Question

Establish relation between current density and drift velocity?

Answer 1

Drift velocity (Vd)is the average velocity with which the free electrons (e) in a circuit drifts towards the positive terminal of the battery, under the inflence of the electric field (E) applied.
It is given by Vd = eET/m, where T is the relaxation time and m is the mass of electrons.

If a current, I, is applied to a conductor of length L and area A, then n electrons with charge, e gets drifted towards the positive terminal of the cell under a potential difference, V, in time, t, then:

Volume of the conductor, V = AL ;

Charge in the conductor, q = volume * n * e = ALne (and)

t = distance travelled / Velocity = L / Vd

We have, I = q/t

Thus,
 = ALne * Vd /  L

(or) I = AneVd ...(i)

Current Density (j) is defined as the current (I) per unit area (A) of the conductor.

ie, j = I/A (or) I = jA ...(ii)

Thus, from (i) and (ii),



=> j = neVd

But q = ne (quantisation of charge)

Hence, j = qVd (or) Vd  = j/q

Answer 2

The relation between current density $J$ and drift velocity ${v_d}$ is $J = ne{v_d}$.

Explanation:

• Consider a wire of length $L$, cross section area $A$, and the density of electrons in the wire is $n$.
• Consider that the electrons are flowing from right to left.
• Consider in time $t$, electrons covers $d$ distance and drift velocity of electrons is ${v_d}$, then,

• ${v_d} = \frac{d}{t}$                                      ...(i)

• The total number of electrons contained in $d$ length is

       $N = volume \times density = Adn$

• The total charge contained in distance $d$ is

        $Q = Ne = Adne$                 ...(ii)

• The current, i.e., the charge flowing in time $t$ in wire is

        $I = \frac{Q}{t}$                                ...(iii)

• Putting the value of $Q$ from equation (ii) and the value of $t$ from equation (i) in the equation (iii), we get,

        $I = \frac{{Adne}}{{{\raise0.7ex\hbox{ d } \!\mathord{\left/ {\vphantom {d {{v_d}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{ {{v_d}} }}}} = Ane{v_d}$           ...(iv)

• Therefore, the current density can be given as

       $\begin{gathered} J = \frac{I}{A} \hfill \\ J = \frac{{Ane{v_d}}}{A} \hfill \\ J = ne{v_d} \hfill \\ \end{gathered}$

• Therefore, the required relation is $J = ne{v_d}$.

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