Introduction to Ohm’s Law and Resistance | Resistivity | Thermistors
We now have basic idea about Potential difference and Current. Now let us study the relation between them. How potential difference applied across the conductor and current flowing through it are related to each other if certain physical conditions are kept the same. This relation was given by G. S. Ohm in 1828 even before the physical mechanism responsible for flow of current was discovered. We will start with Introduction to Ohm’s Law along with resistance, resistivity, specific uses of conducting materials, thermistors.Contents:

## Resistivity

${\rm{R}} \propto {\rm{L}}$ and ${\rm{R}} \propto {1 \over {\rm{A}}}$ So,$\quad {\rm{R}} \propto {{\rm{L}} \over {\rm{A}}}\quad$ or $\quad {\rm{R}} = \rho {{\rm{L}} \over {\rm{A}}}$ where, $\rho$ is called resistivity or specific resistance.In terms of microscopic quantities $E = \rho J$ so resistivity is numerically equal to ratio of magnitude of electric field to current density.$\rho = {{RA} \over L}$ So, if $L = 1{\rm{m}},{\rm{A}} = 1{{\rm{m}}^2}$ then $\rho = {\rm{R}}.$ Specific resistivity is numerically equal to resistance of substance having unit area of cross-section and unit length.It is a scalar with unit ohm-meter $(\Omega – m)$ and dimensions ${M^1 }{L^3}{T^{ – 3}}{{\rm{A}}^{ – 2}}$ .The reciprocal of resistivity is called conductivity or specific conductance with units mho/m and dimensions ${{\rm{M}}^{ – 1}}{{\rm{L}}^{ – 3}}{{\rm{T}}^3}{{\rm{A}}^2},\quad \sigma = {1 \over \rho } = {{{\rm{n}}{{\rm{e}}^2}\tau } \over {\rm{m}}} =$ ne\muImportant Points
1. The resistivity is independent of shape and size of body and depends on nature of material of body. The resistivity is the property of material while resistance is property of object.
2. The temperature dependence of resistivity is given by relation $\rho = {\rho _0}(1 + \infty \Delta \theta )$ where $\propto$is temperature coefficient of resistivity and $\Delta \theta$ is change is temperature. For metals $\propto$ is positive so resistivity increases with temperature while for non-metals $\infty$ is negative so resistivity decreases with temperature.

## Specific uses of conducting materials

1. The heating element of devices like heater, geyser, press etc. are made of nichrome because it has high resistivity and high melting point. It does not react with air and acquires steady state when red hot at $800^{\circ} \mathrm{C}$ .
2. Fuse wire is made of tin lead alloy because it has low melting point and low resistivity. The fuse is used in series, and melts to produce open circuit when current exceeds the safety limit.
3. Resistances of resistance box are made of Manganin or Constantan because they have moderate resistivity and very small temperature coefficient of resistance. The resistivity is nearly independent of temperature.
4. The filament of bulb is made up of tungsten because it has low resistivity, high melting point of 3300 K and gives light at 2400 K. The bulb is filled with inert gas because at high temperature it reacts with air forming oxide.
5. The connection wires are made of copper because it has low resistance and resistivity.
Q. Discuss the effect on the resistance R of wire when mass is kept constant and(a) length is increased n times(b) radius is increased n times(c) cross-sectional area is increased n times.

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Sol.
(a) ${\rm{R}} = {{\rho {\rm{L}}} \over {\rm{A}}} = {{\rho {\rm{L}}} \over {\rm{A}}}{{{\rm{xL}}} \over {{\rm{xL}}}} = {{\rho {{\rm{L}}^2}} \over {\rm{V}}} = {{\rho {{\rm{L}}^2}{\rm{d}}} \over {\rm{m}}}$ where d is density, m is mass and V is volume of wire.When length is increased n times the new resistance ${{\rm{R}}^\prime } = {{\rho {{({\rm{nL}})}^2}{\rm{d}}} \over {\rm{m}}}$So, $\quad {{\rm{R}}^\prime } = {{\rm{n}}^2}{\rm{R}}$(b) ${\rm{R}} = {{\rho {\rm{L}}} \over {\rm{A}}} = {{\rho {\rm{L}} \times {\rm{A}}} \over {{\rm{A}} \times {\rm{A}}}} = {{\rho {\rm{V}}} \over {{{\rm{A}}^2}}} = {{\rho {\rm{m}}} \over {{\rm{d}}{\pi ^2}{{\rm{r}}^4}}}$When radius is increased n times new resistance $R = {{\rho m} \over {d{\pi ^2}{{(nr)}^4}}}$ so ${R \over R} = {{{r^4}} \over {{{(nr)}^4}}}$ or $R = {R \over {{n^4}}}$(c) ${\rm{R}} = {{\rho {\rm{L}}} \over {\rm{A}}} = {{\rho {\rm{L}} \times {\rm{A}}} \over {{\rm{A}} \times {\rm{A}}}} = {{\rho {\rm{V}}} \over {{{\rm{A}}^2}}} = {{\rho {\rm{m}}} \over {{\rm{d}}{{\rm{A}}^2}}}$When area is increased n times new resistance ${{\rm{R}}^\prime } = {{\rho {\rm{m}}} \over {{\rm{d}}{{({\rm{nA}})}^2}}}$ so ${{{{\rm{R}}^\prime }} \over {\rm{R}}} = {{{{\rm{A}}^2}} \over {{{({\rm{nA}})}^2}}}\quad$ or $\quad {{\rm{R}}^\prime } = {{\rm{R}} \over {{{\rm{n}}^2}}}$

## Thermistors

A thermistor is a heat sensitive device made of semiconducting material where resistivity changes rapidly with change of temperature. It has a large temperature coefficient. The temperature coefficient may be positive or negative. These can be used over wide range of temperature.

### Important Applications

1. The thermistors are used to detect small temperature changes because they respond rapidly to temperature changes.
2. The thermistors with a large negative temperature coefficient is used in resistance thermometer in very low temperature measurement.
3. These are used in temperature control units of industry.
4. Thermistors are used in protection of windings of generators, motors and transformers.
5. They are used for voltage stabilization, remote sensing and safe guarding filament of picture tube of T.V.