Electrical Conductivity: Definition, Formula, Derivation and Examples

19.5 Electrical Conductivity

19.5.1 Relation between J and E

1. Whenever current flows in a conducting material, we will have a current density J at any point in the material. At the same time, there has to be an electric field of magnitude E existing at any point within that material so that charges are forced to move through the material. Hence, vector J and vector E are found at the same point.

   

   Figure 19.7 Flow of electrical charges under an applied electric field. When an electric field (E) is applied to a conductor, positive charges move in the direction of the field, while negative charges move in the opposite direction. This motion of charges produces an electric current, and the distribution of this current is described by the current density (J), where J = σE.
   
   

2. The magnitude of J is a measure of the amount of charge flowing per unit time through a very small area. The rate of charge flow has to be determined by the electric force acting on the charges. The electric force is in turn determined by the electric field strength E. This implies that J has to be related to E.
3. At constant temperature, the two vectors are related to each other through the relationship:

$J\propto E$

or

$\boxed{J=\sigma E}$

where σ is a constant (at constant temperature) known as the electrical conductivity of the material. J points in the direction of E, and E points in the direction of positive charge flow.

19.5.2 Electrical Conductivity

1. Meaning

Electrical conductivity σ is a physical quantity whose magnitude reflects the ability of a material to conduct electric current. A material like copper, which conducts electricity much better than a material like silicon, has a much higher value of electrical conductivity.

2. Unit

The unit of σ is siemens per metre (S m^-1) or Ω^-1 m^-1.

3. Some values of σ

The values of σ of some materials are shown in the table at the side.

Example 19.7

A constant current of 5.0 A flows in a uniform copper wire of diameter 1.0 mm. Determine the p.d. applied across 10 cm of the wire.

(For copper, σ = 5.7 × 10⁷ S m^-1)

Answer

Cross-sectional area:

$A = \frac{1}{4} \pi d^2 = \frac{1}{4} \pi (1.0 \times 10^{-3})^2 = 7.85 \times 10^{-7} m^2$

Current density:

$J = \frac{I}{A} = \frac{5.0}{(7.85 \times 10^{-7})} = 6.37 \times 10^6 A \ m^{-2}$

Electric field strength:

$E = \frac{J}{σ} = \frac{(6.37 \times 10^6)}{(5.7 \times 10^7)} = 0.11 \ V \ m^{-1}$

But,

$E=\frac{V}{L}$

$V = EL = (0.11)(0.10) = 0.011 \ V$

19.5.3 Equation for σ

Suppose a p.d. is applied across a conductor so that an electric field is produced within the conductor. A conduction electron will accelerate within the conductor since it is acted upon by an electric force.

$F = ma$

where F and m represent the electric force acting on the electron and the mass of the electron respectively. But F is given by:

$F = eE$

Hence,

$ma = eE$

$a = \frac{eE}{m}$

Let the mean time interval, known as the mean free time, between collisions made by conduction electrons be t. Assume that the speed of the conduction electron drops to zero after colliding with an atom and then increases uniformly to v after time t.

$v = at$

The average speed (drift velocity) is:

$v_d = \frac{0 + v}{2} = \frac{1}{2} at = \frac{1}{2} \frac{eE}{m}t$

But,

$v_d = \frac{J}{ne}$

Hence,

$\frac{J}{ne} = \frac{1}{2} \frac{eE}{m}t$

$J = \frac{ne^2t}{2m} E$

For a material which obeys:

$J = \sigma E$

we get:

$\sigma=\frac{ne^2t}{2m}$

Applications of Electrical Conductivity in Daily Life and Industry

Electrical conductivity is a fundamental property that allows materials to carry electric current. It plays a vital role in modern technology, industrial systems, and everyday devices we use daily.

⚡ 1. Electrical Wiring

Materials with high electrical conductivity such as copper and aluminum are widely used in electrical wiring systems. These materials ensure efficient transmission of electricity with minimal energy loss.

🔌 2. Electronics and Circuits

Electrical conductivity is essential in electronic components such as capacitors, resistors, and diodes. These components regulate and control electric current in circuits.

📡 3. Sensors and Measurement Devices

Many sensors operate based on changes in electrical conductivity. Examples include temperature sensors, strain gauges, and conductivity meters used in laboratories and industries.

🏭 4. Industrial Applications

In industry, electrical conductivity is crucial for processes such as electroplating, battery production, and chemical processing. It helps improve efficiency and product quality.

🏠 5. Everyday Products

Electrical conductivity is found in many daily-use items such as touchscreens, heating elements, and electric kettles. These devices rely on conductive materials to function properly.

Conclusion: Electrical conductivity is not just a theoretical concept—it is a key factor in powering modern life, from simple household appliances to complex industrial systems.

Conclusion: Electrical Conductivity

Electrical conductivity is a fundamental property that determines how easily electric current flows through a material. It depends on the presence and mobility of charge carriers such as electrons, making it essential in understanding how electricity works.

J = σE → Current density is directly proportional to the electric field

This relationship shows that materials with high conductivity, such as metals, allow electric charges to move efficiently, while insulating materials resist current flow. As a result, conductivity directly influences energy efficiency and power transmission.

In physics and engineering, electrical conductivity is widely applied in electrical systems, electronic devices, and modern technologies. It also explains important phenomena such as resistance, heat generation, and energy transfer.

• ✔ Determines how well a material conducts electricity
• ✔ Influences efficiency in electrical systems
• ✔ Essential in electronics and industrial applications

Frequently Asked Questions (FAQ) – Electrical Conductivity

What is electrical conductivity?

Electrical conductivity is the ability of a material to conduct electric current. It depends on the number and mobility of charge carriers such as electrons.

What is the SI unit of electrical conductivity?

The SI unit of electrical conductivity is siemens per meter (S/m), also written as ohm⁻¹ m⁻¹.

What is the formula for electrical conductivity?

The relationship between current density and electric field is given by J = σE, where σ represents electrical conductivity.

What factors affect electrical conductivity?

Electrical conductivity is affected by temperature, material type, number of charge carriers, and impurities.

Why are metals good conductors?

Metals are good conductors because they contain many free electrons that can move easily under an electric field.

Keywords: electrical conductivity, conductivity formula, current density

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