A cube of edge a carries a point charge q at each corner. We have to show that the resultant electric force on any one of the charges is given by

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In the diagram a cube has been drawn with length a of each edge. We have placed a charge of q C at each eight vertices of the cube.

A set of Cartesian co-ordinate axes have been laid out with origin at one of the vertices of the cube as shown in the diagram. Unit vectors along the X,Y, and Z axes are , , and , respectively. We calculate the force on the charge q at the vertex of the cube, which is at the origin of the co-ordinate system. By symmetry force on any other charge due to the other 7 charges and in a direction relative to them will be the same.

By Coulomb’s law

Or

We note that the magnitude of is

As

And the direction of the force on the charge due to the other 7 charges q at the other vertices of the cube will be

That is it is directed along the body diagonal away from the cube.