Finding electric field intensity, power, current given current density

[tessa0508]

Homework Statement

Sheet of wire is biased with an electrical potential at each end of conduct current. Given current density, J, in wire:
J=-x²y a_x +xz a_y + 2xyz a_z

A)Continuity equation to prove its possible
B) current along length of wire
C) Electric field intensity E
D) voltage drop along length of wire
E) Power in wire

Homework Equations

I= $$\int(J) ds$$
E=J / $$\sigma$$
P=IV
V=-$$\int(E) dl$$

The Attempt at a Solution

A) -2xy a_x + 0 + 2xy a_z = 0 , possible

B) = $$\int(2xyz)x dxdy$$ = $$\int2x^{2}yz dxdy$$

= $$\int\frac{2x^{3}yz}{3}dy$$ = $$\frac{x^{3}y^{2}z}{3}$$

C) Do I plug in the equation J or a_z like for B?

D) =$$/int$$(ignore this).

same question for c? if I do like B). Then:

=$$\int\frac{2xyz}{\sigma}$$ = $$\frac{xyz^{2}}{\sigma}$$

E) P=IV = $$\frac{x^{4}y^{3}z^{3}}{3\sigma}$$

 

[mark726]

Hi there! Great question. Let's work through each part together to find the solutions.

A) To prove that the given current density is possible, we need to show that it satisfies the continuity equation. The continuity equation states that the divergence of the current density is equal to the rate of change of charge density. In mathematical notation, this can be written as:

∇⋅J = -∂ρ/∂t

In this case, we are given the current density J = -x^2y ax + xz ay + 2xyz az. To find the divergence, we can use the product rule:

∇⋅J = (-x^2y ax + xz ay + 2xyz az)⋅∂/∂x + (x^2y ax + xz ay + 2xyz az)⋅∂/∂y + (x^2y ax + xz ay + 2xyz az)⋅∂/∂z

= -x^2y⋅∂/∂x + xz⋅∂/∂y + 2xyz⋅∂/∂z

= -x^2y(-2xy) + xz(0) + 2xyz(1)

= 2x^3y^2 + 2x^3yz

Now, we need to find the rate of change of charge density. Since we are not given any information about the charge density, we can assume that it is constant and therefore the rate of change is 0. This means that the continuity equation is satisfied and the given current density is possible.

B) To find the current along the length of the wire, we can use the integral form of Ohm's law:

I = ∫J⋅ds

In this case, we are given J = -x^2y ax + xz ay + 2xyz az and ds = dxdy. Substituting these values into the integral, we get:

I = ∫(-x^2y ax + xz ay + 2xyz az)⋅dxdy

= ∫(-x^2y)(1) + (xz)(0) + (2xyz)(0)⋅dxdy

= ∫-x^2y