Current density: If the current density in a wire or radius R is given by J = kr, 0

Question

Current density: If the current density in a wire or radius R is given by J = kr, 0 < r < R, what is the current in the wire?

Quizplus · Accepted Answer

The current density J given by J = kr implies that the current density increases linearly with the radius r. To find the total current I in the wire, we integrate the current density over the cross-sectional area of the wire. The differential area dA is given by $2\pi r dr$, and the current through this differential area is $J dA = kr \cdot 2\pi r dr$. Integrating from 0 to R gives the total current $I = \int_0^R 2\pi kr^2 dr$, which evaluates to $\frac{2\pi k R^3}{3}$, matching choice A.

Current Density: If the Current Density in a Wire or Radius