A metal bar has rectangular cross section 5.0 cm by 10 cm, as shown in Fig. 24.18. The bar has

step 1 So let's say for this rectangular region that we break it up into thin strips and we integrate o

A metal bar has rectangular cross section 5.0 cm by 10 cm,...

A metal bar has rectangular cross section $5.0 \mathrm{~cm}$ by $10 \mathrm{~cm}$, as shown in Fig. $24.18$. The bar has a nonuniform conductivity, and as a result the current density increases linearly from zero at the bottom to $0.10 \mathrm{~A} / \mathrm{cm}^{2}$ at the top. Find the total current in the bar.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Current Density

Current density is a measure of the electric current per unit area of a cross-section. It indicates how much charge is moving through a given area and is fundamental in determining how current flows in materials, particularly when the distribution of current is not uniform.

Surface Integration of a Variable Field

When current density varies over a cross-sectional area, the total current is obtained by integrating the current density over that area. This process involves summing the contributions of infinitesimal area elements across the entire cross-section, accounting for spatial variations.

Linear Variation in Physical Quantities

A linear variation implies that a physical quantity changes at a constant rate with respect to some spatial coordinate. In the context of current density, if the density increases linearly along one direction, it provides a predictable pattern that simplifies the integration process to yield an average or total value over the area.

Rectangular Cross-Section Geometry

Rectangular cross sections are common in practical applications and provide a straightforward context for setting up integrations. Understanding the geometry allows one to define appropriate limits and differential area elements, facilitating the calculation of total current when the current density distribution is known.

Transcript

Please give Ace some feedback