Is current density a vector quantity or scalar quantity? Deduce the relation between current density and potential difference across a current carrying conductor of length l, area of cross-section A and number density of free electrons n. How does the current density in a conductor vary with
(a) increase in potential gradient,
(b) increase in temperature,
(c) increase in area of cross-section?

Current-density is a vector quantity.
If potential difference V is applied across a conductor of length 'l' and area of cross-section 'A' having number of free electrons per unit volume n.
Then, the charge in the conductor

and current,

I = \frac{q}{t}

= neA (\frac{l}{t})

Therefore,
Current per unit area i.e., current density

J = \frac{I}{A} = {nev}_{d}

(\frac{l}{t} = ν_{d} drift speed)

J = ne \frac{eV}{ml} . τ

J = \frac{{ne}^{2} Vτ}{m . l}

where m is mass of an electron and

τ

is the relaxation time.

(a) current density increases with increase in potential gradient

(\frac{V}{l})

.

(b) when temperature increases, relaxation time (τ) decreases and current density decreases.

(c) when area of cross-section increases the current density

J = \frac{I}{A}

decreases.

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