Solve each problem. Current FlowIn electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

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Identify the relationship: The resistance \( R \) varies inversely as the square of the diameter \( d \). This can be expressed as \( R = \frac{k}{d^2} \), where \( k \) is a constant.

Use the given information to find \( k \): Substitute \( R = 0.4 \) ohms and \( d = 0.01 \) inches into the equation \( R = \frac{k}{d^2} \) to solve for \( k \).

Calculate \( k \) using the equation \( 0.4 = \frac{k}{(0.01)^2} \).

Once \( k \) is found, use it to find the resistance of the wire with diameter 0.03 inches: Substitute \( d = 0.03 \) inches into the equation \( R = \frac{k}{d^2} \).

Solve for \( R \) to find the resistance of the wire with the new diameter, rounding to the nearest ten-thousandth of an ohm.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. This relationship is fundamental in understanding how voltage, current, and resistance interact in electrical circuits, allowing for the calculation of one variable when the others are known.

Resistance and Diameter Relationship

The resistance of a wire is influenced by its physical dimensions and material properties. Specifically, resistance varies inversely with the square of the diameter of the wire, meaning that as the diameter increases, the resistance decreases significantly. This relationship is crucial for solving problems involving different wire sizes and their respective resistances.

Calculating Resistance

To calculate the resistance of a wire based on its diameter, one can use the formula R = k / d^2, where R is resistance, k is a constant that depends on the material and length of the wire, and d is the diameter. This formula allows for the determination of resistance for wires of varying diameters, provided the material and length remain constant.

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