NEIEP Mechanics Practice Exam

How does increasing the cross-sectional area of a wire by four times affect its resistance?

1. The resistance is reduced by half

2. The resistance remains the same

3. The resistance is one fourth of what it was originally

4. The resistance is doubled

The correct answer is: The resistance is one fourth of what it was originally

The resistance of a wire is inversely proportional to its cross-sectional area, as described by the formula for resistance, which is \( R = \frac{\rho L}{A} \). In this equation, \( R \) is the resistance, \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. When the cross-sectional area of a wire is increased, the resistance will decrease. Specifically, if the cross-sectional area is increased four times (meaning \( A' = 4A \)), the new resistance \( R' \) can be calculated as follows: \[ R' = \frac{\rho L}{A'} = \frac{\rho L}{4A} = \frac{1}{4} \cdot \frac{\rho L}{A} = \frac{R}{4} \] This shows that the new resistance is one fourth of the original resistance, confirming that when the area increases by a factor of four, the resistance decreases to a quarter of its original value. Thus, the correct answer accurately reflects this relationship in electrical resistance.