A metallic conductor wire of cross-sectional area 2.5 \times 10^{-6}\,\text{m}^2 has a res… — JAMB UTME

A metallic conductor wire of cross-sectional area $2.5 \times 10^{-6}\,\text{m}^2$ has a resistance of $10.0\,\Omega$ at $0\,^\circ\text{C}$ . The temperature coefficient of resistance of the material is $5.0 \times 10^{-3}\,^\circ\text{C}^{-1}$ . The conductor contains a free-electron density of $5.0 \times 10^{28}\,\text{m}^{-3}$ . When the wire is heated to $100\,^\circ\text{C}$ and connected across a potential difference of $60\,\text{V}$ , what is the drift velocity of the conduction electrons in the wire in millimeters per second ( $\text{mm/s}$ )? (Take elementary charge $e = 1.6 \times 10^{-19}\,\text{C}$ .)

Answer: 0.2 mm/s

Answer

The drift velocity of the conduction electrons is

0.2\,\text{mm/s}

The resistance increases from

10.0\,\Omega

15.0\,\Omega

when heated from

0\,^\circ\text{C}

100\,^\circ\text{C}

. Applying

60\,\text{V}

results in a current of

4.0\,\text{A}

. Combining this with the cross-sectional area and electron density gives a drift velocity of

2.0 \times 10^{-4}\,\text{m/s}

, which equals

0.2\,\text{mm/s}

Step-by-Step Solution

Calculate the resistance at the operating temperature (

100\,^\circ\text{C}

)

R_{100} = 15.0\,\Omega

Resistance varies with temperature according to

R_T = R_0(1 + \alpha \Delta T)

Find the current in the wire using Ohm's Law

I = 4.0\,\text{A}

Current is given by

I = V / R_{100}

Determine current density

J

J = 1.6 \times 10^6\,\text{A/m}^2

Current density is total current per unit cross-sectional area,

J = I / A

Calculate the electron drift velocity

v_d

v_d = 2.0 \times 10^{-4}\,\text{m/s} = 0.2\,\text{mm/s}

Drift velocity relates to current density by

v_d = J / (n e)

Key Concept

Temperature Dependence of Resistance and Microscopic Model of Electric Current

Estimated Time:2m 0s

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