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Current Electricity — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsCurrent Electricity5 Marks
Question
Establish a relation between electric current and drift velocity.$OR$
Prove that the current density of a metallic conductor is directly proportional to the drift speed of electrons.
✓

Answer

Relation between electric current and drift velocity: Consider a uniform metallic wire $XY$ of length l and cross$-$sectional area A. A potential difference $V$ is applied across the ends $X$ and $Y$ of the wire. This causes an electric field at each point of the wire of strength.

$\text{E}=\frac{\text{V}}{\text{l}}\dots\text{(i)}$ Due to this electric field, the electrons gain a drift velocity $vd$ opposite to direction of electric field. If $q$ be the charge passing through the cross$-$section of wire in $t$ seconds, then $\text{Current in write I}=\frac{\text{q}}{\text{r}}\dots\text{(ii)}$
The distance traversed by each electron in time $t =$ average velocity $\times$ time $= vd\ t$ If we consider two planes $P$ and $Q$ at a distance $vd$ t in a conductor, then the total charge flowing in time $t$ will be equal to the total charge on the electrons present within the cylinder $PQ.$ The volume of this cylinder $=$ cross sectional area $\times$ height $= A\ vd\ t$ If $n$ is the number of free electrons in the wire per unit volume, then the number of free electrons in the cylinder $= n(Avd\ t)$ If charge on each electron is $-e (e = 1.6 \times 10^{-19}C),$ then the total charge flowing through a cross$-$section of the wire $q = (nA_vd t) (-e) = –neA_vd\ t ...(iii)$
$\therefore$ Current flowing in the wire, $\text{I}=\frac{\text{q}}{\text{t}}=\frac{-\text{v}}{\text{t}}$ i.e., current $I = -neAvd ...(iv)$
This is the relation between electric current and drift velocity. Negative sign shows that the direction of current is opposite to the drift velocity. Numerically I$ = -neAvd ...(v)$
$\therefore$ Current density, $\text{J}=\frac{\text{I}}{\text{A}}=\text{d}$
$\Rightarrow\text{J}\propto\text{vd.}$ That is, current density of a metallic conductor is directly proportional to the drift velocity.

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