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

Question

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? A) 2πkR³/3 B) 3πkR³/2 C) kR³/3 D) kR² E) kR²/2

Quizplus · Accepted Answer

The current density J is given as a function of the radial distance r from the center of the wire, J = kr. To find the total current I through the wire, integrate the current density over the cross-sectional area of the wire. The differential area dA is an annulus of radius r and thickness dr, so dA = 2πrdr. The differential current dI is then JdA = k*r*2πrdr. Integrating from 0 to R gives the total current I = ∫(0 to R) k*2πr^2 dr = (2πk/3)*R^3, which matches the format of choice A when considering the symbols used for pi (π), the constant k, the radius R, and the powers and fractions involved.

If the Current Density in a Wire or Radius R