Compute the charge enclosed by a cube of 2m each edge centered at the origin and with the edges parallel to the axes. Given D = 10y3 /3 j.

To calculate the charge enclosed within the cube, we can use Gauss’s law, which relates the electric displacement field ( mathbf{D} ) to the free charge ( Q_{text{enc}} ) enclosed by a surface:

[

Q_{text{enc}} = iint_S mathbf{D} cdot dmathbf{A}

]

Where:

– ( mathbf{D} = frac{10y^3}{3} mathbf{j} )

– ( dmathbf{A} ) is the outward differential area vector on the surface of the cube.

First, we will compute the total surface charge by integrating over the surfaces of the cube. The cube has dimensions of 2m and is centered at the origin, meaning its corners are at coordinates ranging from (-1) to (1) along the x, y, and z axes.

The cube has six faces, and we need to evaluate the surface integral for each face:

1. Top face ((z = 1)):

[

dmathbf{A} = hat{k} , dA quad Rightarrow quad int_A mathbf{D} cdot dmathbf{A} = int_A mathbf{D} cdot 0 = 0

]

2. Bottom face ((z = -1)):