Electrical Resistance

Electrical Resistance

The resistance of a conductor is defined as its ability to oppose the flow of charge through it. It is measured by the ratio of the potential difference V across its ends to the current I flowing through it.

\begin{align*} R &= \frac VI \\ \text {Unit of R is ohm,} \Omega \: \text {in SI-units. So,} \\ \text {one ohm} &= \frac {1\: \text {volt}}{1\: \text {ampere}} \\ \end{align*}

The resistance of a conductor is said to be one ohm if one ampere current flows through it under a potential difference of one volt.

Resistivity or Specific Resistance

It is observed that at a constant temperature, the resistance R of a conductor is directly proportional to its length ‘l’ and inversely proportional to the cross-sectional area A.

Let l be the length of a conductor resistance R and A be its cross-sectional area. Then

\begin{align*} R &\propto l \dots (i) \\ \text {and} \: R &\propto \frac 1A \dots (ii) \\ \text {Combining equations} (i) \text {and} (ii), \text {we get} \\ R &\propto \frac {l}{A} \\ \text {or,} R &= \frac {\rho l}{A} \end{align*}

where ρ is proportionality constant, called resistivity or specific resistance of the material of conductor. Different materials have different values of resistivity.

If l = 1 m, A = 1 m², then R = ρ.

Thus, the resistivity of the material of a conductor is defined as the resistances of the conductor of unit cross-sectional are per unit length. Its unit is ohm meter, W m in SI-units.

Conductance

The reciprocal of resistance of a conductor is called its conductance (C). if a conductor has resistance then its conductance is given by,

$$ C = \frac 1R $$

The unit of conductance is mho or (ohm)^-1 in SI-unit. This is also called Siemen, S.

Conductivity

The reciprocal of resistivity of a conductor is called its conductivity is given by,

$$\sigma = \frac {1}{\rho } $$

The unit of conductivity is ohm^-1 meter^-1 or W^-1m^-1 or Sm^-1 SI-unit.

Relation between J and E

When a potential difference V is applied to a conductor of resistance R, the current flowing through the conductor, \( I = \frac VR\).

If A be the uniform area of cross-section of the conductor and l be its length, then

\begin{align*} R &=\frac {\rho l}{A} = \frac {l}{\sigma A} \\ \text {where} \: \sigma = \frac {1}{\rho } \text {is the conductivity. Hence} \\ I &= \frac VR = \frac {\sigma AV}{l} \\ \therefore \frac IA &= \sigma \frac Vl \\ \end{align*}

But current density \(J = \frac IA\) and electric intensity, \(E = \frac Vl\)

Hence, \( J = \sigma E\)

This equation gives the relation between J and E.

Effect of Temperature on Resistance

The resistance of a conductor is observed to increase with the increase in the temperature of the conductor. If a conductor has resistance R₁ at θ₁^oC and R₂ at θ₂^oC , then the increase in resistance (R₂ – R₁) is directly proportional to initial resistance R₁ and rise in temperature (θ₂-θ₁). So,

\begin{align*} (R_2 - R_1) &\propto R_1(\theta _2 - \theta _1) \\ \text {or,}\: (R_2 - R_1) &= \infty R_1(\theta _2 - \theta _1) \\ \text {or,} \: R_1 &= R_1(1+ \infty \Delta \theta) \dots (i) \\ \end{align*} where \( \infty\) is a constant of proportionality, called temperature coefficient of resistance. \begin{align*} \text {The equation}\: (i) \: \text {can be written as} \\ \infty &= \frac {R_2 – R_1}{R_1\theta } \dots (ii) \\ &= \frac {\text {increase in resistance}}{\text {original resistance} \times \text {rise in temperature}} \\ \end{align*}

Hence, the temperature coefficient of resistance of a conductor is the increase in resistance per unit original resistance per ^oC rise in temperature. So,

$$\infty = \frac {\Delta R}{R_1\theta} $$

Superconductor

Superconductors are materials which have zero electrical resistance. Metals can show this property at very low temperature. Superconductors can be classified into two groups:

1. Type-I (or soft} super conductors
2. Type-II (or hard) super conductors

The superconductors in which the magnetic field is totally excluded from the interior of the superconductors below a certain magnetizing field Hc, and at Hc the material loses super conductivity and the magnetic field penetrates fully are termed as type I or soft superconductors.

The superconductors in which the material loses magnetization gradually rather than suddenly are termed as type II or hard superconductors

Super conductors have many applications in various fields. Some of them are as follows:

1. Superconductors are used for the generation and transmission of electrical power.
2. They are also used in medical diagnosis (MRI).
3. They are used in the super computer.
4. They are used in magnetically levitated world’s fastest trains.

Perfect Conductors

A perfect conductor is an electrical conductor with no resistivity. All known perfect conductors are also superconductors: in addition to having no electrical resistance they exhibit quantum effects such as the Meissner effect and quantization of magnetic flux through closed loops. A perfect conductor without the special quantum properties of real superconductors is known as a classical superconductor, but that phrase is ambiguous as it is also used to distinguish between conventional superconductors and high-temperature superconductors.

Ohmic and Non-Ohmic Conductors

The conductors which follow Ohm’s law (I α V) are called ohmic conductors. Most of the metals such are copper, silver, gold are ohmic-conductors. The current-voltage graph of ohmic-conductors is a straight line passing through the origin.

The conductors which do not follow Ohm’s law (I α V) are called non-ohmic conductors. Electrolytes, vacuum tubes, junction diodes, transistors etc are some examples of non-ohmic conductors. The graph of non-ohmic conductors is curved.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.