measured resistance of a metal at t_{1}^{o}C and this is R_{1}. If we know the inferred zero resistance temperature i.e. t_{0} of that particular metal, then we can easily calculate any unknown resistance R_{2} at any temperature t_{2}°C from the above equation.

The resistance variation with temperature is often used for determining temperature variation of any electrical machine. For example, in temperature rise test of transformer, for determining winding temperature rise, the above equation is applied. This is impossible to access winding inside the an electrical power transformer insulation system for measurement of temperature but we are lucky enough that we have resistance variation with temperature graph in our hand. After measuring electrical resistance of the winding both at the beginning and end of the test run of the transformer, we can easily determine the temperature rise in the transformer winding during test run.

20°C is adopted as standard reference temperature for mentioning resistance. That means if we say resistance of any substance is 20 Ω that means this resistance is measured at the temperature of 20°C.

### Video Resistance Variation with Temperature

As we discussed in the page under title

resistance variation with temperaturethat electrical resistance of every substance changes with change in its temperature.

**Temperature coefficient of resistance**is the measure of change in electrical resistance of any substance per degree of temperature rise.

Let a conductor having a resistance of R_{0} at 0°C and R_{t} at t°C respectively.

From the equation of resistance variation with temperature we get

This α_{o} is called **temperature coefficient of resistance** of that substance at 0°C.

From the above equation, it is clear that the change in electrical resistance of any substance due to temperature rise mainly depends upon three factors-

- The value of resistance at initial temperature,
- Rise of temperature and
- the α
_{o}.

This α_{o} is different for different materials, so effect on resistance at different temperature are different in different materials.

So the temperature coefficient of resistance __at 0°C__ of any substance is the reciprocal of the inferred zero resistance temperature of that substance.

So far we have discussed about the materials that resistance increases with increase in temperature, but there are many materials that's electrical resistance decreases with decrease in temperature. Actually in metal if temperature is increased, the random motion of charged particles inside the materials increases which results to more collisions. More collision resist smooth flow of electrons through the metal, hence the resistance of the metal increases with the increase in temperature. So, temperature coefficient of resistance is considered as positive for metal.

But in case of semiconductor or other non - metal, the number of free electrons increases with increase in temperature. That means if temperature increases, more number of electrons comes to the conduction bands from valance band by crossing the forbidden energy gap. As the number of free electrons increases, the resistance of this type of non-metallic substance decreases with increase of temperature. Hence temperature coefficient of resistance is negative for non-metallic substances and semiconductors.

If there is approximately no change in resistance with temperature, the value of this coefficient is considered as zero. Such as alloys like constantan and manganin that's temperature coefficient of resistance is nearly zero.

The value of this coefficient is not constant, it depends upon the initial temperature on which the increment of resistance is based. When the increment is based on initial temperature of 0°C, the value of this coefficient is α_{o} - which is nothing but the reciprocal of the respective inferred zero resistance temperature of the substance. But at any other temperature , temperature coefficient of electrical resistance is not same as this α_{o}. Actually for any material, the value of this coefficient is maximum at 0°C temperature. Say the value of this coefficient of any material at any t°C is α_{t}, then its value can be determined by the following equation,

The value of this coefficient at a temperature of t_{2}°C in the term of the same at t_{1}°C is given as,