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Ch 27: Current and Resistance
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 27: Current and ResistanceProblem 54
Chapter 27, Problem 54

You need to design a 1.0 A fuse that 'blows' if the current exceeds 1.0 A. The fuse material in your stockroom melts at a current density of 500 A/cm2. What diameter wire of this material will do the job?

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1
Determine the relationship between current density \( J \) and current \( I \). The formula is \( J = \frac{I}{A} \), where \( A \) is the cross-sectional area of the wire.
Rearrange the formula to solve for the cross-sectional area \( A \): \( A = \frac{I}{J} \). Substitute \( I = 1.0 \; \text{A} \) and \( J = 500 \; \text{A/cm}^2 \) into the equation.
Convert the current density \( J \) from \( \text{A/cm}^2 \) to \( \text{A/m}^2 \) for consistency in SI units: \( 1 \; \text{cm}^2 = 10^{-4} \; \text{m}^2 \), so \( J = 500 \times 10^4 \; \text{A/m}^2 \).
Once \( A \) is calculated, recall that the cross-sectional area of a circular wire is given by \( A = \pi r^2 \), where \( r \) is the radius. Solve for \( r \): \( r = \sqrt{\frac{A}{\pi}} \).
Finally, calculate the diameter of the wire using \( d = 2r \). This will give the required diameter of the wire that ensures the fuse 'blows' at 1.0 A.

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Key Concepts

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

Current and Current Density

Current is the flow of electric charge, measured in amperes (A), while current density is the amount of current flowing per unit area of a conductor, expressed in amperes per square centimeter (A/cm²). In this context, understanding the relationship between the total current flowing through the fuse and the area of the wire is crucial for determining the appropriate wire diameter that will safely carry the specified current without exceeding the material's melting point.
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Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This principle is essential for understanding how the resistance of the wire affects the current flow and helps in calculating the necessary dimensions of the wire to ensure it operates within safe limits.
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Cross-sectional Area of a Wire

The cross-sectional area of a wire is a measure of the size of the wire's circular section, typically expressed in square centimeters (cm²). This area is critical in determining how much current the wire can safely carry without overheating. By calculating the required cross-sectional area based on the current and current density, one can derive the necessary diameter of the wire to ensure it 'blows' at the specified current of 1.0 A.
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Textbook Question

The resistivity of a metal increases slightly with increased temperature. This can be expressed as ρ=ρ₀[1+α(T−T₀)] , where T₀ is a reference temperature, usually 20°C, and α is the temperature coefficient of resistivity. For copper, α=3.9×10−3 °C−1. Suppose a 2.5-m-long, 0.40-mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at 20°C?

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Textbook Question

A hollow metal cylinder has inner radius a, outer radius b, length L, and conductivity σ. The current I is radially outward from the inner surface to the outer surface. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if a=1.0 cm, b=2.5 cm, L=10 cm, and I=25 A.

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Textbook Question

The total amount of charge in coulombs that has entered a wire at time t is given by the expression Q=4t−t2, where t is in seconds and t≥0. Find an expression for the current in the wire at time .

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Textbook Question

Electrical engineers sometimes use a wire's conductance, G=σA/L, instead of its resistance. A 1.5 A current flows through the wire of part b. What is the potential difference between the ends of the wire?

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Textbook Question

Variations in the resistivity of blood can give valuable clues about changes in various properties of the blood. Suppose a medical device inserts microelectrodes into a 1.5-mm-diameter vein at positions 5.0 cm apart. What is the blood resistivity if a 9.0 V potential difference causes a 230 μA current through the blood in the vein?

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Textbook Question

Electrical engineers sometimes use a wire's conductance, G=σA/L, instead of its resistance. What is the conductance of a 5.4-cm-long, 0.15-mm-diameter tungsten wire?

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