A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying along the x, y, and z axes, respectively (Fig. 25–36). Determine the resistance for current that passes through the solid in the y direction, (Assume the resistivity is ρ = 3.0 x 10⁻⁵ Ω•m).

Verified step by step guidance

1. Identify the formula for resistance in terms of resistivity: \( R = \frac{\rho \cdot L}{A} \), where \( R \) is the resistance, \( \rho \) is the resistivity, \( L \) is the length of the material in the direction of current flow, and \( A \) is the cross-sectional area perpendicular to the current flow.
2. Determine the length \( L \) in the direction of current flow. Since the current flows in the y-direction, \( L = 2.0 \; \text{cm} \) or \( 0.02 \; \text{m} \).
3. Calculate the cross-sectional area \( A \) perpendicular to the y-direction. The cross-section is formed by the sides along the x and z axes, so \( A = \text{length along x} \times \text{length along z} = 1.0 \; \text{cm} \times 4.0 \; \text{cm} = 4.0 \; \text{cm}^2 \) or \( 4.0 \times 10^{-6} \; \text{m}^2 \).
4. Substitute the given values into the resistance formula: \( R = \frac{\rho \cdot L}{A} \), where \( \rho = 3.0 \times 10^{-5} \; \Omega \cdot \text{m} \), \( L = 0.02 \; \text{m} \), and \( A = 4.0 \times 10^{-6} \; \text{m}^2 \).
5. Simplify the expression to find the resistance \( R \). Ensure the units are consistent throughout the calculation to obtain the resistance in ohms (\( \Omega \)).

Key Concepts

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol ρ (rho) and is measured in ohm-meters (Ω·m). The resistivity of a material depends on its composition and temperature, and it plays a crucial role in determining the resistance of a conductor when its dimensions are known.

Resistance

Resistance is a measure of the opposition to the flow of electric current in a conductor. It is calculated using the formula R = ρ(L/A), where R is resistance, ρ is resistivity, L is the length of the conductor, and A is its cross-sectional area. The resistance varies with the shape and size of the conductor, as well as the material's resistivity.

Geometric Orientation

Geometric orientation refers to the alignment and dimensions of a solid object in relation to the flow of current. In this case, the rectangular solid's dimensions along the x, y, and z axes affect how current flows through it. Understanding the orientation is essential for calculating resistance, as it determines the effective length and cross-sectional area through which the current passes.