In this lecture, we are going to learn about the electrical conductivity of metals, which is derived from the free electron theory of metals, which we have discussed in the previous lecture. If you want to learn about it, we have given the link below.

Read about:Free Electron Theory of Metals

**What is Electrical Conductivity**

**Electrical conductivity definition**

**Electrical Conductivity**is defined as the rate of charge flows across a unit area in a conductor per unit potential gradient ( voltage difference).

- The Electrical conductivity can be expressed as \mathbf{\sigma}.

**Electrical conductivity Formula**

- Electrical Conductivity can be expressed as,

\boxed{\mathbf{\sigma = \frac{J}{E}}}

- The electrical Conductivity unit is
**mho-m**or \Omega^{-1}m^{-1}.^{-1}

**Expression for Electrical Conductivity**

The electrical conductivity of a conductor is the property by which it allows the flow of electric current. In a metal, the valence electrons are not attached to the individual atoms and are we to move about within the lattice. Hence, the valence electrons are also called free electrons or conduction electrons.

In the absence of an electric field, the motion of the free electrons is completely random like those of the molecules of a gas in a container. But, when an electric field is applied to a metal, the electrons modify their random motion in such a way that they drift slowly in the opposite direction to that of the applied field with an average velocity called the drift velocity v_{d}.

When an electric field E is applied, the free electrons in a metal experiences a force eE. Due to this force, the acceleration ‘a’ gained by the electron is

F = eE

ma = eE \; [\therefore F = ma]

or, a = \frac{eE}{m} …… Eq.(1)

Consider an electron that has just collided with an ion core. The collision momentarily destroys the tendency to drift and the electron will have a truly random direction after this collision. In the next collision, the electron velocity would have changed to an average value v_{d} given by,

v_d=a\tau …… Eq.(2)

where \tau is called the mean free time.

Substituting Eq.(1) in Eq.(2), we get

v_d=\frac{eE\tau}{m}

The current density is given by, J=nev_d, where n is the number of free electrons per unit volume.

From the above equations we get,

\mathbf{J = ne \times \frac{eE\tau}{m}}

\mathbf{J = \frac{ne^2\tau}{m}.E}

But J is also expressed as, \mathbf{J= \sigma E}.

So we can express electrical conductivity \sigma as,

\boxed{\mathbf{\sigma = \frac{ne^2\tau}{m}}}

Thus, the above expression relates the electrical conductivity to the number of free electrons per unit volume.

Also Read:Thermal Conductivity

## What is the Conductivity of Water?

The degree to which water conducts or transmits electricity or heat or sound is called the conductivity of water.

**The conductivity of Water Units**

- The conductivity of water can be measured using multiple units. Some of them are:

Units | |

SI units | Siemens per meter [S/m] |

U.S units | millimhos per centimeter [mmho/cm] |

**Value of conductivity of water **

- The value of conductivity of water for various types of water is given below:

Types of water | Conductivity Value |

Pure distilled and Deionized water | 0.05 µS/cm |

Seawater | 50 mS/cm |

Drinking water | 200 to 800 µS/cm. |

Rain or Snow water | 2 to 100 µS/cm |