A uniform metallic conductor of length 200\,\text{m} and cross-sectional area 2.0 \times 1… — JAMB UTME

A uniform metallic conductor of length $200\,\text{m}$ and cross-sectional area $2.0 \times 10^{-6}\,\text{m}^2$ has a resistivity of $1.6 \times 10^{-8}\,\Omega\cdot\text{m}$ at an initial temperature of $20\,^\circ\text{C}$ . The temperature coefficient of resistivity for the material is $5.0 \times 10^{-3}\,^\circ\text{C}^{-1}$ . If the operating temperature of the conductor increases to $120\,^\circ\text{C}$ while it is connected across a constant potential difference of $12\,\text{V}$ , what is the magnitude of the electric current flowing through the conductor in amperes?

Answer: 5 A

Answer

The electric current flowing through the conductor is

5.0\,\text{A}

The temperature change of

100\,^\circ\text{C}

increases the resistivity of the material from

1.6 \times 10^{-8}\,\Omega\cdot\text{m}

2.4 \times 10^{-8}\,\Omega\cdot\text{m}

via

\rho = \rho_0(1 + \alpha \Delta T)

. Substituting this updated resistivity into

R = \frac{\rho L}{A}

gives a resistance of

2.4\,\Omega

. Applying Ohm's Law

I = \frac{V}{R}

with a potential difference of

12\,\text{V}

yields

5.0\,\text{A}

Step-by-Step Solution

Calculate the temperature difference

\Delta T = 100\,^\circ\text{C}

The temperature change relative to the reference temperature dictates the change in resistivity.

Calculate the resistivity at the final temperature

\rho = 2.4 \times 10^{-8}\,\Omega\cdot\text{m}

Resistivity depends on temperature according to

\rho = \rho_0(1 + \alpha \Delta T)

Calculate the total electrical resistance of the conductor

R = 2.4\,\Omega

Resistance is related to physical geometry and resistivity by

R = \frac{\rho L}{A}

Apply Ohm's law to solve for the current

I = 5.0\,\text{A}

Electric current is determined by potential difference divided by resistance (

I = V / R

Key Concept

Temperature dependence of resistivity and Ohm's Law

Estimated Time:2m 0s

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