If a given charge distribution has noticeable planes of symmetry or antisymmetry, then the following principles will help us determine the direction of the electric field that it creates.

A given plane is said to be a plane of symmetry of a charge distribution if the two halves of the distribution are mirror images of one another and carry the same exact amount of charge. For a given plane of symmetry, the electric field at any point of that plane lies within the plane.

A given plane is said to be a plane of antisymmetry of a charge distribution if the two halves of the distribution are mirror images of one another and carry the exact opposite amount of charge. For a given plane of antisymmetry, the electric field at any point of that plane is perpendicular to the plane.

To illustrate these principles, we apply them to the classic charge distributions below to derive the direction of the electric fields they create.

Infinite cylinder of charge:

Consider an infinite insulating cylinder with uniformly distributed charge throughout its volume. For such a cylinder, because it is infinite, any horizontal plane such as the one shown below is a plane of symmetry of the charge distribution. In addition, any vertical plane containing the axis of revolution of the cylinder is a plane a of symmetry of the charge distribution too.

Thus, at any point along the line created by the intersection of these two planes of symmetry, the electric field must lie simultaneously in both planes and therefore must lie along the line itself (pointing outward if we assume the cylinder is positive). Overall, the electric field created by an infinite cylinder of charge is radial, directed outward if the cylinder is positively charged and inward if the cylinder is negatively charged.

Infinite sheet of charge:

Consider an infinite sheet with uniformly distributed charge over its surface. For such a sheet, because it is infinite, any vertical plane such as the ones shown below is a plane of symmetry of the charge distribution.

Thus, at any point along the line created by the intersection of these two planes of symmetry, the electric field must lie simultaneously in both planes and therefore must lie along the line itself (pointing outward if we assume the sheet is positive). Overall, the electric field created by an infinite sheet of charge is perpendicular to the sheet, directed away from the sheet if it is positively charged and toward the sheet if it is negatively charged.

Solid sphere of charge:

Consider an insulating sphere with charge uniformly distributed throughout its volume. For such a sphere, any plane containing its center is a plane of symmetry of the charge distribution.

Thus, at any point along the line created by the intersection of the two planes of symmetry shown above, the electric field must lie simultaneously in both planes and therefore must lie along the line itself (pointing outward if we assume the sphere is positive).

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