Resistance, Resistivity and Conductivity: Relationship, Formula- Embibe
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  • Written By Anum
  • Last Modified 17-05-2022
  • Written By Anum
  • Last Modified 17-05-2022

Study Resistance, Resistivity and Conductivity: SI Unit and Examples

Resistance, Resistivity and Conductivity: We deal with different kinds of materials. Some are metals, some are plastic, some are wood, and some are fluids. We are often cautioned before touching a switch with wet hands or touching a wire outside during rain, but such warnings are not given while touching plastic or wood objects! Why is that? It is because some materials allow the current to pass through them easily, while others do not. Why is it that current flows easily through some objects while in some other materials, it is severely restricted?

Some materials are conductors, some are insulators, and almost all of them are resistors. The nature of the material, its size and its dimensions may play an important role in deciding its current flow. The nature of a material can be judged based on its resistance or conductance.  Let us learn In detail about the concept of resistance, resistivity and conductivity.

Study Factors Affecting Resistance

Resistance

Resistance can be defined as the measure of the opposition of electric current through a circuit or the flow of electrons through a conductor. Thus resistance controls the amount of current flowing through a circuit. Every material offers some amount of resistance to the flow of electrons through it. Physicist Georg Simon Ohm discovered the relation between the current flowing through a metallic wire and the potential difference applied across the wire. According to him, With the temperature remaining constant, the current flowing through the given metallic wire in an electric circuit is directly proportional to the potential difference across its terminals. Thus, if \(V\) is the potential difference across the ends and \(I\) is current, then:

\(V \propto I\)

\(V = IR\,\,\,\,…\left( 1 \right)\)

Here. \(R\) is the constant of proportionality called resistance.

\(R = \frac{V}{I}\)

Thus, resistance is inversely proportional to the current flowing through the circuit. Therefore, the higher the value of resistance, the lesser the current flow; the lesser is the value of resistance, the higher the amount of current flow in the circuit.

Equation (1) is known as Ohm’s law. The SI unit of resistance is the Ohm, and its represented by the letter \(\Omega \).

Resistance and Resistivity

Resistance and Resistivity

Consider a single piece of electrical conductor of length \(L\), uniform area of cross-section \(A\).

Resistance and Resistivity

If the resistance of the conductor is \(R\) and there is a current \(i\) flowing through it, then from Ohm’s law,

\(i = V/R\)

Now, if two such conductors are joined end to end, i.e. in a series combination:

Resistance and Resistivity

The effective length of the conductor increases while the area of the cross-section remains unchanged. In this series combination of conductors,

Length of the conductor: \(L + L = 2L\)

Area of conductor: \(A\)

Since the resistors are joined in series, the effective resistance, \(R + R = 2R\)

Thus doubling the length of the conductor doubled the resistance across it. This means that the resistance of a conductor is directly proportional to its length or \(R \propto L\).

Thus, for a conductor of a given area, increasing the length would increase the resistance across it and vice versa.

If two conductors are joined along the length, i.e. in a parallel combination:

Resistance and Resistivity

The effective length of the conductor remains unchanged, although the effective area of the cross-section increases. In parallel combination of conductors:

Length of the conductor: \(L\)

Area of conductor: \(A + A = 2A\)

Since the resistors are joined in parallel, the effective resistance, \(\frac{1}{{\left[ {\left( {\frac{1}{R}} \right) + \left( {\frac{1}{R}} \right)} \right]}} = R/2\)

Thus doubling the area of cross-section of the conductor halved the resistance across it. This means that the resistance of a conductor is inversely proportional to its area or \(R \propto \frac{1}{A}\)

Thus, for a conductor of a given length, increasing the cross-section area decreases its resistance and vice versa.

Thus, from above \(R \propto \frac{l}{A}\)

or, \(R = \frac{{\rho l}}{A}\)

Here, is the resistivity of the material. It depends on the nature of the material because different conductive materials have different physical and electrical properties.

Resistivity

Resistivity

The electrical resistivity is a measure of opposition to the flow of electric current through a given conductor. It allows us to compare how efficiently different materials allow or restrict the flow of current through them. Resistivity is also known as “specific resistance.” It gives us an idea of the resistance offered to the current flow through a conductor. Higher is the resistivity of a conductor; higher will be the resistance offered by it. 

The resistivity of a substance can be defined as the resistance offered by a cube made of that substance having edges of unit length with the current flowing normally through the opposite faces of the cube and distributed uniformly all over them.

Or in simple words, the electrical resistivity is the electrical resistance per unit length and per unit cross-sectional area offered by a conductor at a specific temperature. Mathematically, 

\(\rho  = \frac{{RA}}{l} = \frac{{{\rm{ohm}} \times {{\left( {{\rm{metre}}} \right)}^2}}}{{{\rm{metre}}}}\)

The SI unit of Resistivity is Ohm-metre.

Materials that allow the electric current to pass through them easily are called conductors. These have a low value of resistivity, for example, copper is a conductor, and its resistivity is \(1.72 \times {10^{ – 8}}\Omega  – m\), making copper and aluminium ideal materials to make electric wires and cables. Although silver and gold have much lower values of resistivity, these metals are too expensive to be used as electrical wires.

