Figure P 27.8 represents a section of a circular conductor...

Figure $P 27.8$ represents a section of a circular conductor of nonuniform diameter carrying a current of $5.00 \mathrm{A}$. The radius of cross section $A_{1}$ is $0.400 \mathrm{cm} .$ (a) What is the magnitude of the current density across $A_{1} ?$ (b) If the current density across $A_{2}$ is one-fourth the value across $A_{1}$ what is the radius of the conductor at $A_{2} ?$ (FIGURE CANNOT COPY)

Key Concepts

Scaling Relations in Conduction

Scaling relations in conduction refer to how electrical properties such as current density adjust when the dimensions of a conductor change. Since the area of a circular cross section scales with the square of the radius, any change in the conductor’s radius leads to a quadratic change in area, which in turn affects the magnitude of the current density if the current is conserved.

Conservation of Electric Current

In steady-state conditions, the principle of conservation of electric current states that the total current remains constant along a conductor regardless of changes in its cross-sectional area. This concept is crucial because it allows us to relate current densities at different sections by considering the changes in area while the overall current remains unchanged.

Current Density

Current density is defined as the amount of electric current flowing per unit area of a conductor's cross section. It is given by the formula J = I/A, where I is the current and A is the area through which the current flows. This concept is critical for understanding how current distributes in conductors, especially when their cross-sectional geometries vary.

Cross-Sectional Area

The cross-sectional area of a conductor, especially when it has a circular cross section, is determined using the formula A = ?r². This calculation is fundamental when relating geometric dimensions of a conductor to its electrical properties such as current density.

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