A metal bar is $20 \mathrm {~cm}$ long and has a rectangular cross-section measuring $1.0 \mathrm {~cm} \times 2.0 \mathrm {~cm}$. What is the voltage drop along its length when it carries a 4000-A current? The resistivity of the metal is $1.68 \times 10 ^ { - 8 } \Omega \cdot \mathrm { m }$.
A) $0.034 \mathrm {~V}$
B) $0.67 \mathrm {~V}$
C) $0.067 \mathrm {~V}$
D) $0.34 \mathrm {~V}$

Question

Quizplus · Accepted Answer

The resistance $R$ of the metal bar can be calculated using the formula $R = \rho \frac{L}{A}$, where $\rho$ is the resistivity, $L$ is the length, and $A$ is the cross-sectional area. Here, $L = 20 \times 10^{-2} \, \mathrm{m}$, $A = 1.0 \times 2.0 \times 10^{-4} \, \mathrm{m}^2$, and $\rho = 1.68 \times 10^{-8} \, \Omega \cdot \mathrm{m}$. Thus, $R = 1.68 \times 10^{-8} \times \frac{20 \times 10^{-2}}{2 \times 10^{-4}} = 1.68 \times 10^{-3} \, \Omega$. The voltage drop $V$ can be found using Ohm's law $V = IR$, where $I = 4000 \, \mathrm{A}$. Therefore, $V = 4000 \times 1.68 \times 10^{-3} = 6.72 \, \mathrm{V}$, which is not an option due to a calculation mistake. Correctly calculating $R$ and $V$ should yield an answer that matches one of the options provided. Re-evaluating the calculation correctly: $R = 1.68 \times 10^{-8} \Omega \cdot m \times \frac{0.2 m}{2 \times 10^{-4} m^2} = 0.168 \Omega$, and $V = I \times R = 4000 A \times 0.168 \Omega = 672 V$, which indicates a mistake in the initial calculation. The correct approach is to use the given values accurately in the formula for resistance and then calculate the voltage drop correctly. Given the error in the initial explanation, the correct calculation should align with the provided options, suggesting a recalculation is necessary. Correctly recalculating: $R = 1.68 \times 10^{-8} \Omega \cdot m \times \frac{0.2 m}{2 \times 10^{-4} m^2} = 0.84 \Omega$, leading to a voltage drop $V = 4000 A \times 0.84 \Omega = 3360 V$, which still does not match the provided options, indicating a persistent error in the recalculation process. The correct calculation involves accurately applying the resistivity, length, and area in the resistance formula, followed by using the current value in Ohm's law to find the voltage drop, ensuring the result aligns with one of the given choices.

A Metal Bar Is 20 cm20 \mathrm {~cm}20 cm Long and Has a Rectangular Cross-Section Measuring

A Metal Bar Is $20 \mathrm {~cm}$ Long and Has a Rectangular Cross-Section Measuring