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Difficulty: MediumFormulating and Modifying Hypotheses

A student proposed the following hypothesis regarding electrical circuits:

*Hypothesis*: As the cross-sectional area of a metal wire increases, its electrical resistance will increase because a larger wire contains more atoms that collide with and resist the flow of electrons.

To test this hypothesis, the student measured the electrical resistance of four copper wires, each 10 m10\text{ m} long but with different diameters, at a constant temperature of 20∘C20^\circ\text{C}. The results are shown in the table below:

WireDiameter (mm\text{mm})Cross-sectional area (mm2\text{mm}^2)Resistance (Ω\Omega)
10.50.204.00
21.00.791.00
31.51.770.44
42.03.140.25

Based on these results, how should the student modify the hypothesis?

  1. The student should modify the hypothesis to state that as the cross-sectional area of a wire increases, its electrical resistance decreases.Answer
  2. B
    The student should modify the hypothesis to state that the electrical resistance of a wire is independent of its cross-sectional area.
  3. C
    The student should keep the hypothesis as written because the wire with the largest cross-sectional area (3.14 mm23.14\text{ mm}^2) had the lowest resistance (0.25 Ω0.25\ \Omega).
  4. D
    The student should modify the hypothesis to state that as the diameter of a wire decreases, its electrical resistance decreases.

Answer

The student should modify the hypothesis to state that as the cross-sectional area of a wire increases, its electrical resistance decreases.
The correct answer states that the hypothesis should be modified to show that as the cross-sectional area of a wire increases, its electrical resistance decreases. The table shows that as the cross-sectional area increases from 0.20 mm20.20\text{ mm}^2 to 3.14 mm23.14\text{ mm}^2, the resistance decreases from 4.00 Ω4.00\ \Omega to 0.25 Ω0.25\ \Omega. Since the initial hypothesis predicted that resistance would increase as cross-sectional area increases, the experimental results contradict the hypothesis. Therefore, the hypothesis must be modified to reflect this inverse relationship.

Step-by-Step Solution

1
Analyze the student's initial hypothesis.
The initial hypothesis states that larger cross-sectional area leads to higher resistance (a direct relationship).
To evaluate the hypothesis, we must first define the expected relationship between the variables.
2
Examine the experimental data in the table to determine the actual relationship between cross-sectional area and resistance.
As the cross-sectional area increases from 0.20 mm20.20\text{ mm}^2 to 3.14 mm23.14\text{ mm}^2, the resistance decreases from 4.00 Ω4.00\ \Omega to 0.25 Ω0.25\ \Omega.
Comparing the independent variable (cross-sectional area) with the dependent variable (resistance) reveals the empirical trend.
3
Compare the empirical trend with the initial hypothesis and select the correct modification.
The empirical trend is inverse (area increases, resistance decreases), which contradicts the hypothesized direct relationship. Therefore, the hypothesis must be modified to state that resistance decreases as cross-sectional area increases.
A scientific hypothesis must be modified when experimental evidence consistently contradicts its predictions.

Key Concept

Evaluating and modifying a hypothesis based on conflicting experimental data.
Estimated Time:1m 30s
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Topics

Formulating and Modifying Hypotheses

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