Physics > Current Electricity > 2.0 Conduction of current in a metal
Current Electricity
1.0 Introduction
2.0 Conduction of current in a metal
3.0 Ohm's law
3.1 Temperature dependence of resistance
3.2 Resistivities of different materials
3.3 Limitations of ohm's law
4.0 Combination of Resistors
5.0 Electromotive force $\left( \xi \right)$
6.0 Heating effect of current
7.0 Wheatstone bridge
8.0 Metre Bridge Or Slide wire bridge
9.0 Potentiometer
9.1 Comparison of emfs of two primary cells.
9.2 Determination of Internal resistance of a cell using potentiometer
10.0 Electrical devices
2.1 Expression for drift velocity
3.2 Resistivities of different materials
3.3 Limitations of ohm's law
9.2 Determination of Internal resistance of a cell using potentiometer
Force experienced by electron in presence of electric field, $\vec F$= -e$\vec E$
Accelaration acquired by an electron , $$\vec a = \frac{{ - e\vec E}}{m}$$ where $m$ is the mass of an electron.
If $N$ electrons having random thermal velocities ${\vec u_1},{\vec u_2}.....{\vec u_n}$ and accelerates for time ${t_1},{t_2}.....{t_n}$ respectively, then final velocities are, $$\begin{equation} \begin{aligned} {{\vec v}_1} = {{\vec u}_1} + \vec a{t_1} \\ {{\vec v}_2} = {{\vec u}_2} + \vec a{t_2} \\ .... \\ .... \\ {{\vec v}_n} = {{\vec u}_n} + \vec a{t_n} \\\end{aligned} \end{equation} $$
Drift velocity of $N$ electrons, $$\begin{equation} \begin{aligned} {{\vec v}_d} = \frac{{{{\vec v}_1} + {{\vec v}_2} + .....+{{\vec v}_n}}}{N} \\ \Rightarrow \frac{{\left( {{{\vec u}_1} + \vec a{t_1}} \right) + \left( {{{\vec u}_2} + \vec a{t_2}} \right) + .....+\left( {{{\vec u}_n} + \vec a{t_n}} \right)}}{N} \\ \Rightarrow \frac{{{{\vec u}_1} + {{\vec u}_2} + .....+{{\vec u}_n}}}{N} + \frac{{\vec a({t_1} + {t_2} + .....+{t_n})}}{N} \\ \Rightarrow 0 + \vec a\tau \\ \Rightarrow \vec a\tau \\\end{aligned} \end{equation} $$
where $\tau$ is relaxation time. It is given as, $$\tau = \frac{{({t_1} + {t_2} + .....+{t_n})}}{N}$$
Thus, $$\begin{equation} \begin{aligned} \Rightarrow {{\vec v}_d} = \vec a\tau \\ \Rightarrow {{\vec v}_d} = \frac{{ - e\vec E\tau }}{m} \\\end{aligned} \end{equation} $$
where negative sign shows that the direction of drift velocity is opposite to that of $\vec E $
Relation between current $(I)$ and drift velocity ${({\vec v}_d})$
Consider a conductor of length $L$ and area of cross-section $A$.
Let $n$ = number of electron per unit volume,
${{v_d}}$ = drift velocity of an electron,
Number of electrons in length $L$ of a conductor is given by $=nAL$
Total charge in length $L=neAL$
Time taken by electron to pass the length $L$, $$t = \frac{L}{{{v_d}}}$$
Current $(I)$, $$\begin{equation} \begin{aligned} \Rightarrow I = \frac{q}{t} = \frac{{neAL}}{{\frac{L}{{{v_d}}}}} \\ \Rightarrow I = neA{v_d} \\\end{aligned} \end{equation} $$
Current density $(j)$, $$\begin{equation} \begin{aligned} \Rightarrow \overrightarrow j = \frac{I}{A} = \frac{{neA{v_d}}}{A} \\ \Rightarrow \vec j = ne{{\vec v}_d} \\\end{aligned} \end{equation} $$
