(a) Define 'current density'. Is it a scalar or a vector? An electr...

Question

(a) Define 'current density'. Is it a scalar or a vector? An electric field $\vec E$ is maintained in a metallic conductor. If $n$ is the number of electrons (mass $m$, charge $-e$) per unit volume in the conductor and $	au$ is their relaxation time, show that the current density $\vec j=\alpha\vec E$, where $\alpha = \dfrac{ne^2}{m}	au$.

Accepted Answer

Step 1: Current density is the current flowing per unit cross-sectional area, in the direction of current flow; it is a VECTOR quantity.

Step 2: In an electric field $E$, each electron experiences force $eE$, giving acceleration $a=eE/m$.

Step 3: Average drift velocity gained between collisions: $v_d = a\tau = \dfrac{eE\tau}{m}$.

Step 4: Current density $j = nev_d = ne\times\dfrac{eE\tau}{m} = \dfrac{ne^2\tau}{m}E$.

Step 5: So $\vec j = \alpha \vec E$ where $\alpha = \dfrac{ne^2\tau}{m}$ (this $\alpha$ is the conductivity $\sigma$ of the material).

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