An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the current density J. Place your results in a table for easy viewing.

검증된 단계별 안내

Determine the formula for current density. The current density \( J \) is defined as the current \( I \) divided by the cross-sectional area \( A \) of the wire. Mathematically, \( J = \frac{I}{A} \).

Identify the current \( I \) in the wire. From the problem, the current in the top segment is given as \( I = 10 \; \text{A} \). This current is the same for all three segments since the wire is continuous and the current is conserved.

Calculate the cross-sectional area \( A \) for each segment. The cross-sectional area is related to the diameter \( d \) of the wire by the formula \( A = \pi \left( \frac{d}{2} \right)^2 \). Use the given diameters for each segment to compute \( A \).

Substitute the values of \( I \) and \( A \) into the formula \( J = \frac{I}{A} \) for each segment. Perform this calculation for all three segments to find the current density \( J \) for each one.

Organize the results into a table. The table should include the segment number, the diameter \( d \), the cross-sectional area \( A \), and the current density \( J \) for each segment.

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영상 길이:

12m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Current Density

Current density (J) is defined as the amount of electric current flowing per unit area of a cross-section of a conductor. It is expressed in amperes per square meter (A/m²). The formula to calculate current density is J = I/A, where I is the current in amperes and A is the cross-sectional area in square meters. Understanding current density is crucial for analyzing how current distributes across different segments of a wire.

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. It is mathematically expressed as V = IR. This law is fundamental in understanding how voltage, current, and resistance interact in electrical circuits, which is essential for calculating current density in different wire segments.

Cross-Sectional Area

The cross-sectional area of a wire is the area of its cut surface perpendicular to its length. It is a critical factor in determining the current density, as a larger cross-sectional area allows more current to flow through it without increasing the current density. The area can be calculated using geometric formulas depending on the wire's shape, such as A = πr² for a circular wire. This concept is vital for understanding how the physical dimensions of a wire affect its electrical properties.