3.0 Ohm's law

  Current Electricity

The current $(I)$ flowing through a conductor is directly proportional to the potential difference $(V)$ applied across its end, at constant temperature $(T)$ and pressure $(P)$.

Thus, $$\begin{equation} \begin{aligned} \Rightarrow V \propto I\quad \quad ({\text{At constant }}T{\text{ }}P) \\ \Rightarrow V = RI \\ \Rightarrow \frac{V}{I} = R \\\end{aligned} \end{equation} $$

where, $R$ is the constant of propotionality and is known as electric resistance of conductor.

Graph between $V$ and $I$ is a straight line.

Deduction of ohm's law

When potential difference $V$ is applied across a conductor of length $L$, an electric field is produced in the conductor.

Mathematically it is given as, $$E = \frac{V}{L}$$

Drift velocity in terms of $V$ is, $${v_d} = \frac{{eE\tau }}{m} = \frac{{eV\tau }}{{mL}}$$

If $n$ is number of electrons per unit volume, and $A$ is the area of cross-section of conductor, then current through the conductor will be, $$I = neA{v_d} = enA\frac{{eV\tau }}{{ml}}$$

or, $$\frac{V}{I} = \frac{{ml}}{{n{e^2}\tau A}}$$

At constant temperature, quantities $m$, $l$, $n$, $e$, $\tau$ and $A$ are constant.

Therefore, $$\frac{V}{I} = const=R$$

This proves ohm's law is valid at constant $(T)$ and constant pressure $(P)$.

Resistance of conductor,$$R = \frac{{ml}}{{n{e^2}\tau A}}$$

Resistance

The property of a substance by virtue of which it opposes the flow of charges through it.

It is equal to the ratio of a potential difference $(V)$ applied across the conductor to the current $(I)$ flowing through it. $$ \Rightarrow R = \frac{V}{I}$$

SI unit of resistance is $ohm$. It is represented by the symbol $\Omega $.

One ohm $\left( {1\Omega } \right)$

The resistance of a conductor is said to be one ohm if a current of $1A$ flows through it on applying a potential difference of $1V$ across its end. $$1\Omega = 1\frac{V}{A}$$

Cause of resistance of conductor

It is due to collisions of electrons with the ions of conductor while drifting towards the positive end of the conductor.

Factors affecting the resistance of conductors are,

• Directly proportional to the length of a conductor $$R \propto L$$
• Inversely proportional to the area of cross-section $$R \propto \frac{1}{A}$$
• Depends on the nature of a material

Combining first two factors, $$\begin{equation} \begin{aligned} \Rightarrow R \propto \frac{L}{A} \\ \Rightarrow R = \rho \frac{L}{A} \\\end{aligned} \end{equation} $$

where $\rho $ is known as the constant of proportionality and is called Resistivity.

Resistivity

If $L=1m$ and $A = 1{m^2}$, then resistance $R$ is, $$R = \rho $$

Thus , resistivity is defined as resistance of a conductor of unit length and unit area of cross-section.

SI unit is $\Omega m$

Conductance $(G)$

Inverse of resistance is known as conductance. $$ \Rightarrow G = \frac{1}{R}$$

SI unit of conductance is ${\Omega ^{ - 1}}$ or $mho$ or Siemens $(S)$.

Conductivity $(\sigma )$

Inverse of the resistivity of a material is called its conductivity. $$ \Rightarrow \sigma = \frac{1}{\rho }$$

SI unit of conductivity is ${\Omega ^{ - 1}}{m^{ - 1}}$ or $\frac{{mho}}{m}$ or $\frac{S}{m}$.

Resistivity in terms of electron density and relaxation time

Resistance of conductor of length $L$ , area of cross-section $A$ and resistivity $\rho $ is$$R = \rho \frac{L}{A}$$

But, $$R = \frac{{mL}}{{n{e^2}\tau A}}$$

Comparing the above two equations, $$\rho = \frac{m}{{n{e^2}\tau }}$$

Thus, $\rho$ is independent of dimensions of the conductor but depends on,

1. Electron density $(n)$ of the conductor

2. Relaxation time $(\tau)$

Relation between $(\vec j$, $\sigma $, $\vec E)$

For electron ,$$q = - e$$

where, ${\vec v_d} = \frac{{ - e\vec E\tau}}{m}$

$$\vec j = nq{\vec v_d} = n( - e)\frac{{ - e\vec E\tau }}{m} = \frac{{n{e^2}\tau }}{m}\vec E$$

where, $$\frac{1}{\rho } = \frac{{n{e^2}\tau }}{m}\vec E = \sigma $$

Thus, $$\begin{equation} \begin{aligned} \Rightarrow \vec j = \sigma \vec E \\ \Rightarrow \vec E = \rho \vec j \\\end{aligned} \end{equation} $$

The above relation is also known as the vector form of Ohm's law.

Classification of material in terms of resistivity

1. Conductor: have low resistivity in the range ${10^{ - 8}}\Omega $ to ${10^{ - 6}}\Omega $

2. Insulator: have very high resistivity in the range ${10^{4}}\Omega $ to ${10^{8}}\Omega $

3. Semiconductor: Have resistivity between conductor and insulator i.e in the range ${10^{ - 6}}\Omega $ to ${10^{4}}\Omega $