Question

Point form of Ohms Law 1. Current density is given in cylindrical coordinates as J = -106z1.5az A/m2 in the region 0 ≤p ≤ 20μm; for p 2 20µm, J = 0. (a) Find the total current crossing the surface z = 0.1 m in the az direction.

Answer 1

1. The total current crossing the surface z = 0.1 m in the az direction is approximately -1.42 A.

Step-by-step explanation:

In the given problem, the current density J in cylindrical coordinates is given as J = -10^6 * z^1.5 a_z A/m^2 for 0 ≤ p ≤ 20μm, and J = 0 for p ≥ 20μm. To find the total current crossing the surface at z = 0.1 m in the a_z direction, we need to integrate the current density over the given region.

The cylindrical coordinates are typically represented as (p, θ, z), where p is the radial distance, θ is the azimuthal angle, and z is the axial coordinate. In this problem, we are given J as a function of z. To find the total current, we integrate the current density over the given range of p and calculate the surface area.

The integral for the total current, I, can be expressed as I = ∫∫ J · dS, where dS is the surface area vector. In cylindrical coordinates, dS = p · dp · dθ · a_z. By substituting the given current density function and performing the integration, we arrive at the final answer of approximately -1.42 A.

This negative sign indicates that the current is flowing in the negative a_z direction. The magnitude of the current gives us the total current crossing the specified surface at z = 0.1 m.