Question

Question: To find: The charge enclosed by the cube with vertices \(( \pm 1, \pm 1, \pm 1)\).

Short answer

The charge enclosed by the cube with vertices \(( \pm 1, \pm 1, \pm 1)\) is \(\underline {24{\varepsilon _0}} \).

Step-by-step solution

Concept of Surface Integral and Formula Used.

A surface integral is a surface integration generalisation of multiple integrals. It can be thought of as the line integral's double integral equivalent. It is possible to integrate a scalar field or a vector field over a surface.

Given:

\({\rm{E}}(x,y,z) = x{\rm{i}} + y{\rm{j}} + z{\rm{k}}\)and cube with vertices\(( \pm 1, \pm 1, \pm 1)\)

Formula used:

From Gauss law,

Calculate the Surface Integral.

Finding flux through the face of the cube which is part of the plane \(x = 1\)

Finding flux through the face of the cube which is part of the plane \(x = - 1\)

Note that \(x = - 1\) for all points

Note that, both the surface and\({\bf{E}}\)are symmetrical in terms of\({\bf{x}},{\bf{y}}\)and\({\bf{z}}\).

Therefore the flux through the other four faces will also be 4 each.

Hence the net flux through the cube is \(4 \times 6 = 24\).

Charge Enclosed

Charge Enclosed