Resistance is a physical property of a material due to which, the material resists the flow of electricity through it. Resistance depends on the physical dimensions of the material, its composition, and temperature. The fundamental property resistivity or specific resistance is a measure of the resistance offered by a material. The resistance of a conductor is very low whereas insulators have very high resistances. The resistance of a conducting wire is proportional to its length and inversely proportional to its cross-sectional area. Resistance is a scalar quantity and it is expressed using a number with appropriate units. The SI unit of resistance is Ohm.

Ohm’s law states that the current flowing through a conductor is proportional to the potential difference between the two ends of the conductor, given that the temperature and other physical quantities remain constant. Mathematically, if the potential difference between the ends of a conductor is V, the current I flowing through it is,

I∝V

The proportionality constant is given by,

\[\frac{V}{I}\] = R

R is called the resistance of the conducting wire, which depends on the physical state and composition of the constituent material.

Image will be uploaded soon

SI unit of resistance is Ohm(\[\Omega\]) , named after Georg Ohm. If 1 A of current flows through the ends of a conductor when the two ends are kept at 1 V of potential difference, the resistance of the conductor is defined as 1 Ohm.

1 \[\Omega\] = \[\frac{1V}{1A}\]

In terms of fundamental units, Ohm can be expressed as,

\[\Omega\] = \[\frac{J}{S.A^{2}}\] = kg . \[m^{2}\] . \[s^{-3}\] . \[A^{-2}\]

The dimension of resistance is [ \[ML^{2}\] \[T^{-3}\] \[I^{-2}\] ].

The resistance of a column of mercury, which has a temperature of melting ice, uniform cross-sectional area, length of 106.3 cm, and mass of 14.4521 g, is called 1 Ohm.

Electrical conductance is defined as the reciprocal of resistance. It is a property of a material that quantifies how easily current can conduct through the material. The SI unit of conductance is siemens (S) or mho, which is the inverse of ohm, the SI unit of electrical resistance.

mho = \[\frac{1}{ohm}\] = \[\frac{A}{V}\]

The resistance R of a conductor depends on its length L, cross-section A , and its composition. For a fixed cross-section, the resistance is proportional to the length of the conductor. Whereas the resistance is inversely proportional to the cross-section for a fixed length. These two dependencies can be written down combinedly as,

R\[\infty\] \[\frac{L}{A}\]

R = \[\frac{pL}{A}\]

Here, is a proportionality constant, which is known as the specific resistance. The resistance of a homogeneous chunk of a material of unit length and unit cross-section is defined as the resistivity or specific resistance of the material. Quantitatively,

p = \[\frac{RA}{L}\]

The SI unit of specific resistance is Ohmm (\[\Omega\] . m) .

CGS unit of resistance is esu of resistance or statohm (stat \[\Omega\]). It is related to ohm as,

1 \[\Omega\] = \[\frac{1V}{1A}\]

= \[\frac{1/300 stat V}{3 × 10^{9} stat A}\]

= \[\frac{1}{9}\] X \[10^{11}\] stat \[\Omega\]

1 stat \[\Omega\] = 9 X \[10^{11}\] \[\Omega\]

Another unit of resistance is emu of resistance and it is related to ohm as,

1 emu of resistance = \[\frac{1 emu of potential}{1 emu of current}\]

= \[\frac{10^{-8} V}{10A}\]

= \[10^{-9}\] \[\Omega\]

A 5 mm diameter wire is produced from a chunk of metal. Another wire of diameter 1 cm is produced from an identical chunk. What is the ratio of the resistance of the two wires?

Solution: Resistance of a wire of length L and cross-section A is \[\frac{pL}{A}\], where is the resistivity of the material. The mass and density of the chunk are m and D respectively. If the diameter of a wire of volume V is d,

A = \[\frac{\pi d^{2}}{4}\]

V= \[\frac{m}{D}\]

L = \[\frac{V}{A}\]

Therefore, the resistance of a wire of diameter d is,

R = \[\frac{16pm}{\pi Dd^{4}}\]

According to the problem, mass and density of the two wires of diameters \[d_{1}\] = 5 mm=0.5cm and \[d_{2}\]=1 cm are same such that the ratio of resistance is,

\[\frac{R_{1}}{R_{2}}\] = \[\left ( \frac{d_{2}}{d_{1}} \right )^{4}\]

\[\frac{R_{1}}{R_{2}}\] = \[\left ( \frac{1cm}{0.5cm} \right )^{4}\]

\[\frac{R_{1}}{R_{2}}\] = 16

The ratio of the resistance of the wires is 16:1.

A parallel combination of two wires, made up of the same material, is connected to a battery. If the ratio of lengths and radii of the two wires are \[\frac{4}{3}\] and \[\frac{2}{3}\] respectively, what is the ratio of currents flowing through the wires?

Solution: Resistance of a wire of length L and radius r is,

R = \[\frac{pL}{A}\] = \[\frac{pL}{\pi r^{2}}\]

where, is the resistivity of the material.

The ratio of lengths \[L_{1}\] and \[L_{2}\] of the two wires is,

\[\frac{L_{1}}{L_{2}}\] = \[\frac{4}{3}\]

The ratio of radii \[r_{1}\] and \[r_{2}\] of the two wires is,

\[\frac{r_{1}}{r_{2}}\] = \[\frac{2}{3}\]

Since the two wires are made up of the same material, the values of are the same for both of them. The ratio of resistance \[r_{1}\] and \[r_{2}\] is,

\[\frac{r_{1}}{r_{2}}\] = \[\left ( \frac{L_{1}}{L_{2}} \right )\] \[\left ( \frac{r_{2}}{r_{1}} \right )^{2}\]

\[\left ( \frac{4}{3} \right )\] \[\left ( \frac{3}{2} \right )^{2}\]

=3

In a parallel combination of resistors, the current flowing through one conductor is inversely proportional to its resistance i.e. the ratio of currents \[L_{1}\] and \[L_{2}\] is,

\[\frac{I_{1}}{I_{2}}\] = \[\frac{R_{2}}{R_{1}}\] = \[\frac{1}{3}\]

The ratio of magnitudes of current flowing through the wires is 1:3.

Resistance depends on temperature. If other physical quantities are held constant, resistance increases with increasing temperature for metals. For glass, however, at very high temperatures, the resistivity decreases considerably.

Superconductors have zero resistance in the superconducting state (at very low temperature).

The resistivity of semiconductors decreases with increasing temperature.

FAQ (Frequently Asked Questions)

1. What is meant by resistance and resistivity?

Both resistance and resistivity quantify the amount of difficulty provided by a material to the flow of electricity. The resistance between two ends of a wire is the ratio of the potential difference of the two points to the current flowing through the conductor. Resistivity or specific resistance of a material is defined as the resistance of unit length and unit cross-section of that material.

2. What is the unit of resistance and resistivity?

In SI units, the unit of resistance is Ohm (Ω) and the unit of resistivity is Ohmm (Ωm). For practical purposes, 1 ohm is a very small unit to work with. Kilo ohm (1 k=10^{3} Ω) and Mega ohm (1 MΩ=10^{6} Ω) are some commonly used units for electrical circuits. Some smaller units are stat ohm and emu ohm.

3. Define the unit of resistance.

The SI unit of resistance Ohm can be defined as the resistance of a conductor when a current of 1 A flows through it due to a potential difference of 1 V. The international definition is as follows,

The resistance of a column of mercury, which has a temperature of melting ice, uniform cross-sectional area, length of 106.3 cm, and mass of 14.4521 g, is called 1 Ohm.