Ohm's law determines the relationship between electric field and current density. The relationship between electric field and current density can be quiet complex but for some materials (mostly metals) the current density is nearly directly proportional to the electric field. Such an empirical relationship is called Ohm's law, that is

\[J = \sigma E \tag{1} \label{1}\]

where $\sigma$ is constant called conductivity. Good conductors of electricity have larger conductivities and insulators have smaller conductivities. The reciprocal of conductivity is resistivity $\rho$. Ohm's law is valid in certain situations only for some materials, so it is not the fundamental law of nature. More specific statement of Ohm's law is

OHM'S LAW: For some materials (mostly metals) the ratio of electric field to current density is a constant independent of the electric field.

The ratio of electric field to the current density is called resistivity $\rho$ which is also the reciprocal of conductivity $\sigma$,

\[\rho = \frac{E}{J} \tag{2}\]

Greater the electric field to produce a given current density, greater the resistivity. Or if current density is smaller for a given electric field, the resistivity is greater. Good conductors of electricity have smaller resistivities and insulators have larger resistivities.

Electrical conduction is analogous to thermal conduction. Good conductors of electricity have larger conductivities and are also good thermal conductors. It is because the free electrons also take part in thermal conduction. The electrical conductivities of insulators are very small, so we can enclose electric current in a specific path but the the thermal conductivities are not small enough to hold heat in a predefined path. We can not restrict heat currents in a specific path for longer time.

A potential difference has been applied to a conductor of length L of uniform cross-sectional area A. The potential difference also called voltage across the length of the conductor is $V = EL$. We know from Equation \eqref{1}, $J = \sigma E$ and we can write,

\[V = \frac{J}{\sigma}L\]

We know $J = I/A$ and the potential difference is

\[V = \frac{L}{\sigma A}I \tag{3}\]

The quantity $L/\sigma A$ is called electrical resistance $R$, so we have

\[V = RI \tag{4} \label{4}\]

The Equation \eqref{4} is used many times in electrical circuits. Ohm's law is the direct proportionality of current density to the electric field. The above equation is also sometimes refereed to as Ohm's law but it is not valid to do so. For example, junction diodes do not obey Ohm's law but the above equation correctly determines the resistance.

The SI unit of resistance is Ohm denoted by $\Omega$. The resistance is directly proportional to the length of the conductor, inversely proportional to the cross-sectional area and also directly proportional to the resistivity of the material of the conductor. Greater the length of a conductor, greater the resistance and greater the cross-sectional area smaller the resistance. Materials with greater resistivities have greater resistances.

Ohm's law is the direct proportionality of current density to the electric field for some materials not the direct proportionality electric current to the potential difference (voltage) but if the resistance is constant the relationship $V = RI$ is equivalent to Ohm's law. Considering this relationship as Ohm's law is a mistake if you don't strictly believe the resistance is constant. So, we call it the relationship between voltage, current and resistance not Ohm's law.

Conductors which obey Ohm's law are called ohmic or linear conductors and those that do not obey ohm's law are non-onmic or non-linear conductors. Note that in real Ohm's law the conductivity or resistivity does not depend on electric field.

## Resistivity, Temperature and Electrical Resistance

Resistivity increases with increasing temperature. As the temperature increases the ions in a conductor start to vibrate with greater amplitudes which causes the difficulty to the flow of charge in a particular direction. The collisions of electrons with the massive vibrating ions increases the resistivity of the conductor. It is difficult for electrons to flow through vibrating ions with greater amplitudes than with smaller amplitudes.

In case of the semiconductors such as germanium, silicon, graphite etc. the resistivity decreases with increasing temperature. The increase in temperature of semiconductors generates more free electrons available for conduction. The resistivity to temperature graphs are shown in the figures above.

For a small temperature range (up to $100{{\kern 1pt} ^ \circ }{\rm{C}}$), the resistivity of a conductor changes with temperature expressed by the following relationship:

\[\rho = \rho_0[1+\alpha(T - T_0)] \tag{5} \label{5}\]

The temperature $T_0$ is the reference temperature often taken as $0{{\kern 1pt} ^ \circ }{\rm{C}}$ or $20{{\kern 1pt} ^ \circ }{\rm{C}}$ and the factor $\alpha$ is the temperature coefficient of resistivity which is

\[\alpha = \frac{\Delta \rho}{\rho_0 \Delta T}\]

where $\Delta \rho = \rho - \rho_0$ is the change in resistivity in time interval $\Delta T = T - T_0$. The resistance, that is $R = L/\sigma A = \rho L/A$, is proportional to the resistivity, so the effect of temperature change in resistance can be expressed by the similar expression

\[R = R_0[1+\alpha(T - T_0)] \tag{6} \label{6}\]

where $R_0$ is the initial resistance at temperature $T_0$ usually taken as $0{{\kern 1pt} ^ \circ }{\rm{C}}$ or $20{{\kern 1pt} ^ \circ }{\rm{C}}$. The temperature coefficient $\alpha$ is the same as with resistivity. If we neglect the change in dimensions of a material with temperature, that is the change in $L$ and $A$, the above equation accurately determines the resistance at the final temperature (the initial resistance $R_0$ at the initial temperature $T_0$ plus the change in resistance $R_0\alpha(T-T_0)$). In most of the cases the change in $L$ and $A$ is negligibly small.

## Superconductivity

There is a situation when you continuously decrease the temperature of some materials, the resistivity suddenly reduces to zero. Such a situation when the resistivity decreases to zero is called superconductivity. In Figure 1 above the resistivity is continuously decreasing with decreasing temperature. But for some materials mostly metals and metal alloys, the resistivity suddenly drops to zero as shown in Figure 2 below. The temperature at which the resistivity is zero is called critical temperature $T_c$.

The first superconductivity was discovered by a Dutch physicist H. Kamerlingh Onnes in 1911 and found that the resistivity of mercury dropped to zero at the temperature $4.2 \text{K}$. The usefulness of superconductivities at higher temperatures is extraordinary. The superconductivity discovering race at higher critical temperatures started after that time . The next superconductivity was discovered after 75 years and the value of critical temperature $T_c$ obtained was $20 \text{K}$. In 2015 the superconductivity seen was at around $203 \text{K}$ for $\text{H}_\text{2} \text{S}$ but at very high pressure condition.