Resistivity

Materials that do not allow current to pass through them easily are called insulators. These have a high value of resistivity, for example, the air is a poor conductor(or insulator), and its resistivity is \(1.5 \times {10^{14}}\Omega  – m\).

Temperature Dependence of Resistivity

The variation of resistance and hence resistivity of a conductor depends on various factors, and temperature is one of the most important characteristics that affect the resistivity of a conductor.

The resistivity of a given metallic conductor increases with the rise of temperature. When a conductor is heated, its constituent atoms start to vibrate with a greater amplitude. This, in turn, results in the increase of collision frequency between ions and electrons. Due to this, the average time between the two successive collisions decreases, resulting in a reduction of drift velocity. So this increase in collisions with the temperature rise will lead to an increase in resistivity.

The formula of resistivity of metals for small temperature variations can be given from the equation:

\(\rho \left( T \right) = \rho \left( {{T_0}} \right)\left[ {1 + \alpha \left( {T – {T_0}} \right)} \right]\)

Here,

\(\rho \left( T \right):\) resistivity of the material at temperature \(T\)

\(\rho \left( {{T_0}} \right):\) resistivity of the material at the temperature \({T_0}\)

\(\alpha :\) It is the constant for a given material, known as the coefficient of resistivity

Temperature variation of the resistance can be given from the equation:

\(R\left( T \right) = R\left( {{T_0}} \right)\left[ {1 + \alpha \left( {T – {T_0}} \right)} \right]\)

The value of resistivity of alloys also increases with temperature, but this increase is much smaller than metals, although the resistivity of non-metals decreases with the rise in temperature. Similar behaviour is observed for semi-conductors, the temperature coefficient of resistivity is negative for semi-conductors and non-metals, and its value is often large for semi-conductor materials.

Conductivity

The conductivity of a material is a measure of the ease with which electric current can flow through a material. It is also known as specific conductance. The conductivity of a substance is the inverse of its resistivity.  Higher is the value of resistivity; lower will be the value of conductivity and vice versa.

It is represented by \(\sigma \).

\(\sigma  = \frac{1}{\rho }\)

Where \(\rho \) is the resistivity.

The SI unit of conductivity is the inverse of the SI unit of resistivity. Thus, the SI unit of conductivity is \({\Omega ^{ – 1}}{m^{ – 1}}\) or Siemens/metre or \(S/m\).

The conductivity of a material is closely related to the property of conductance, and the conductance of a given material is the reciprocal of electrical resistance.

We know that the electrical resistance \(R\) and specific resistance \(\rho \) depend on the physical nature of the given material; its dimensions or physical shape are expressed in terms of its length \(L\) and area of cross-section A. Thus, the conductance of a material is a function of the nature and physical properties of the given substance.

The conductance of a substance is equal to the reciprocal of electrical resistance. It is represented by \(G\).

\(G = \frac{1}{R}\)

Where \(R\) is the resistance.

The SI unit of conductance is Seimens (S) and is represented by an inverted ohm and is represented by ℧ (mho).

Just as resistance gives an idea about the resistance to the flow of current, conductance gives an idea regarding the ease of current flow through a substance. Thus, good conductors like copper and aluminium have large conductance values, while insulators like plastic and wood have low conductance values. 

Summary

Resistance can be defined as the measure of the opposition of electric current through a circuit or the flow of electrons through a conductor. By Ohm’s law, resistance is inversely proportional to the current flowing through the circuit. The resistance can be given as:

\(R = \frac{{\rho l}}{A}\)

Here, \(\rho \) is the resistivity of the material. It depends on the nature of the material because different conductive materials have different physical and electrical properties.

Electrical resistivity is a measure of opposition to the flow of electric current through a given conductor. It allows us to compare how efficiently different materials allow or restrict the flow of current through them.

The conductivity of a material is a measure of the ease with which electric current can flow through a material. It is also known as specific conductance. The conductivity can be given as:

\(\sigma  = \frac{1}{\rho }\)

Where \(\rho \) is the resistivity. The SI unit of conductivity is the inverse of the SI unit of resistivity. Thus, the SI unit of conductivity is \({\Omega ^{ – 1}}{{\rm{m}}^{ – 1}}\) or Siemens/metre or \({\rm{S/m}}\).

FAQs on Resistance, Resistivity and Conductivity

Below here we have provided some of the most asked questions related to Resistivity and Conductivity:

Q.1. What is the SI unit of resistivity?
Ans: The SI unit of resistivity is ohm-metre.

Q.2. Define resistance.
Ans: Resistance is the opposition to the flow of electric current through a conductor.

Q.3. What are the factors on which the resistance of a material depends?
Ans: It depends on:
1. Nature of material
2. Area of the material
3. Length of material
4. Temperature

Q.4. What is the relation between resistivity and conductivity of a material?
Ans: The resistivity of a material is equal to the inverse of its conductivity.

Q.5. What is another name for conductivity?
Ans: Another name for conductivity is specific conductance.

Study Combination of Resistances Here

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